Autocatalysis in Reaction Networks

11 October, 2013

guest post by Manoj Gopalkrishnan

Since this is my first time writing a blog post here, let me start with a word of introduction. I am a computer scientist at the Tata Institute of Fundamental Research, broadly interested in connections between Biology and Computer Science, with a particular interest in reaction networks. I first started thinking about them during my Ph.D. at the Laboratory for Molecular Science. My fascination with them has been predominantly mathematical. As a graduate student, I encountered an area with rich connections between combinatorics and dynamics, and surprisingly easy-to-state and compelling unsolved conjectures, and got hooked.

There is a story about Richard Feynman that he used to take bets with mathematicians. If any mathematician could make Feynman understand a mathematical statement, then Feynman would guess whether or not the statement was true. Of course, Feynman was in a habit of winning these bets, which allowed him to make the boast that mathematics, especially in its obsession for proof, was essentially irrelevant, since a relative novice like himself could after a moment’s thought guess at the truth of these mathematical statements. I have always felt Feynman’s claim to be unjust, but have often wondered what mathematical statement I would put to him so that his chances of winning were no better than random.

Today I want to tell you of a result about reaction networks that I have recently discovered with Abhishek Deshpande. The statement seems like a fine candidate to throw at Feynman because until we proved it, I would not have bet either way about its truth. Even after we obtained a short and elementary proof, I do not completely ‘see’ why it must be true. I am hoping some of you will be able to demystify it for me. So, I’m just going to introduce enough terms to be able to make the statement of our result, and let you think about how to prove it.

John and his colleagues have been talking about reaction networks as Petri nets in the network theory series on this blog. As discussed in part 2 of that series, a Petri net is a diagram like this:

Following John’s terminology, I will call the aqua squares ‘transitions’ and the yellow circles ‘species’. If we have some number #rabbit of rabbits and some number #wolf of wolves, we draw #rabbit many black dots called ‘tokens’ inside the yellow circle for rabbit, and #wolf tokens inside the yellow circle for wolf, like this:

Here #rabbit = 4 and #wolf = 3. The predation transition consumes one ‘rabbit’ token and one ‘wolf’ token, and produces two ‘wolf’ tokens, taking us here:

John explained in parts 2 and 3 how one can put rates on different transitions. For today I am only going to be concerned with ‘reachability:’ what token states are reachable from what other token states. John talked about this idea in part 25.

By a complex I will mean a population vector: a snapshot of the number of tokens in each species. In the example above, (#rabbit, #wolf) is a complex. If y, y' are two complexes, then we write

y \to y'

if we can get from y to y' by a single transition in our Petri net. For example, we just saw that

(4,3)\to (3,4)

via the predation transition.

Reachability, denoted \to^*, is the transitive closure of the relation \to. So y\to^* y' (read y' is reachable from y) iff there are complexes

y=y_0,y_1,y_2,\dots,y_k =y'

such that

y_0\to y_1\to\cdots\to y_{k-1}\to y_k.

For example, here (5,1) \to^* (1, 5) by repeated predation.

I am very interested in switches. After all, a computer is essentially a box of switches! You can build computers by connecting switches together. In fact, that’s how early computers like the Z3 were built. The CMOS gates at the heart of modern computers are essentially switches. By analogy, the study of switches in reaction networks may help us understand biochemical circuits.

A siphon is a set of species that is ‘switch-offable’. That is, if there are no tokens in the siphon states, then they will remain absent in future. Equivalently, the only reactions that can produce tokens in the siphon states are those that require tokens from the siphon states before they can fire. Note that no matter how many rabbits there are, if there are no wolves, there will continue to be no wolves. So {wolf} is a siphon. Similarly, {rabbit} is a siphon, as is the union {rabbit, wolf}. However, when Hydrogen and Oxygen form Water, {Water} is not a siphon.

For another example, consider this Petri net:

The set {HCl, NaCl} is a siphon. However, there is a conservation law: whenever an HCl token is destroyed, an NaCl token is created, so that #HCl + #NaCl is invariant. If both HCl and NaCl were present to begin with, the complexes where both are absent are not reachable. In this sense, this siphon is not ‘really’ switch-offable. As a first pass at capturing this idea, we will introduce the notion of ‘critical set’.

A conservation law is a linear expression involving numbers of tokens that is invariant under every transition in the Petri net. A conservation law is positive if all the coefficients are non-negative. A critical set of states is a set that does not contain the support of a positive conservation law.

For example, the support of the positive conservation law #HCl + #NaCl is {HCl, NaCl}, and hence no set containing this set is critical. Thus {HCl, NaCl} is a siphon, but not critical. On the other hand, the set {NaCl} is critical but not a siphon. {HCl} is a critical siphon. And in our other example, {Wolf, Rabbit} is a critical siphon.

Of particular interest to us will be minimal critical siphons, the minimal sets among critical siphons. Consider this example:

Here we have two transitions:

X \to 2Y

and

2X \to Y

The set \{X,Y\} is a critical siphon. But so is the smaller set \{X\}. So, \{X,Y\} is not minimal.

We define a self-replicable set to be a set A of species such that there exist complexes y and y' with y\to^* y' such that for all i \in A we have

y'_i > y_i

So, there are transitions that accomplish the job of creating more tokens for all the species in A. In other words: these species can ‘replicate themselves’.

We define a drainable set by changing the > to a <. So, there are transitions that accomplish the job of reducing the number of tokens for all the species in A. These species can ‘drain away’.

Now here comes the statement:

Every minimal critical siphon is either drainable or self-replicable!

We prove it in this paper:

• Abhishek Deshpande and Manoj Gopalkrishnan, Autocatalysis in reaction networks.

But first note that the statement becomes false if the critical siphon is not minimal. Look at this example again:

The set \{X,Y\} is a critical siphon. However \{X,Y\} is neither self-replicable (since every reaction destroys X) nor drainable (since every reaction produces Y). But we’ve already seen that \{X,Y\} is not minimal. It has a critical subsiphon, namely \{X\}. This one is minimal—and it obeys our theorem, because it is drainable.

Checking these statements is a good way to make sure you understand the concepts! I know I’ve introduced a lot of terminology here, and it takes a while to absorb.

Anyway: our proof that every minimal critical siphon is either drainable or self-replicable makes use of a fun result about matrices. Consider a real square matrix with a sign pattern like this:

\left( \begin{array}{cccc} <0 & >0 & \cdots & > 0 \\ >0 & <0 & \cdots &> 0 \\ \vdots & \vdots & <0 &> 0 \\ >0 & >0 & \cdots & <0 \end{array} \right)

If the matrix is full-rank then there is a positive linear combination of the rows of the matrix so that all the entries are nonzero and have the same sign. In fact, we prove something stronger in Theorem 5.9 of our paper. At first, we thought this statement about matrices should be equivalent to one of the many well-known alternative statements of Farkas’ lemma, like Gordan’s theorem.

However, we could not find a way to make this work, so we ended up proving it by a different technique. Later, my colleague Jaikumar Radhakrishnan came up with a clever proof that uses Farkas’ lemma twice. However, so far we have not obtained the stronger result in Theorem 5.9 with this proof technique.

My interest in the result that every minimal critical siphon is either drainable or self-replicable is not purely aesthetic (though aesthetics is a big part of it). There is a research community of folks who are thinking of reaction networks as a programming language, and synthesizing molecular systems that exhibit sophisticated dynamical behavior as per specification:

International Conference on DNA Computing and Molecular Programming.

Foundations of Nanoscience: Self-Assembled Architectures and Devices.

Molecular Programming Architectures, Abstractions, Algorithms and Applications.

Networks that exhibit some kind of catalytic behavior are a recurring theme among such systems, and even more so in biochemical circuits.

Here is an example of catalytic behavior:

A + C \to B + C

The ‘catalyst’ C helps transform A to B. In the absence of C, the reaction is turned off. Hence, catalysts are switches in chemical circuits! From this point of view, it is hardly surprising that they are required for the synthesis of complex behaviors.

In information processing, one needs amplification to make sure that a signal can propagate through a circuit without being overwhelmed by errors. Here is a chemical counterpart to such amplification:

A + C \to 2C

Here the catalyst C catalyzes its own production: it is an ‘autocatalyst’, or a self-replicating species. By analogy, autocatalysis is key for scaling synthetic molecular systems.

Our work deals with these notions on a network level. We generalize the notion of catalysis in two ways. First, we allow a catalyst to be a set of species instead of a single species; second, its absence can turn off a reaction pathway instead of a single reaction. We propose the notion of self-replicable siphons as a generalization of the notion of autocatalysis. In particular, ‘weakly reversible’ networks have critical siphons precisely when they exhibit autocatalytic behavior. I was led to this work when I noticed the manifestation of this last statement in many examples.

Another hope I have is that perhaps one can study the dynamics of each minimal critical siphon of a reaction network separately, and then somehow be able to answer interesting questions about the dynamics of the entire network, by stitching together what we know for each minimal critical siphon. On the synthesis side, perhaps this could lead to a programming language to synthesize a reaction network that will achieve a specified dynamics. If any of this works out, it would be really cool! I think of how abelian group theory (and more broadly, the theory of abelian categories, which includes categories of vector bundles) benefits from a fundamental theorem that lets you break a finite abelian group into parts that are easy to study—or how number theory benefits from a special case, the fundamental theorem of arithmetic. John has also pointed out that reaction networks are really presentations of symmetric monoidal categories, so perhaps this could point the way to a Fundamental Theorem for Symmetric Monoidal Categories.

And then there is the Global Attractor Conjecture, a
long-standing open problem concerning the long-term behavior of solutions to the rate equations. Now that is a whole story by itself, and will have to wait for another day.


Quantum Network Theory (Part 2)

13 August, 2013

guest post by Tomi Johnson

Last time I told you how a random walk called the ‘uniform escape walk’ could be used to analyze a network. In particular, Google uses it to rank nodes. For the case of an undirected network, the steady state of this random walk tells us the degrees of the nodes—that is, how many edges come out of each node.

Now I’m going to prove this to you. I’ll also exploit the connection between this random walk and a quantum walk, also introduced last time. In particular, I’ll connect the properties of this quantum walk to the degrees of a network by exploiting its relationship with the random walk.

This is pretty useful, considering how tricky these quantum walks can be. As the parts of the world that we model using quantum mechanics get bigger and have more complicated structures, like biological network, we need all the help in understanding quantum walks that we can get. So I’d better start!

Flashback

Starting with any (simple, connected) graph, we can get an old-fashioned ‘stochastic’ random walk on this graph, but also a quantum walk. The first is the uniform escape stochastic walk, where the walker has an equal probability per time of walking along any edge leaving the node they are standing at. The second is the related quantum walk we’re going to study now. These two walks are generated by two matrices, which we called S and Q. The good thing is that these matrices are similar, in the technical sense.

We studied this last time, and everything we learned is summarized here:

Diagram outlining the main concepts (again)

where:

G is a simple graph that specifies

A the adjacency matrix (the generator of a quantum walk) with elements A_{i j} equal to unity if nodes i and j are connected, and zero otherwise (A_{i i} = 0), which subtracted from

D the diagonal matrix of degrees D_{i i} = \sum_j A_{i j} gives

L = D - A the symmetric Laplacian (generator of stochastic and quantum walks), which when normalized by D returns both

S = L D^{-1} the generator of the uniform escape stochastic walk and

Q = D^{-1/2} L D^{-1/2} the quantum walk generator to which it is similar!

Now I hope you remember where we are. Next I’ll talk you through the mathematics of the uniform escape stochastic walk S and how it connects to the degrees of the nodes in the large-time limit. Then I’ll show you how this helps us solve aspects of the quantum walk generated by Q.

Stochastic walk

The uniform escape stochastic walk generated by S is popular because it has a really useful stationary state.

To recap from Part 20 of the network theory series, a stationary state of a stochastic walk is one that does not change in time. By the master equation

\displaystyle{ \frac{d}{d t} \psi(t) = -S \psi(t)}

the stationary state must be an eigenvector of S with eigenvalue 0.

A fantastic pair of theorems hold:

• There is always a unique (up to multiplication by a positive number) positive eigenvector \pi of S with eigenvalue 0. That is, there is a unique stationary state \pi.

• Regardless of the initial state \psi(0), any solution of the master equation approaches this stationary state \pi in the large-time limit:

\displaystyle{ \lim_{t \rightarrow \infty} \psi(t) = \pi }

To find this unique stationary state, consider the Laplacian L, which is both infinitesimal stochastic and symmetric. Among other things, this means the rows of L sum to zero:

\displaystyle{ \sum_j L_{i j} = 0 }

Thus, the ‘all ones’ vector \mathbf{1} is an eigenvector of L with zero eigenvalue:

L \mathbf{1} = 0

Inserting the identity I = D^{-1} D into this equation we then find D \mathbf{1} is a zero eigenvector of S:

L \mathbf{1} =  ( L D^{-1} ) ( D \mathbf{1} ) = S ( D \mathbf{1} ) = 0

Therefore we just need to normalize this to get the large-time stationary state of the walk:

\displaystyle{ \pi = \frac{D \mathbf{1}}{\sum_i D_{i i}} }

If we write i for the basis vector that is zero except at the ith node of our graph, and 1 at that node, the inner product \langle i , \pi \rangle is large-time probability of finding a walker at that node. The equation above implies this is proportional to the degree D_{i i} of node i.

We can check this for the following graph:

Illustration of a simple graph

We find that \pi is

\displaystyle{ \left( \begin{matrix} 1/6 \\ 1/6 \\ 1/4 \\ 1/4 \\ 1/6 \end{matrix} \right) }

which implies large-time probability 1/6 for nodes 1, 2 and 5, and 1/4 for nodes 3 and 4. Comparing this to the original graph, this exactly reflects the arrangement of degrees, as we knew it must.

Math works!

The quantum walk

Next up is the quantum walk generated by Q. Not a lot is known about quantum walks on networks of arbitrary geometry, but below we’ll see some analytical results are obtained by exploiting the similarity of S and Q.

Where to start? Well, let’s start at the bottom, what quantum physicists call the ground state. In contrast to stochastic walks, for a quantum walk every eigenvector \phi_k of Q is a stationary state of the quantum walk. (In Puzzle 5, at the bottom of this page, I ask you to prove this). The stationary state \phi_0 is of particular interest physically and mathematically. Physically, since eigenvectors of the Q correspond to states of well-defined energy equal to the associated eigenvalue, \phi_0 is the state of lowest energy, energy zero, hence the name ‘ground state’. (In Puzzle 3, I ask you to prove that all eigenvalues of Q are non-negative, so zero really does correspond to the ground state.)

Mathematically, the relationship between eigenvectors implied by the similarity of S and Q means

\phi_0 \propto D^{-1/2} \pi \propto  D^{1/2} \mathbf{1}

So in the ground state, the probability of our quantum walker being found at node i is

| \langle i , \phi_0 \rangle |^2 \propto | \langle i , D^{1/2} \rangle \mathbf{1} |^2 = D_{i i}

Amazingly, this probability is proportional to the degree and so is exactly the same as \langle i , \pi \rangle, the probability in the stationary state \pi of the stochastic walk!

In short: a zero energy quantum walk Q leads to exactly the same distribution of the walker over the nodes as in the large-time limit of the uniform escape stochastic walk S. The classically important notion of degree distribution also plays a role in quantum walks!

This is already pretty exciting. What else can we say? If you are someone who feels faint at the sight of quantum mechanics, well done for getting this far, but watch out for what’s coming next.

What if the walker starts in some other initial state? Is there some quantum walk analogue of the unique large-time state of a stochastic walk?

In fact, the quantum walk in general does not converge to a stationary state. But there is a probability distribution that can be thought to characterize the quantum walk in the same way as the large-time state characterizes the stochastic walk. It’s the large-time average probability vector P.

If you didn’t know the time that had passed since the beginning of a quantum walk, then the best estimate for the probability of your measuring the walker to be at node i would be the large-time average probability

\displaystyle{ \langle i , P \rangle = \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T | \psi_i (t) |^2 d t }

There’s a bit that we can do to simplify this expression. As usual in quantum mechanics, let’s start with the trick of diagonalizing Q. This amounts to writing

\displaystyle{  Q= \sum_k \epsilon_k \Phi_k }

where \Phi_k are projectors onto the eigenvectors $\phi_k$ of Q, and \epsilon_k are the corresponding eigenvalues of Q. If we insert this equation into

\psi(t)  = e^{-Q t} \psi(0)

we get

\displaystyle{  \psi(t)  = \sum_k e^{-\epsilon_k t} \Phi_k \psi(0) }

and thus

\displaystyle{ \langle i , P \rangle = \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T | \sum_k e^{-i \epsilon_k t} \langle i, \Phi_k \psi (0) \rangle |^2 d t }

Due to the integral over all time, the interference between terms corresponding to different eigenvalues averages to zero, leaving:

\displaystyle{ \langle i , P \rangle = \sum_k | \langle i, \Phi_k \psi(0) \rangle |^2 }

The large-time average probability is then the sum of terms contributed by the projections of the initial state onto each eigenspace.

So we have a distribution that characterizes a quantum walk for a general initial state, but it’s a complicated beast. What can we say about it?

Our best hope of understanding the large-time average probability is through the term | \langle i, \Phi_0 \psi (0) \rangle |^2 associated with the zero energy eigenspace, since we know everything about this space.

For example, we know the zero energy eigenspace is one-dimensional and spanned by the eigenvector \phi_0. This means that the projector is just the usual outer product

\Phi_0 = | \phi_0 \rangle \langle \phi_0 | = \phi_0 \phi_0^\dagger

where we have normalized \phi_0 according to the inner product \langle \phi_0, \phi_0\rangle = 1. (If you’re wondering why I’m using all these angled brackets, well, they’re a notation named after Dirac that is adored by quantum physicists.)

The zero eigenspace contribution to the large-time average probability then breaks nicely into two:

\begin{array}{ccl} | \langle i, \Phi_0 \psi (0) \rangle |^2 &=& | \langle i, \phi_0\rangle \; \langle \phi_0,  \psi (0) \rangle |^2  \\  \\  &=& | \langle i, \phi_0\rangle |^2 \; | \langle \phi_0 , \psi (0) \rangle |^2 \\   \\  &=& \langle i ,  \pi \rangle \; | \langle \phi_0 ,  \psi (0) \rangle |^2 \end{array}

This is just the product of two probabilities:

• first, the probability \langle i , \pi \rangle for a quantum state in the zero energy eigenspace to be at node i, as we found above,

and

• second, the probability | \langle \phi_0, \psi (0)\rangle |^2 of being in this eigenspace to begin with. (Remember, in quantum mechanics the probability of measuring the system to have an energy is the modulus squared of the projection of the state onto the associated eigenspace, which for the one-dimensional zero energy eigenspace means just the inner product with the ground state.)

This is all we need to say something interesting about the large-time average probability for all states. We’ve basically shown that we can break the large-time probability vector P into a sum of two normalized probability vectors:

P = (1- \eta) \pi + \eta \Omega

the first \pi being the stochastic stationary state associated with the zero energy eigenspace, and the second $\Omega$ associated with the higher energy eigenspaces, with

\displaystyle{ \langle i , \Omega \rangle = \frac{ \sum_{k\neq 0} | \langle i, \Phi_k  \psi (0) \rangle |^2  }{ \eta} }

The weight of each term is governed by the parameter

\eta =  1 - | \langle \phi_0, \psi (0)\rangle |^2

which you could think of as the quantumness of the result. This is one minus the probability of the walker being in the zero energy eigenspace, or equivalently the probability of the walker being outside the zero energy eigenspace.

So even if we don’t know \Omega, we know its importance is controlled by a parameter \eta that governs how close the large-time average distribution P of the quantum walk is to the corresponding stochastic stationary distribution \pi.

What do we mean by ‘close’? Find out for yourself:

Puzzle 1. Show, using a triangle inequality, that the trace distance between the two characteristic stochastic and quantum distributions \{ \langle i , P \rangle \}_i and \{ \langle i , \pi \rangle \}_i is upper-bounded by 2 \eta.

Can we say anything physical about when the quantumness \eta is big or small?

Because the eigenvalues of Q have a physical interpretation in terms of energy, the answer is yes. The quantumness \eta is the probability of being outside the zero energy state. Call the next lowest eigenvalue \Delta = \min_{k \neq 0} \epsilon_k the energy gap. If the quantum walk is not in the zero energy eigenspace then it must be in an eigenspace of energy greater or equal to \Delta. Therefore the expected energy E of the quantum walker must bound the quantumness E \ge \eta \Delta.

This tells us that a quantum walk with a low energy is similar to a stochastic walk in the large-time limit. We already knew this was exactly true in the zero energy limit, but this result goes further.

So little is known about quantum walks on networks of arbitrary geometry that we were very pleased to find this result. It says there is a special case in which the walk is characterized by the degree distribution of the network, and a clear physical parameter that bounds how far the walk is from this special case.

Also, in finding it we learned that the difficulties of the initial state dependence, enhanced by the lack of convergence to a stationary state, could be overcome for a quantum walk, and that the relationships between quantum and stochastic walks extend beyond those with shared generators.

What next?

That’s all for the latest bit of idea sharing at the interface between stochastic and quantum systems.

I hope I’ve piqued your interest about quantum walks. There’s so much still left to work out about this topic, and your help is needed!

Other questions we have include: What holds analytically about the form of the quantum correction? Numerically it is known that the so-called quantum correction \Omega tends to enhance the probability of being found on nodes of low degree compared to \pi. Can someone explain why? What happens if a small amount of stochastic noise is added to a quantum walk? Or a lot of noise?

It’s difficult to know who is best placed to answer these questions: experts in quantum physics, graph theory, complex networks or stochastic processes? I suspect it’ll take a bit of help from everyone.

Background reading

A couple of textbooks with comprehensive sections on non-negative matrices and continuous-time stochastic processes are:

• Peter Lancaster and Miron Tismenetsky, The Theory of Matrices: with Applications, 2nd edition, Academic Press, San Diego, 1985.

• James R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1997.

There is, of course, the book that arose from the Azimuth network theory series, which considers several relationships between quantum and stochastic processes on networks:

• John Baez and Jacob Biamonte, A Course on Quantum Techniques for Stochastic Mechanics, 2012.

Another couple of books on complex networks are:

• Mark Newman, Networks: An Introduction, Oxford University Press, Oxford, 2010.

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford University Press, Oxford, 2011. Note that the first chapter is available free online.

There are plenty more useful references in our article on this topic:

• Mauro Faccin, Tomi Johnson, Jacob Biamonte, Sabre Kais and Piotr Migdał, Degree distribution in quantum walks on complex networks.

Puzzles for the enthusiastic

Sadly I didn’t have space to show proofs of all the theorems I used. So here are a few puzzles that guide you to doing the proofs for yourself:

Stochastic walks and stationary states

Puzzle 2. (For the hard core.) Prove there is always a unique positive eigenvector for a stochastic walk generated by S. You’ll need the assumption that the graph G is connected. It’s not simple, and you’ll probably need help from a book, perhaps one of those above by Lancaster and Tismenetsky, and Norris.

Puzzle 3. Show that the eigenvalues of S (and therefore Q) are non-negative. A good way to start this proof is to apply the Perron-Frobenius theorem to the non-negative matrix M = - S + I \max_i S_{i i}. This implies that M has a positive eigenvalue r equal to its spectral radius

r = \max_k | \lambda_k |

where \lambda_k are the eigenvalues of M, and the associated eigenvector v is positive. Since S = - M + I \max_i S_{i i}, it follows that S shares the eigenvectors of M and the associated eigenvalues are related by inverted translation:

\epsilon_k = - \lambda_k + \max_i S_{i i}

Puzzle 4. Prove that regardless of the initial state \psi(0), the zero eigenvector \pi is obtained in the large-time limit \lim_{t \rightarrow \infty} \psi(t) = \pi of the walk generated by S. This breaks down into two parts:

(a) Using the approach from Puzzle 5, to show that S v  = \epsilon_v v, the positivity of v and the infinitesimal stochastic property \sum_i S_{i j} = 0 imply that \epsilon_v = \epsilon_0 = 0 and thus v = \pi is actually the unique zero eigenvector and stationary state of S (its uniqueness follows from puzzle 4, you don’t need to re-prove it).

(b) By inserting the decomposition S = \sum_k \epsilon_k \Pi_k into e^{-S t} and using the result of puzzle 5, complete the proof.

(Though I ask you to use the diagonalizability of S, the main results still hold if the generator is irreducible but not diagonalizable.)

Quantum walks

Here are a couple of extra puzzles for those of you interested in quantum mechanics:

Puzzle 5. In quantum mechanics, probabilities are given by the moduli squared of amplitudes, so multiplying a state by a number of modulus unity has no physical effect. By inserting

\displaystyle{ Q= \sum_k \epsilon_k \Phi_k }

into the quantum time evolution matrix e^{-Q t}, show that if

\psi(0) = \phi_k

then

\psi(t)  = e^{ - i \epsilon_k t} \psi(0)

hence \phi_k is a stationary state in the quantum sense, as probabilities don’t change in time.

Puzzle 6. By expanding the initial state \psi(0) in terms of the complete orthogonal basis vectors \phi_k show that for a quantum walk \psi(t) never converges to a stationary state unless it began in one.


Quantum Network Theory (Part 1)

5 August, 2013

guest post by Tomi Johnson

If you were to randomly click a hyperlink on this web page and keep doing so on each page that followed, where would you end up?

As an esteemed user of Azimuth, I’d like to think you browse more intelligently, but the above is the question Google asks when deciding how to rank the world’s web pages.

Recently, together with the team (Mauro Faccin, Jacob Biamonte and Piotr Migdał) at the ISI Foundation in Turin, we attended a workshop in which several of the attendees were asking a similar question with a twist. “What if you, the web surfer, behaved quantum mechanically?”

Now don’t panic! I have no reason to think you might enter a superposition of locations or tunnel through a wall. This merely forms part of a recent drive towards understanding the role that network science can play in quantum physics.

As we’ll find, playing with quantum networks is fun. It could also become a necessity. The size of natural systems in which quantum effects have been identified has grown steadily over the past few years. For example, attention has recently turned to explaining the remarkable efficiency of light-harvesting complexes, comprising tens of molecules and thousands of atoms, using quantum mechanics. If this expansion continues, perhaps quantum physicists will have to embrace the concepts of complex networks.

To begin studying quantum complex networks, we found a revealing toy model. Let me tell you about it. Like all good stories, it has a beginning, a middle and an end. In this part, I’ll tell you the beginning and the middle. I’ll introduce the stochastic walk describing the randomly clicking web surfer mentioned above and a corresponding quantum walk. In part 2 the story ends with the bounding of the difference between the two walks in terms of the energy of the walker.

But for now I’ll start by introducing you to a graph, this time representing the internet!

If this taster gets you interested, there are more details available here:

• Mauro Faccin, Tomi Johnson, Jacob Biamonte, Sabre Kais and Piotr Migdał, Degree distribution in quantum walks on complex networks, arXiv:1305.6078 (2013).

What does the internet look like from above?

As we all know, the idea of the internet is to connect computers to each other. What do these connections look like when abstracted as a network, with each computer a node and each connection an edge?

The internet on a local scale, such as in your house or office, might look something like this:

 

Local network

with several devices connected to a central hub. Each hub connects to other hubs, and so the internet on a slightly larger scale might look something like this:

 

Regional network

What about the full global, not local, structure of the internet? To answer this question, researchers have developed representations of the whole internet, such as this one:

 

Global network

While such representations might be awe inspiring, how can we make any sense of them? Or are they merely excellent desktop wallpapers and new-age artworks?

In terms of complex network theory, there’s actually a lot that can be said that is not immediately obvious from the above representation.

For example, we find something very interesting if we plot the number of web pages with different incoming links (called degree) on a log-log axis. What is found for the African web is the following:

Power law degree distribution

This shows that very few pages are linked to by a very large number others, while a very large number of pages receive very few links. More precisely, what this shows is a power law distribution, the signature of which is a straight line on a log-log axis.

In fact, power law distributions arise in a diverse number of real world networks, human-built networks such as the internet and naturally occurring networks. It is often discussed alongside the concept of the preferential attachment; highly connected nodes seem to accumulate connections more quickly. We all know of a successful blog whose success had led to an increased presence and more success. That’s an example of preferential attachment.

It’s clear then that degree is an important concept in network theory, and its distribution across the nodes a useful characteristic of a network. Degree gives one indication of how important a node is in a network.

And this is where stochastic walks come in. Google, who are in the business of ranking the importance of nodes (web pages) in a network (the web), use (up to a small modification) the idealized model of a stochastic walker (web surfer) who randomly hops to connected nodes (follows one of the links on a page). This is called the uniform escape model, since the total rate of leaving any node is set to be the same for all nodes. Leaving the walker to wander for a long while, Google then takes the probability of the walker being on a node to rank the importance of that node. In the case that the network is undirected (all links are reciprocated) this long-time probability, and therefore the rank of the node, is proportional to the degree of the node.

So node degrees and the uniform escape model play an important role in the fields of complex networks and stochastic walks. But can they tell us anything about the much more poorly understood topics of quantum networks and quantum walks? In fact, yes, and demonstrating that to you is the purpose of this pair of articles.

Before we move on to the interesting bit, the math, it’s worth just listing a few properties of quantum walks that make them hard to analyze, and explaining why they are poorly understood. These are the difficulties we will show how to overcome below.

No convergence. In a stochastic walk, if you leave the walker to wander for a long time, eventually the probability of finding a walker at a node converges to a constant value. In a quantum walk, this doesn’t happen, so the walk can’t be characterized so easily by its long-time properties.

Dependence on initial states. In some stochastic walks the long-time properties of the walk are independent of the initial state. It is possible to characterize the stochastic walk without referring to the initialization of the walker. Such a characterization is not so easy in quantum walks, since their evolution always depends on the initialization of the walker. Is it even possible then to say something useful that applies to all initializations?

Stochastic and quantum generators differ. Those of you familiar with the network theory series know that some generators produce both stochastic and quantum walks (see part 16 for more details). However, most stochastic walk generators, including that for the uniform escape model, do not generate quantum walks and vice versa. How do we then compare stochastic and quantum walks when their generators differ?

With the task outlined, let’s get started!

Graphs and walks

In the next couple of sections I’m going to explain the diagram below to you. If you’ve been following the network theory series, in particular part 20, you’ll find parts of it familiar. But as it’s been a while since the last post covering this topic, let’s start with the basics.

Diagram outlining the main concepts

A simple graph G can be used to define both stochastic and quantum walks. A simple graph is something like this:

Illustration of a simple graph

where there is at most one edge between any two nodes, there are no edges from a node to itself and all edges are undirected. To avoid complications, let’s stick to simple graphs with a finite number n of nodes. Let’s also assume you can get from every node to every other node via some combination of edges i.e. the graph is connected.

In the particular example above the graph represents a network of n = 5 nodes, where nodes 3 and 4 have degree (number of edges) 3, and nodes 1, 2 and 5 have degree 2.

Every simple graph defines a matrix A, called the adjacency matrix. For a network with n nodes, this matrix is of size n \times n, and each element A_{i j} is unity if there is an edge between nodes i and j, and zero otherwise (let’s use this basis for the rest of this post). For the graph drawn above the adjacency matrix is

\left( \begin{matrix} 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \end{matrix} \right)

By construction, every adjacency matrix is symmetric:

A =A^T

(the T means the transposition of the elements in the node basis) and further, because each A is real, it is self-adjoint:

A=A^\dagger

(the \dagger means conjugate transpose).

This is nice, since (as seen in parts 16 and 20) a self-adjoint matrix generates a continuous-time quantum walk.

To recap from the series, a quantum walk is an evolution arising from a quantum walker moving on a network.

A state of a quantum walk is represented by a size n complex column vector \psi. Each element \langle i , \psi \rangle of this vector is the so-called amplitude associated with node i and the probability of the walker being found on that node (if measured) is the modulus of the amplitude squared |\langle i , \psi \rangle|^2. Here i is the standard basis vector with a single non-zero ith entry equal to unity, and \langle u , v \rangle = u^\dagger v is the usual inner product.

A quantum walk evolves in time according to the Schrödinger equation

\displaystyle{ \frac{d}{d t} \psi(t)= - i H \psi(t) }

where H is called the Hamiltonian. If the initial state is \psi(0) then the solution is written as

\psi(t) = \exp(- i t H) \psi(0)

The probabilities | \langle i , \psi (t) \rangle |^2 are guaranteed to be correctly normalized when the Hamiltonian H is self-adjoint.

There are other matrices that are defined by the graph. Perhaps the most familiar is the Laplacian, which has recently been a topic on this blog (see parts 15, 16 and 20 of the series, and this recent post).

The Laplacian L is the n \times n matrix

L = D - A

where the degree matrix D is an n \times n diagonal matrix with elements given by the degrees

\displaystyle{ D_{i i}=\sum_{j} A_{i j} }

For the graph drawn above, the degree matrix and Laplacian are:

\left( \begin{matrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} \right)  \qquad \mathrm{and} \qquad \left( \begin{matrix} 2 & -1 & 0 & -1 & 0 \\ -1 & 2 & -1 & 0 & 0 \\ 0 & -1 & 3 & -1 & -1 \\ -1 & 0 & -1 & 3 & -1 \\ 0 & 0 & -1 & -1 & 2 \end{matrix} \right)

The Laplacian is self-adjoint and generates a quantum walk.

The Laplacian has another property; it is infinitesimal stochastic. This means that its off diagonal elements are non-positive and its columns sum to zero. This is interesting because an infinitesimal stochastic matrix generates a continuous-time stochastic walk.

To recap from the series, a stochastic walk is an evolution arising from a stochastic walker moving on a network.

A state of a stochastic walk is represented by a size n non-negative column vector \psi. Each element \langle i , \psi \rangle of this vector is the probability of the walker being found on node i.

A stochastic walk evolves in time according to the master equation

\displaystyle{ \frac{d}{d t} \psi(t)= - H \psi(t) }

where H is called the stochastic Hamiltonian. If the initial state is \psi(0) then the solution is written

\psi(t) = \exp(- t H) \psi(0)

The probabilities \langle i , \psi (t) \rangle are guaranteed to be non-negative and correctly normalized when the stochastic Hamiltonian H is infinitesimal stochastic.

So far, I have just presented what has been covered on Azimuth previously. However, to analyze the important uniform escape model we need to go beyond the class of (Dirichlet) generators that produce both quantum and stochastic walks. Further, we have to somehow find a related quantum walk. We’ll see below that both tasks are achieved by considering the normalized Laplacians: one generating the uniform escape stochastic walk and the other a related quantum walk.

Normalized Laplacians

The two normalized Laplacians are:

• the asymmetric normalized Laplacian S = L D^{-1} (that generates the uniform escape Stochastic walk) and

• the symmetric normalized Laplacian Q = D^{-1/2} L D^{-1/2} (that generates a Quantum walk).

For the graph drawn above the asymmetric normalized Laplacian S is

\left( \begin{matrix} 1 & -1/2 & 0 & -1/3 & 0 \\ -1/2 & 1 & -1/3 & 0 & 0 \\ 0 & -1/2 & 1 & -1/3 & -1/2 \\ -1/2 & 0 & -1/3 & 1 & -1/2 \\ 0 & 0 & -1/3 & -1/3 & 1 \end{matrix} \right)

The identical diagonal elements indicates that the total rates of leaving each node are identical, and the equality within each column of the other non-zero elements indicates that the walker is equally likely to hop to any node connected to its current node. This is the uniform escape model!

For the same graph the symmetric normalized Laplacian Q is

\left( \begin{matrix} 1 & -1/2 & 0 & -1/\sqrt{6}  & 0 \\ -1/2 & 1 & -1/\sqrt{6}  & 0 & 0 \\ 0 & -1/\sqrt{6}  & 1 & -1/3 & -1/\sqrt{6}  \\ -1/\sqrt{6} & 0 & -1/3 & 1 & -1/\sqrt{6}  \\ 0 & 0 & -1/\sqrt{6}  & -1/\sqrt{6}  & 1 \end{matrix} \right)

That the diagonal elements are identical in the quantum case indicates that all nodes are of equal energy, this is type of quantum walk usually considered.

Puzzle 1. Show that in general S is infinitesimal stochastic but not self-adjoint.

Puzzle 2. Show that in general Q is self-adjoint but not infinitesimal stochastic.

So a graph defines two matrices: one S that generates a stochastic walk, and one Q that generates a quantum walk. The natural question to ask is whether these walks are related. The answer is that they are!

Underpinning this relationship is the mathematical property that S and Q are similar. They are related by the following similarity transformation

S = D^{1/2} Q D^{-1/2}

which means that any eigenvector \phi_k of Q associated to eigenvalue \epsilon_k gives a vector

\pi_k \propto D^{1/2} \phi_k

that is an eigenvector of S with the same eigenvalue! To show this, insert the identity I = D^{-1/2} D^{1/2} into

Q \phi_k = \epsilon_k \phi_k

and multiply from the left with D^{1/2} to obtain

\begin{aligned} (D^{1/2}  Q D^{-1/2} ) (D^{1/2} \phi_k) &= \epsilon_k ( D^{1/2} \phi_k ) \\ S \pi_k &= \epsilon_k \pi_k  \end{aligned}

The same works in the opposite direction. Any eigenvector \pi_k of S gives an eigenvector

\phi_k \propto D^{-1/2} \pi_k

of Q with the same eigenvalue \epsilon_k.

The mathematics is particularly nice because Q is self-adjoint. A self-adjoint matrix is diagonalizable, and has real eigenvalues and orthogonal eigenvectors.

As a result, the symmetric normalized Laplacian can be decomposed as

Q = \sum_k \epsilon_k \Phi_k

where \epsilon_k is real and \Phi_k are orthogonal projectors. Each \Phi_k acts as the identity only on vectors in the space spanned by \phi_k and as zero on all others, such that

\Phi_k \Phi_\ell = \delta_{k \ell} \Phi_k.

Multiplying from the left by D^{1/2} and the right by D^{-1/2} results in a similar decomposition for S:

S = \sum_k \epsilon_k \Pi_k

with orthogonal projectors

\Pi_k  = D^{1/2} \Phi_k D^{-1/2}

I promised above that I would explain the following diagram:

Diagram outlining the main concepts (again)

Let’s summarize what it represents now:

G is a simple graph that specifies

A the adjacency matrix (generator of a quantum walk), which subtracted from

D the diagonal matrix of the degrees gives

L the symmetric Laplacian (generator of stochastic and quantum walks), which when normalized by D returns both

S the generator of the uniform escape stochastic walk and

Q the quantum walk generator to which it is similar!

What next?

Sadly, this is where we’ll finish for now.

We have all the ingredients necessary to study the walks generated by the normalized Laplacians and exploit the relationship between them.

Next time, in part 2, I’ll talk you through the mathematics of the uniform escape stochastic walk S and how it connects to the degrees of the nodes in the long-time limit. Then I’ll show you how this helps us solve aspects of the quantum walk generated by Q.

In other news

Before I leave you, let me tell you about a workshop the ISI team recently attended (in fact helped organize) at the Institute of Quantum Computing, on the topic of quantum computation and complex networks. Needless to say, there were talks on papers related to quantum mechanics and networks!

Some researchers at the workshop gave exciting talks based on numerical examinations of what happens if a quantum walk is used instead of a stochastic walk to rank the nodes of a network:

• Giuseppe Davide Paparo and Miguel Angel Martín-Delgado, Google in a quantum network, Sci. Rep. 2 (2012), 444.

• Eduardo Sánchez-Burillo, Jordi Duch, Jesús Gómez-Gardenes and David Zueco, Quantum navigation and ranking in complex networks, Sci. Rep. 2 (2012), 605.

Others attending the workshop have numerically examined what happens when using quantum computers to represent the stationary state of a stochastic process:

• Silvano Garnerone, Paolo Zanardi and Daniel A. Lidar, Adiabatic quantum algorithm for search engine ranking, Phys. Rev. Lett. 108 (2012), 230506.

It was a fun workshop and we plan to organize/attend more in the future!


Quantum Techniques for Reaction Networks

11 June, 2013

Fans of the network theory series might like to look at this paper:

• John Baez, Quantum techniques for reaction networks.

and I would certainly appreciate comments and corrections.

This paper tackles a basic question we never got around to discussing: how the probabilistic description of a system where bunches of things randomly interact and turn into other bunches of things can reduce to a deterministic description in the limit where there are lots of things!

Mathematically, such systems are given by ‘stochastic Petri nets’, or if you prefer, ‘stochastic reaction networks’. These are just two equivalent pictures of the same thing. For example, we could describe some chemical reactions using this Petri net:

but chemists would use this reaction network:

C + O2 → CO2
CO2 + NaOH → NaHCO3
NaHCO3 + HCl → H2O + NaCl + CO2

Making either of them ‘stochastic’ merely means that we specify a ‘rate constant’ for each reaction, saying how probable it is.

For any such system we get a ‘master equation’ describing how the probability of having any number of things of each kind changes with time. In the class I taught on this last quarter, the students and I figured out how to derive from this an equation saying how the expected number of things of each kind changes with time. Later I figured out a much slicker argument… but either way, we get this result:

Theorem. For any stochastic reaction network and any stochastic state \Psi(t) evolving in time according to the master equation, then

\displaystyle{ \frac{d}{dt} \langle N \Psi(t) \rangle } =  \displaystyle{\sum_{\tau \in T}} \, r(\tau) \,  (s(\tau) - t(\tau)) \;  \left\langle N^{\underline{s(\tau)}}\, \Psi(t) \right\rangle

assuming the derivative exists.

Of course this will make no sense yet if you haven’t been following the network theory series! But I explain all the notation in the paper, so don’t be scared. The main point is that \langle N \Psi(t) \rangle is a vector listing the expected number of things of each kind at time t. The equation above says how this changes with time… but it closely resembles the ‘rate equation’, which describes the evolution of chemical systems in a deterministic way.

And indeed, the next big theorem says that the master equation actually implies the rate equation when the probability of having various numbers of things of each kind is given by a product of independent Poisson distributions. In this case \Psi(t) is what people in quantum physics call a ‘coherent state’. So:

Theorem. Given any stochastic reaction network, let
\Psi(t) be a mixed state evolving in time according to the master equation. If \Psi(t) is a coherent state when t = t_0, then \langle N \Psi(t) \rangle obeys the rate equation when t = t_0.

In most cases, this only applies exactly at one moment of time: later \Psi(t) will cease to be a coherent state. Then we must resort to the previous theorem to see how the expected number of things of each kind changes with time.

But sometimes our state \Psi(t) will stay coherent forever! For one case where this happens, see the companion paper, which I blogged about a little while ago:

• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.

We wrote this first, but logically it comes after the one I just finished now!

All this material will get folded into the book I’m writing with Jacob Biamonte. There are just a few remaining loose ends that need to be tied up.


Network Theory (Part 29)

23 April, 2013

I’m talking about electrical circuits, but I’m interested in them as models of more general physical systems. Last time we started seeing how this works. We developed an analogy between electrical circuits and physical systems made of masses and springs, with friction:

Electronics Mechanics
charge: Q position: q
current: I = \dot{Q} velocity: v = \dot{q}
flux linkage: \lambda momentum: p
voltage: V = \dot{\lambda} force: F = \dot{p}
inductance: L mass: m
resistance: R damping coefficient: r  
inverse capacitance: 1/C   spring constant: k

But this is just the first of a large set of analogies. Let me list some, so you can see how wide-ranging they are!

More analogies

People in system dynamics often use effort as a term to stand for anything analogous to force or voltage, and flow as a general term to stand for anything analogous to velocity or electric current. They call these variables e and f.

To me it’s important that force is the time derivative of momentum, and velocity is the time derivative of position. Following physicists, I write momentum as p and position as q. So, I’ll usually write effort as \dot{p} and flow as \dot{q}.

Of course, ‘position’ is a term special to mechanics; it’s nice to have a general term for the thing whose time derivative is flow, that applies to any context. People in systems dynamics seem to use displacement as that general term.

It would also be nice to have a general term for the thing whose time derivative is effort… but I don’t know one. So, I’ll use the word momentum.

Now let’s see the analogies! Let’s see how displacement q, flow \dot{q}, momentum p and effort \dot{p} show up in several subjects:

displacement:    q flow:      \dot q momentum:      p effort:           \dot p
Mechanics: translation position velocity momentum force
Mechanics: rotation angle angular velocity angular momentum torque
Electronics charge current flux linkage voltage
Hydraulics volume flow pressure momentum pressure
Thermal Physics entropy entropy flow temperature momentum temperature
Chemistry moles molar flow chemical momentum chemical potential

We’d been considering mechanics of systems that move along a line, via translation, but we can also consider mechanics for systems that turn round and round, via rotation. So, there are two rows for mechanics here.

There’s a row for electronics, and then a row for hydraulics, which is closely analogous. In this analogy, a pipe is like a wire. The flow of water plays the role of current. Water pressure plays the role of electrostatic potential. The difference in water pressure between two ends of a pipe is like the voltage across a wire. When water flows through a pipe, the power equals the flow times this pressure difference—just as in an electrical circuit the power is the current times the voltage across the wire.

A resistor is like a narrowed pipe:

An inductor is like a heavy turbine placed inside a pipe: this makes the water tend to keep flowing at the same rate it’s already flowing! In other words, it provides a kind of ‘inertia’ analogous
to mass.

A capacitor is like a tank with pipes coming in from both ends, and a rubber sheet dividing it in two lengthwise:

When studying electrical circuits as a kid, I was shocked when I first learned that capacitors don’t let the electrons through: it didn’t seem likely you could do anything useful with something like that! But of course you can. Similarly, this gizmo doesn’t let the water through.

A voltage source is like a compressor set up to maintain a specified pressure difference between the input and output:

Similarly, a current source is like a pump set up to maintain a specified flow.

Finally, just as voltage is the time derivative of a fairly obscure quantity called ‘flux linkage’, pressure is the time derivative of an even more obscure quantity which has no standard name. I’m calling it ‘pressure momentum’, thanks to the analogy

momentum: force :: pressure momentum: pressure

Just as pressure has units of force per area, pressure momentum has units of momentum per area!

People invented this analogy back when they were first struggling to understand electricity, before electrons had been observed:

Hydraulic analogy, Wikipedia.

The famous electrical engineer Oliver Heaviside pooh-poohed this analogy, calling it the “drain-pipe theory”. I think he was making fun of William Henry Preece. Preece was another electrical engineer, who liked the hydraulic analogy and disliked Heaviside’s fancy math. In his inaugural speech as president of the Institution of Electrical Engineers in 1893, Preece proclaimed:

True theory does not require the abstruse language of mathematics to make it clear and to render it acceptable. All that is solid and substantial in science and usefully applied in practice, have been made clear by relegating mathematic symbols to their proper store place—the study.

According to the judgement of history, Heaviside made more progress in understanding electromagnetism than Preece. But there’s still a nice analogy between electronics and hydraulics. And I’ll eventually use the abstruse language of mathematics to make it very precise!

But now let’s move on to the row called ‘thermal physics’. We could also call this ‘thermodynamics’. It works like this. Say you have a physical system in thermal equilibrium and all you can do is heat it up or cool it down ‘reversibly’—that is, while keeping it in thermal equilibrium all along. For example, imagine a box of gas that you can heat up or cool down. If you put a tiny amount dE of energy into the system in the form of heat, then its entropy increases by a tiny amount dS. And they’re related by this equation:

dE = TdS

where T is the temperature.

Another way to say this is

\displaystyle{ \frac{dE}{dt} = T \frac{dS}{dt} }

where t is time. On the left we have the power put into the system in the form of heat. But since power should be ‘effort’ times ‘flow’, on the right we should have ‘effort’ times ‘flow’. It makes some sense to call dS/dt the ‘entropy flow’. So temperature, T, must play the role of ‘effort’.

This is a bit weird. I don’t usually think of temperature as a form of ‘effort’ analogous to force or torque. Stranger still, our analogy says that ‘effort’ should be the time derivative of some kind of ‘momentum’, So, we need to introduce temperature momentum: the integral of temperature over time. I’ve never seen people talk about this concept, so it makes me a bit nervous.

But when we have a more complicated physical system like a piston full of gas in thermal equilibrium, we can see the analogy working. Now we have

dE = TdS - PdV

The change in energy dE of our gas now has two parts. There’s the change in heat energy TdS, which we saw already. But now there’s also the change in energy due to compressing the piston! When we change the volume of the gas by a tiny amount dV, we put in energy -PdV.

Now look back at the first chart I drew! It says that pressure is a form of ‘effort’, while volume is a form of ‘displacement’. If you believe that, the equation above should help convince you that temperature is also a form of effort, while entropy is a form of displacement.

But what about the minus sign? That’s no big deal: it’s the result of some arbitrary conventions. P is defined to be the outward pressure of the gas on our piston. If this is positive, reducing the volume of the gas takes a positive amount of energy, so we need to stick in a minus sign. I could eliminate this minus sign by changing some conventions—but if I did, the chemistry professors at UCR would haul me away and increase my heat energy by burning me at the stake.

Speaking of chemistry: here’s how the chemistry row in the analogy chart works. Suppose we have a piston full of gas made of different kinds of molecules, and there can be chemical reactions that change one kind into another. Now our equation gets fancier:

\displaystyle{ dE = TdS - PdV + \sum_i  \mu_i dN_i }

Here N_i is the number of molecules of the ith kind, while \mu_i is a quantity called a chemical potential. The chemical potential simply says how much energy it takes to increase the number of molecules of a given kind. So, we see that chemical potential is another form of effort, while number of molecules is another form of displacement.

But chemists are too busy to count molecules one at a time, so they count them in big bunches called ‘moles’. A mole is the number of atoms in 12 grams of carbon-12. That’s roughly

602,214,150,000,000,000,000,000

atoms. This is called Avogadro’s constant. If we used 1 gram of hydrogen, we’d get a very close number called ‘Avogadro’s number’, which leads to lots of jokes:

(He must be desperate because he looks so weird… sort of like a mole!)

So, instead of saying that the displacement in chemistry is called ‘number of molecules’, you’ll sound more like an expert if you say ‘moles’. And the corresponding flow is called molar flow.

The truly obscure quantity in this row of the chart is the one whose time derivative is chemical potential! I’m calling it chemical momentum simply because I don’t know another name.

Why are linear and angular momentum so famous compared to pressure momentum, temperature momentum and chemical momentum?

I suspect it’s because the laws of physics are symmetrical
under translations and rotations. When the assumptions of Noether’s theorem hold, this guarantees that the total momentum and angular momentum of a closed system are conserved. Apparently the laws of physics lack the symmetries that would make the other kinds of momentum be conserved.

This suggests that we should dig deeper and try to understand more deeply how this chart is connected to ideas in classical mechanics, like Noether’s theorem or symplectic geometry. I will try to do that sometime later in this series.

More generally, we should try to understand what gives rise to a row in this analogy chart. Are there are lots of rows I haven’t talked about yet, or just a few? There are probably lots. But are there lots of practically important rows that I haven’t talked about—ones that can serve as the basis for new kinds of engineering? Or does something about the structure of the physical world limit the number of such rows?

Mildly defective analogies

Engineers care a lot about dimensional analysis. So, they often make a big deal about the fact that while effort and flow have different dimensions in different rows of the analogy chart, the following four things are always true:

pq has dimensions of action (= energy × time)
\dot{p} q has dimensions of energy
p \dot{q} has dimensions of energy
\dot{p} \dot{q} has dimensions of power (= energy / time)

In fact any one of these things implies all the rest.

These facts are important when designing ‘mixed systems’, which combine different rows in the chart. For example, in mechatronics, we combine mechanical and electronic elements in a single circuit! And in a hydroelectric dam, power is converted from hydraulic to mechanical and then electric form:

One goal of network theory should be to develop a unified language for studying mixed systems! Engineers have already done most of the hard work. And they’ve realized that thanks to conservation of energy, working with pairs of flow and effort variables whose product has dimensions of power is very convenient. It makes it easy to track the flow of energy through these systems.

However, people have tried to extend the analogy chart to include ‘mildly defective’ examples where effort times flow doesn’t have dimensions of power. The two most popular are these:

displacement:    q flow:      \dot q momentum:      p effort:           \dot p
Heat flow heat heat flow temperature momentum temperature
Economics inventory product flow economic momentum product price

The heat flow analogy comes up because people like to think of heat flow as analogous to electrical current, and temperature as analogous to voltage. Why? Because an insulated wall acts a bit like a resistor! The current flowing through a resistor is a function the voltage across it. Similarly, the heat flowing through an insulated wall is about proportional to the difference in temperature between the inside and the outside.

However, there’s a difference. Current times voltage has dimensions of power. Heat flow times temperature does not have dimensions of power. In fact, heat flow by itself already has dimensions of power! So, engineers feel somewhat guilty about this analogy.

Being a mathematical physicist, a possible way out presents itself to me: use units where temperature is dimensionless! In fact such units are pretty popular in some circles. But I don’t know if this solution is a real one, or whether it causes some sort of trouble.

In the economic example, ‘energy’ has been replaced by ‘money’. So other words, ‘inventory’ times ‘product price’ has units of money. And so does ‘product flow’ times ‘economic momentum’! I’d never heard of economic momentum before I started studying these analogies, but I didn’t make up that term. It’s the thing whose time derivative is ‘product price’. Apparently economists have noticed a tendency for rising prices to keep rising, and falling prices to keep falling… a tendency toward ‘conservation of momentum’ that doesn’t fit into their models of rational behavior.

I’m suspicious of any attempt to make economics seem like physics. Unlike elementary particles or rocks, people don’t seem to be very well modelled by simple differential equations. However, some economists have used the above analogy to model economic systems. And I can’t help but find that interesting—even if intellectually dubious when taken too seriously.

An auto-analogy

Beside the analogy I’ve already described between electronics and mechanics, there’s another one, called ‘Firestone’s analogy':

• F.A. Firestone, A new analogy between mechanical and electrical systems, Journal of the Acoustical Society of America 4 (1933), 249–267.

Alain Bossavit pointed this out in the comments to Part 27. The idea is to treat current as analogous to force instead of velocity… and treat voltage as analogous to velocity instead of force!

In other words, switch your p’s and q’s:

Electronics Mechanics          (usual analogy) Mechanics      (Firestone’s analogy)
charge position: q momentum: p
current velocity: \dot{q} force: \dot{p}
flux linkage momentum: p position: q
voltage force: \dot{p} velocity: \dot{q}

This new analogy is not ‘mildly defective': the product of effort and flow variables still has dimensions of power. But why bother with another analogy?

It may be helpful to recall this circuit from last time:

It’s described by this differential equation:

L \ddot{Q} + R \dot{Q} + C^{-1} Q = V

We used the ‘usual analogy’ to translate it into classical mechanics problem, and we got a problem where an object of mass L is hanging from a spring with spring constant 1/C and damping coefficient R, and feeling an additional external force F:

m \ddot{q} + r \dot{q} + k q = F

And that’s fine. But there’s an intuitive sense in which all three forces are acting ‘in parallel’ on the mass, rather than in series. In other words, all side by side, instead of one after the other.

Using Firestone’s analogy, we get a different classical mechanics problem, where the three forces are acting in series. The spring is connected to source of friction, which in turn is connected to an external force.

This may seem a bit mysterious. But instead of trying to explain it, I’ll urge you to read his paper, which is short and clearly written. I instead want to make a somewhat different point, which is that we can take a mechanical system, convert it to an electrical one following the usual analogy, and then convert back to a mechanical one using Firestone’s analogy. This gives us an ‘auto-analogy’ between mechanics and itself, which switches p and q.

And although I haven’t been able to figure out why from Firestone’s paper, I have other reasons for feeling sure this auto-analogy should contain a minus sign. For example:

p \mapsto q, \qquad q \mapsto -p

In other words, it should correspond to a 90° rotation in the (p,q) plane. There’s nothing sacred about whether we rotate clockwise or counterclockwise; we can equally well do this:

p \mapsto -q, \qquad q \mapsto p

But we need the minus sign to get a so-called symplectic transformation of the (p,q) plane. And from my experience with classical mechanics, I’m pretty sure we want that. If I’m wrong, please let me know!

I have a feeling we should revisit this issue when we get more deeply into the symplectic aspects of circuit theory. So, I won’t go on now.

References

The analogies I’ve been talking about are studied in a branch of engineering called system dynamics. You can read more about it here:

• Dean C. Karnopp, Donald L. Margolis and Ronald C. Rosenberg, System Dynamics: a Unified Approach, Wiley, New York, 1990.

• Forbes T. Brown, Engineering System Dynamics: a Unified Graph-Centered Approach, CRC Press, Boca Raton, 2007.

• Francois E. Cellier, Continuous System Modelling, Springer, Berlin, 1991.

System dynamics already uses lots of diagrams of networks. One of my goals in weeks to come is to explain the category theory lurking behind these diagrams.


Petri Net Programming (Part 3)

19 April, 2013

guest post by David Tanzer

The role of differential equations

Last time we looked at stochastic Petri nets, which use a random event model for the reactions. Individual entities are represented by tokens that flow through the network. When the token counts get large, we observed that they can be approximated by continuous quantities, which opens the door to the application of continuous mathematics to the analysis of network dynamics.

A key result of this approach is the “rate equation,” which gives a law of motion for the expected sizes of the populations. Equilibrium can then be obtained by solving for zero motion. The rate equations are applied in chemistry, where they give the rates of change of the concentrations of the various species in a reaction.

But before discussing the rate equation, here I will talk about the mathematical form of this law of motion, which consists of differential equations. This form is naturally associated with deterministic systems involving continuous magnitudes. This includes the equations of motion for the sine wave:

and the graceful ellipses that are traced out by the orbits of the planets around the sun:

This post provides some mathematical context to programmers who have not worked on scientific applications. My goal is to get as many of you on board as possible, before setting sail with Petri net programming.

Three approaches to equations: theoretical, formula-based, and computational

Let’s first consider the major approaches to equations in general. We’ll illustrate with a Diophantine equation

x^9 + y^9 + z^9 = 2

where x, y and z are integer variables.

In the theoretical approach (aka “qualitative analysis”), we start with the meaning of the equation and then proceed to reason about its solutions. Here are some simple consequences of this equation. They can’t all be zero, can’t all be positive, can’t all be negative, can’t all be even, and can’t all be odd.

In the formula-based approach, we seek formulas to describe the solutions. Here is an example of a formula (which does not solve our equation):

\{(x,y,z) | x = n^3, y = 2n - 4, z = 4 n | 1 \leq n \leq 5 \}

Such formulas are nice to have, but the pursuit of them is diabolically difficult. In fact, for Diophantine equations, even the question of whether an arbitrarily chosen equation has any solutions whatsoever has been proven to be algorithmically undecidable.

Finally, in the computational approach, we seek algorithms to enumerate or numerically approximate the solutions to the equations.

The three approaches to differential equations

Let’s apply the preceding classification to differential equations.

Theoretical approach

A differential equation is one that constrains the rates at which the variables are changing. This can include constraints on the rates at which the rates are changing (second-order equations), etc. The equation is ordinary if there is a single independent variable, such as time, otherwise it is partial.

Consider the equation stating that a variable increases at a rate equal to its current value. The bigger it gets, the faster it increases. Given a starting value, this determines a process — the solution to the equation — which here is exponential growth.

Let X(t) be the value at time t, and let’s initialize it to 1 at time 0. So we have:

X(0) = 1

X'(t) = X(t)

These are first-order equations, because the derivative is applied at most once to any variable. They are linear equations, because the terms on each side of the equations are linear combinations of either individual variables or derivatives (in this case all of the coefficients are 1). Note also that a system of differential equations may in general have zero, one, or multiple solutions. This example belongs to a class of equations which are proven to have a unique solution for each initial condition.

You could imagine more complex systems of equations, involving multiple dependent variables, all still depending on time. That includes the rate equations for a Petri net, which have one dependent variable for each of the population sizes. The ideas for such systems are an extension of the ideas for a single-variable system. Then, a state of the system is a vector of values, with one component for each of the dependent variables. For first-order systems, such as the rate equations, where the derivatives appear on the left-hand sides, the equations determine, for each possible state of the system, a “direction” and rate of change for the state of the system.

Now here is a simple illustration of what I called the theoretical approach. Can X(t) ever become negative? No, because it starts out positive at time 0, and in order to later become negative, it must be decreasing at a time t_1 when it is still positive. That is to say, X(t_1) > 0, and X'(t_1) < 0. But that contradicts the assumption X'(t) = X(t). The general lesson here is that we don’t need a solution formula in order to make such inferences.

For the rate equations, the theoretical approach leads to substantial theorems about the existence and structure of equilibrium solutions.

Formula-based approach

It is natural to look for concise formulas to solve our equations, but the results of this overall quest are largely negative. The exponential differential equation cannot be solved by any formula that involves a finite combination of simple operations. So the solution function must be treated as a new primitive, and given a name, say \exp(t). But even when we extend our language to include this new symbol, there are many differential equations that remain beyond the reach of finite formulas. So an endless collection of primitive functions is called for. (As standard practice, we always include exp(t), and its complex extensions to the trigonometric functions, as primitives in our toolbox.)

But the hard mathematical reality does not end here, because even when solution formulas do exist, finding them may call for an ace detective. Only for certain classes of differential equations, such as the linear ones, do we have systematic solution methods.

The picture changes, however, if we let the formulas contain an infinite number of operations. Then the arithmetic operators give a far-reaching base for defining new functions. In fact, as you can verify, the power series

X(t) = 1 + t + t^2/2! + t^3/3! + ...

which we view as an “infinite polynomial” over the time parameter t, exactly satisfies our equations for exponential motion, X(0) = 1 and X'(t) = X(t). This power series therefore defines \exp(t). By the way, applying it to the input 1 produces a definition for the transcendental number e:

e = X(1) = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... \approx 2.71828

Computational approach

Let’s leave aside our troubles with formulas, and consider the computational approach. For broad classes of differential equations, there are approximation algorithms that be successfully applied.

For starters, any power series that satisfies a differential equation may work for a simple approximation method. If a series is known to converge over some range of inputs, then one can approximate the value at those points by stopping the computation after a finite number of terms.

But the standard methods work directly with the equations, provided that they can be put into the right form. The simplest one is called Euler’s method. It works over a sampling grid of points separated by some small number \epsilon. Let’s take the case where we have a first-order equation in explicit form, which means that X'(t) = f(X(t)) for some function f.

We begin with the initial value X(0). Applying f, we get X'(0) = f(X(0)). Then for the interval from 0 to \epsilon,we use a linear approximation for X(t), by assuming that the derivative remains constant at X'(0). That gives X(\epsilon) = X(0) + \epsilon \cdot X'(0). Next, X'(\epsilon) = f(X(\epsilon)), and X(2 \epsilon) = X(\epsilon) + \epsilon \cdot X'(\epsilon), etc. Formally,

X(0) = \textrm{initial}

X((n+1) \epsilon) = X(n \epsilon) + \epsilon f(X(n \epsilon))

Applying this to our exponential equation, where f(X(t)) = X(t), we get:

X(0) = 1

X((n+1) \epsilon) = X(n \epsilon) + \epsilon X(n \epsilon) = X(n \epsilon) (1 + \epsilon)

Hence:

X(n \epsilon) = (1 + \epsilon) ^ n

So the approximation method gives a discrete exponential growth, which converges to a continuous exponential in the limit as the mesh size goes to zero.

Note, the case we just considered has more generality than might appear at first, because (1) the ideas here are easily extended to systems of explicit first order equations, and (2) higher-order equations that are “explicit” in an extended sense—meaning that the highest-order derivative is expressed as a function of time, of the variable, and of the lower-order derivatives—can be converted into an equivalent system of explicit first-order equations.

The challenging world of differential equations

So, is our cup half-empty or half-full? We have no toolbox of primitive formulas for building the solutions to all differential equations by finite compositions. And even for those which can be solved by formulas, there is no general method for finding the solutions. That is how the cookie crumbles. But on the positive side, there is an array of theoretical tools for analyzing and solving important classes of differential equations, and numerical methods can be applied in many cases.

The study of differential equations leads to some challenging problems, such as the Navier-Stokes equations, which describe the flow of fluids.

These are partial differential equations involving flow velocity, pressure, density and external forces (such as gravity), all of which vary over space and time. There are non-linear (multiplicative) interactions between these variables and their spatial and temporal derivatives, which leads to complexity in the solutions.

At high flow rates, this complexity can produce chaotic solutions, which involve complex behavior at a wide range of resolution scales. This is turbulence. Here is an insightful portrait of turbulence, by Leonardo da Vinci, whose studies in turbulence date back to the 15th Century.

Turbulence, which has been described by Richard Feynman as the most important unsolved problem of classical physics, also presents a mathematical puzzle. The general existence of solutions to the Navier-Stokes equations remains unsettled. This is one of the “Millennium Prize Problems”, for which a one million dollar prize is offered: in three dimensions, given initial values for the velocity and scalar fields, does there exist a solution that is smooth and everywhere defined? There are also complications with grid-based numerical methods, which will fail to produce globally accurate results if the solutions contain details at a smaller scale than the grid mesh. So the ubiquitous phenomenon of turbulence, which is so basic to the movements of the atmosphere and the seas, remains an open case.

But fortunately we have enough traction with differential equations to proceed directly with the rate equations for Petri nets. There we will find illuminating equations, which are the subject of both major theorems and open problems. They are non-linear and intractable by formula-based methods, yet, as we will see, they are well handled by numerical methods.


Network Theory (Part 28)

10 April, 2013

Last time I left you with some puzzles. One was to use the laws of electrical circuits to work out what this one does:

If we do this puzzle, and keep our eyes open, we’ll see an analogy between electrical circuits and classical mechanics! And this is the first of a huge set of analogies. The same math shows up in many different subjects, whenever we study complex systems made of interacting parts. So, it should become part of any general theory of networks.

This simple circuit is very famous: it’s called a series RLC circuit, because it has a resistor of resistance R, an inductor of inductance L, and a capacitor of capacitance C, all hooked up ‘in series’, meaning one after another. But understand this circuit, it’s good to start with an even simpler one, where we leave out the voltage source:

This has three edges, so reading from top to bottom there are 3 voltages V_1, V_2, V_3, and 3 currents I_1, I_2, I_3, one for each edge. The white and black dots are called ‘nodes’, and the white ones are called ‘terminals': current can flow in or out of those.

The voltages and currents obey a bunch of equations:

• Kirchhoff’s current law says the current flowing into each node that’s not a terminal equals the current flowing out:

I_1 = I_2 = I_3

• Kirchhoff’s voltage law says there are potentials \phi_0, \phi_1, \phi_2, \phi_3, one for each node, such that:

V_1 = \phi_0 - \phi_1

V_2 = \phi_1 - \phi_2

V_3 = \phi_2 - \phi_3

In this particular problem, Kirchhoff’s voltage law doesn’t say much, since we can always find potentials obeying this, given the voltages. But in other problems it can be important. And even here it suggests that the sum V_1 + V_2 + V_3 will be important; this is the ‘total voltage across the circuit’.

Next, we get one equation for each circuit element:

• The law for a resistor says:

V_1 = R I_1

The law for a inductor says:

\displaystyle{ V_2 = L \frac{d I_2}{d t} }

The law for a capacitor says:

\displaystyle{ I_3 = C \frac{d V_3}{d t} }

These are all our equations. What should we do with them? Since I_1 = I_2 = I_3, it makes sense to call all these currents simply I and solve for each voltage in terms of this. Here’s what we get:

V_1 = R I

\displaystyle{ V_2 = L \frac{d I}{d t} }

\displaystyle {V_3 = C^{-1} \int I \, dt }

So, if we know the current flowing through the circuit we can work out the voltage across each circuit element!

Well, not quite: in the case of the capacitor we only know it up to a constant, since there’s a constant of integration. This may seem like a minor objection, but it’s worth taking seriously. The point is that the charge on the capacitor’s plate is proportional to the voltage across the capacitor:

\displaystyle{V_3 = C^{-1} Q }

When electrons move on or off the plate, this charge changes, and we get a current:

\displaystyle{I = \frac{d Q}{d t} }

So, we can work out the time derivative of V_3 from the current I, but to work out V_3 itself we need the charge Q.

Treat these as definitions if you like, but they’re physical facts too! And they let us rewrite our trio of equations:

V_1 = R I

\displaystyle{ V_2 = L \frac{d I}{d t} }

\displaystyle{V_3 = C^{-1} \int I \, dt }

in terms of the charge, as follows:

V_1 = R \dot{Q}

V_2 = L \ddot{Q}

V_3 = C^{-1} Q

Then if we add these three equations, we get

V_1 + V_2 + V_3 = L \ddot Q + R \dot Q + C^{-1} Q

So, if we define the total voltage by

V = V_1 + V_2 + V_3 = \phi_0 - \phi_3

we get

L \ddot Q + R \dot Q + C^{-1} Q = V

And this is great!

Why? Because this equation is famous! If you’re a mathematician, you know it as the most general second-order linear ordinary differential equation with constant coefficients. But if you’re a physicist, you know it as the damped driven oscillator.

The analogy between electronics and mechanics

Here’s an example of a damped driven oscillator:

We’ve got an object hanging from a spring with some friction, and an external force pulling it down. Here the external force is gravity, so it’s constant in time, but we can imagine fancier situations where it’s not. So in a general damped driven oscillator:

• the object has mass m (and the spring is massless),

• the spring constant is k (this says how strong the spring force is),

• the damping coefficient is r (this says how much friction there is),

• the external force is F (in general a function of time).

Then Newton’s law says

m \ddot{q} + r \dot{q} + k q = F

And apart from the use of different letters, this is exactly like the equation for our circuit! Remember, that was

L \ddot Q + R \dot Q + C^{-1} Q = V

So, we get a wonderful analogy relating electronics and mechanics! It goes like this:

Electronics Mechanics
charge: Q position: q
current: I = \dot{Q} velocity: v = \dot{q}
voltage: V force: F
inductance: L mass: m
resistance: R damping coefficient: r  
inverse capacitance: 1/C   spring constant: k

If you understand mechanics, you can use this to get intuition about electronics… or vice versa. I’m more comfortable with mechanics, so when I see this circuit:

I imagine a current of electrons whizzing along, ‘forced’ by the voltage across the circuit, getting slowed by the ‘friction’ of the resistor, wanting to continue their motion thanks to the inertia or ‘mass’ of the inductor, and getting stuck on the plate of the capacitor, where their mutual repulsion pushes back against the flow of current—just like a spring fights back when you pull on it! This lets me know how the circuit will behave: I can use my mechanical intuition.

The only mildly annoying thing is that the inverse of the capacitance C is like the spring constant k. But this makes perfect sense. A capacitor is like a spring: you ‘pull’ on it with voltage and it ‘stretches’ by building up electric charge on its plate. If its capacitance is high, it’s like a easily stretchable spring. But this means the corresponding spring constant is low.

Besides letting us transfer intuition and techniques, the other great thing about analogies is that they suggest ways of extending themselves. For example, we’ve seen that current is the time derivative of charge. But if we hadn’t, we could still have guessed it, because current is like velocity, which is the time derivative of something important.

Similarly, force is analogous to voltage. But force is the time derivative of momentum! We don’t have momentum on our chart. Our chart is also missing the thing whose time derivative is voltage. This thing is called flux linkage, and sometimes denotes \lambda. So we should add this, and momentum, to our chart:

Electronics Mechanics
charge: Q position: q
current: I = \dot{Q} velocity: v = \dot{q}
flux linkage: \lambda momentum: p
voltage: V = \dot{\lambda} force: F = \dot{p}
inductance: L mass: m
resistance: R damping coefficient: r  
inverse capacitance: 1/C   spring constant: k

Fourier transforms

But before I get carried away talking about analogies, let’s try to solve the equation for our circuit:

L \ddot Q + R \dot Q + C^{-1} Q = V

This instantly tells us the voltage V as a function of time if we know the charge Q as a function of time. So, ‘solving’ it means figuring out Q if we know V. You may not care about Q—it’s the charge of the electrons stuck on the capacitor—but you should certainly care about the current I = \dot{Q}, and figuring out Q will get you that.

Besides, we’ll learn something good from solving this equation.

We could solve it using either the Laplace transform or the Fourier transform. They’re very similar. For some reason electrical engineers prefer the Laplace transform—does anyone know why? But I think the Fourier transform is conceptually preferable, slightly, so I’ll use that.

The idea is to write any function of time as a linear combination of oscillating functions \exp(i\omega t) with different frequencies \omega. More precisely, we write our function f as an integral

\displaystyle{ f(t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \hat{f}(\omega) e^{i\omega t} \, d\omega }

Here the function \hat{f} is called the Fourier transform of f, and it’s given by

\displaystyle{ \hat{f}(\omega) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(t) e^{-i\omega t} \, dt }

There is a lot one could say about this, but all I need right now is that differentiating a function has the effect of multiplying its Fourier transform by i\omega. To see this, we simply take the Fourier transform of \dot{f}:

\begin{array}{ccl}  \hat{\dot{f}}(\omega) &=& \displaystyle{  \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \frac{df(t)}{dt} \, e^{-i\omega t} \, dt } \\  \\  &=& \displaystyle{ -\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(t) \frac{d}{dt} e^{-i\omega t} \, dt } \\  \\  &=& \displaystyle{ i\omega \; \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(t) e^{-i\omega t} \, dt } \\  \\  &=& i\omega \hat{f}(\omega) \end{array}

where in the second step we integrate by parts. So,

\hat{\dot{f}}(\omega) = i\omega \hat{f}(\omega)

The Fourier transform is linear, too, so we can start with our differential equation:

L \ddot Q + R \dot Q + C^{-1} Q = V

and take the Fourier transform of each term, getting

\displaystyle{ \left((i\omega)^2 L + (i\omega) R + C^{-1}\right) \hat{Q}(\omega) = \hat{V}(\omega) }

We can now solve for the charge in a completely painless way:

\displaystyle{  \hat{Q}(\omega) =  \frac{1}{((i\omega)^2 L + (i\omega) R + C^{-1})} \, \hat{V}(\omega) }

Well, we actually solved for \hat{Q} in terms of \hat{V}. But if we’re good at taking Fourier transforms, this is good enough. And it has a deep inner meaning.

To see its inner meaning, note that the Fourier transform of an oscillating function \exp(i \omega_0 t) is a delta function at the frequency \omega = \omega_0. This says that this oscillating function is purely of frequency \omega_0, like a laser beam of one pure color, or a sound of one pure pitch.

Actually there’s a little fudge factor due to how I defined the Fourier transform: if

f(t) = e^{i\omega_0 t}

then

\displaystyle{ \hat{f}(\omega) = \sqrt{2 \pi} \, \delta(\omega - \omega_0) }

But it’s no big deal. (You can define your Fourier transform so the 2\pi doesn’t show up here, but it’s bound to show up somewhere.)

Also, you may wonder how the complex numbers got into the game. What would it mean to say the voltage is \exp(i \omega t)? The answer is: don’t worry, everything in sight is linear, so we can take the real or imaginary part of any equation and get one that makes physical sense.

Anyway, what does our relation

\displaystyle{  \hat{Q}(\omega) =  \frac{1}{((i\omega)^2 L + (i\omega) R + C^{-1})} \hat{V}(\omega) }

mean? It means that if we put an oscillating voltage of frequency \omega_0 across our circuit, like this:

V(t) = e^{i \omega_0 t}

then we’ll get an oscillating charge at the same frequency, like this:

\displaystyle{  Q(t) =  \frac{1}{((i\omega_0)^2 L + (i\omega_0) R + C^{-1})}  e^{i \omega_0 t}  }

To see this, just use the fact that the Fourier transform of \exp(i \omega_0 t) is essentially a delta function at \omega_0, and juggle the equations appropriately!

But the magnitude and phase of this oscillating charge Q(t) depends on the function

\displaystyle{ \frac{1}{((i\omega_0)^2 L + (i\omega_0) R + C^{-1})}  }

For example, Q(t) will be big when \omega_0 is near a pole of this function! We can use this to study the resonant frequency of our circuit.

The same idea works for many more complicated circuits, and other things too. The function up there is an example of a transfer function: it describes the response of a linear, time-invariant system to an input of a given frequency. Here the ‘input’ is the voltage and the ‘response’ is the charge.

Impedance

Taking this idea to its logical conclusion, we can see inductors and capacitors as being resistors with a frequency-dependent, complex-valued resistance! This generalized resistance is called ‘impedance. Let’s see how it works.

Suppose we have an electrical circuit. Consider any edge e of this circuit:

• If our edge e is labelled by a resistor of resistance R:

then

V_e = R I_e

Taking Fourier transforms, we get

\hat{V}_e = R \hat{I}_e

so nothing interesting here: our resistor acts like a resistor of resistance R no matter what the frequency of the voltage and current are!

• If our edge e is labelled by an inductor of inductance L:

then

\displaystyle{ V_e = L \frac{d I_e}{d t} }

Taking Fourier transforms, we get

\hat{V}_e = (i\omega L) \hat{I}_e

This is interesting: our inductor acts like a resistor of resistance i \omega L when the frequency of the current and voltage is \omega. So, we say the ‘impedance’ of the inductor is i \omega L.

• If our edge e is labelled by a capacitor of capacitance C:

we have

\displaystyle{ I_e = C \frac{d V_e}{d t} }

Taking Fourier transforms, we get

\hat{I}_e = (i\omega C) \hat{V}_e

or

\displaystyle{ \hat{V}_e = \frac{1}{i \omega C} \hat{I_e} }

So, our capacitor acts like a resistor of resistance 1/(i \omega C) when the frequency of the current and voltage is \omega. We say the ‘impedance’ of the capacitor is 1/(i \omega L).

It doesn’t make sense to talk about the impedance of a voltage source or current source, since these circuit elements don’t give a linear relation between voltage and current. But whenever an element is linear and its properties don’t change with time, the Fourier transformed voltage will be some function of frequency times the Fourier transformed current. And in this case, we call that function the impedance of the element. The symbol for impedance is Z, so we have

\hat{V}_e(\omega) = Z(\omega) \hat{I}_e(\omega)

or

\hat{V}_e = Z \hat{I}_e

for short.

The big picture

In case you’re getting lost in the details, here are the big lessons for today:

• There’s a detailed analogy between electronics and mechanics, which we’ll later extend to many other systems.

• The study of linear time-independent elements can be reduced to the study of resistors if we generalize resistance to impedance by letting it be a complex-valued function instead of a real number.

One thing we’re doing is preparing for a general study of linear time-independent open systems. We’ll use linear algebra, but the field—the number system in our linear algebra—will consist of complex-valued functions, rather than real numbers.

Puzzle

Let’s not forget our original problem:

This is closely related to the problem we just solved. All the equations we derived still hold! But if you do the math, or use some intuition, you’ll see the voltage source ensures that the voltage we’ve been calling V is a constant. So, the current I flowing around the wire obeys the same equation we got before:

L \ddot Q + R \dot Q + C^{-1} Q = V

where \dot Q = I. The only difference is that now V is constant.

Puzzle. Solve this equation for Q(t).

There are lots of ways to do this. You could use a Fourier transform, which would give a satisfying sense of completion to this blog article. Or, you could do it some other way.


Network Theory (Part 27)

3 April, 2013

This quarter my graduate seminar at UCR will be about network theory. I have a few students starting work on this, so it seems like a good chance to think harder about the foundations of the subject. I’ve decided that bicategories of spans play a basic role, so I want to talk about those.

If you haven’t read the series up to now, don’t worry! Nothing I do for a while will rely on that earlier stuff. I want a fresh start. But just for a minute, I want to talk about the big picture: how the new stuff will relate to the old stuff.

So far this series has been talking about three closely related kinds of networks:

Markov processes
stochastic Petri nets
stochastic reaction networks

but there are many other kinds of networks, and I want to bring some more into play:

circuit diagrams
bond graphs
signal-flow graphs

These come from the world of control theory and engineering—especially electrical engineering, but also mechanical, hydraulic and other kinds of engineering.

My goal is not to tour different formalisms, but to integrate them into a single framework, so we can easily take ideas and theorems from one discipline and apply them to another.

For example, in Part 16 we saw that a special class of Markov processes can also be seen as a special class of circuit diagrams: namely, electrical circuits made of resistors. Also, in Part 17 we saw that stochastic Petri nets and stochastic reaction networks are just two different ways of talking about the same thing. This allows us to take results from chemistry—where they like stochastic reaction networks, which they call ‘chemical reaction networks’—and apply them to epidemiology, where they like stochastic Petri nets, which they call ‘compartmental models’.

As you can see, fighting through the thicket of terminology is half the battle here! The problem is that people in different applied subjects keep reinventing the same mathematics, using terminologies specific to their own interests… making it harder to see how generally applicable their work actually is. But we can’t blame them for doing this. It’s the job of mathematicians to step in, learn all this stuff, and extract the general ideas.

We can see a similar thing happening when writing was invented in ancient Mesopotamia, around 3000 BC. Different trades invented their own numbering systems! A base-60 system, the S system, was used to count most discrete objects, such as sheep or people. But for ‘rations’ such as cheese or fish, they used a base 120 system, the B system. Another system, the ŠE system, was used to measure quantities of grain. There were about a dozen such systems! Only later did they get standardized.

Circuit diagrams

But enough chit-chat; let’s get to work. I want to talk about circuit diagrams—diagrams of electrical circuits. They can get really complicated:

This is a 10-watt audio amplifier with bass boost. It looks quite intimidating. But I’ll start with a simple class of circuit diagrams, made of just a few kinds of parts:

• resistors,
• inductors,
• capacitors,
• voltage sources

and maybe some others later on. I’ll explain how you can translate any such diagram into a system of differential equations that describes how the voltages and currents along the wires change with time.

This is something you’d learn in a basic course on electrical engineering, at least back in the old days before analogue circuits had been largely replaced by digital ones. But my goal is different. I’m not mainly interested in electrical circuits per se: to me the important thing is how circuit diagrams provide a pictorial way of reasoning about differential equations… and how we can use the differential equations to describe many kinds of systems, not just electrical circuits.

So, I won’t spend much time explaining why electrical circuits do what they do—see the links for that. I’ll focus on the math of circuit diagrams, and how they apply to many different subjects, not just electrical circuits.

Let’s start with an example:

This describes a current flowing around a loop of wire with 4 elements on it: a resistor, an inductor, a capacitor, and a voltage source—for example, a battery. Each of these elements is designated by a cute symbol, and each has a real number associated to it:

• This is a resistor:

and it comes with a number R, called its resistance.

• This is an inductor:

and it comes with a number L, called its inductance.

• This is a capacitor:

and it comes with a number C, called its capacitance.

• This is a voltage source:

and it comes with a number V, called its voltage.

You may wonder why inductance got called L instead of I. Well, it’s probably because I stands for ‘current’. And then you’ll ask why current is called I instead of C. I don’t know: maybe because C stands for ‘capacitance’. If every word started with its own unique letter, we wouldn’t have these problems. But then we wouldn’t need words.

Here’s another example:

This example has two new features. First, it has places where wires meet, drawn as black dots. These dots are often called nodes, or sometimes vertices. Since ‘vertex’ starts with V and so does ‘voltage’, let’s call the dots ‘nodes’. Roughly speaking, a graph is a thing with nodes and edges, like this:

This suggests that in our circuit, the wires with elements on them should be seen as edges of a graph. Or perhaps just the wires should be seen as edges, and the elements should be seen as nodes! This is an example of a ‘design decision’ we have to make when formalizing the theory of circuit diagrams. There are also various different precise definitions of ‘graph’, and we need to try to choose the best one.

A second new feature of this example is that it has some white dots called terminals, where wires end. Mathematically these terminals are also vertices in our graph, but they play a special role: they are places where we are allowed to connect this circuit to another circuit. You’ll notice this circuit doesn’t have a voltage source. So, it’s like piece of electrical equipment without its own battery. We need to plug it in for it to do anything interesting!

This is very important. Big complicated electrical circuits are often made by hooking together smaller ones. The pieces are best thought of as ‘open systems’: that is, physical systems that interact with the outside world. Traditionally, a lot of physics focuses on ‘closed systems’, which don’t interact with the outside the world—the part of the world we aren’t modeling. But network theory is all about how we can connect open systems together to form larger open systems (or closed systems). And this is one place where category shows up. As we’ll see, we can think of an open system as a ‘morphism’ going from some inputs to some outputs, and we can ‘compose’ morphisms to get new morphisms by hooking them together.

Differential equations from circuit diagrams

Let me sketch how to get a bunch of ordinary differential equations from a circuit diagram. These equations will say what the circuit does.

We start with a graph having some set N of nodes and some set E of edges. To say how much current is flowing along each edge it will be helpful to give each edge a direction, like this:

So, define a graph to consist of two functions

s,t : E \to N

Then each edge e will have some vertex s(e) as its source, or starting-point, and some vertex t(e) as its target, or endpoint:

(This kind of graph is often called a directed multigraph or quiver, to distinguish it from other kinds, but I’ll just say ‘graph’.)

Next, each edge is labelled by one of four elements: resistor, capacitor, inductor or voltage source. It’s also labelled by a real number, which we call the resistance, capacitance, inductance or voltage of that element. We will make this part prettier later on, so we can easily introduce more kinds of elements without any trouble.

Finally, we specify a subset T \subseteq N and call these nodes terminals.

Our goal now is to write down some ordinary differential equations that say how a bunch of variables change with time. These variables come in two kinds:

• Each edge e has a current running along it, which is a function of time denoted I_e. So, for each e \in E we have a function

I_e : \mathbb{R} \to \mathbb{R}

• Each edge e also has a voltage across it, which is a function of time denoted V_e. So, for each e \in E we have a function

V_e : \mathbb{R} \to \mathbb{R}

We now write down a bunch of equations obeyed by these currents and voltages. First there are some equations called Kirchhoff’s laws:

Kirchhoff’s current law says that for each node that is not a terminal, the total current flowing into that node equals the total current flowing out. In other words:

\displaystyle{ \sum_{e: t(e) = n} I_e = \sum_{e: s(e) = n} I_e }

for each node n \in N - T. We don’t impose Kirchhoff’s current law at terminals, because we want to allow current to flow in or out there!

Kirchhoff’s voltage law says that we can choose for each node a potential \phi_n, which is a function of time:

\phi_n : \mathbb{R} \to \mathbb{R}

such that

V_e = \phi_{s(e)} - \phi_{t(e)}

for each e \in E. In other words, the voltage across each edge is the difference of potentials at the two ends of this edge. This is a slightly nonstandard way to state Kirchhoff’s voltage law, but it’s equivalent to the usual one.

In addition to Kirchhoff’s laws, there’s an equation for each edge, relating the current and voltage on that edge. The details of this equation depends on the element labelling that edge, so we consider the four cases in turn:

• If our edge e is labelled by a resistor of resistance R:

we write the equation

V_e = R I_e

This is called Ohm’s law.

• If our edge e is labelled by an inductor of inductance L:

we write the equation

\displaystyle{ V_e = L \frac{d I_e}{d t} }

I don’t know a name for this equation, but you can read about it here.

• If our edge e is labelled by a capacitor of capacitance C:

we write the equation

\displaystyle{ I_e = C \frac{d V_e}{d t} }

I don’t know a name for this equation, but you can read about it here.

• If our edge e is labelled by a voltage source of voltage V:

we write the equation

V_e = V

This explains the term ‘voltage source’.

Puzzles

Next time we’ll look at some examples and see how we can start polishing up this formalism into something more pretty. But you can get to work now:

Puzzle 1. Starting from the rules above, write down and simplify the equations for this circuit:

Puzzle 2. Do the same for this circuit:

Puzzle 3. If we added a fifth kind of element, our rules for getting equations from circuit diagrams would have more symmetry between voltages and currents. What is this extra element?


Network Theory for Economists

15 January, 2013

Tomorrow I’m giving a talk in the econometrics seminar at U.C. Riverside. I was invited to speak on my work on network theory, so I don’t feel too bad about the fact that I’ll be saying only a little about economics and practically nothing about econometrics. Still, I’ve tried to slant the talk in a way that emphasizes possible applications to economics and game theory. Here are the slides:

Network Theory.

For long-time readers here the fun comes near the end. I explain how reaction networks can be used to describe evolutionary games. I point out that in certain classes of evolutionary games, evolution tends to increase ‘fitness’, and/or lead the players to a ‘Nash equilibrium’. For precise theorems you’ll have to click the links in my talk and read the references!

I conclude with an example: a game with three strategies and 7 Nash equilibria. Here evolution makes the proportion of these three strategies follow these flow lines, at least in the limit of large numbers of players:

This picture is from William Sandholm’s nice expository paper:

• William H. Sandholm, Evolutionary game theory, 2007.

I mentioned it before in Information Geometry (Part 12), en route to showing a proof that some quantity always decreases in a class of evolutionary games. Sometime I want to tell the whole story linking:

reaction networks
evolutionary games
the 2nd law of thermodynamics

and

Fisher’s fundamental theorem of natural selection.

But not today! Think of these talk slides as a little appetizer.


Network Theory (Part 26)

15 January, 2013

Last time I described the reachability problem for reaction networks, which has the nice feature of connecting chemistry, category theory, and computation.

Near the end I raised a question. Luca Cardelli, who works on biology and computation at Microsoft, answered it.


His answer is interesting enough that I thought you should all read it. I imagined an arbitrary chemical reaction network and said:

We could try to use these reactions to build a ‘chemical computer’. But how powerful can such a computer be? I don’t know the answer.

Cardelli replied:

The answer to that question is known. Stochastic Petri Nets (and equivalently chemical reaction networks) are not Turing-powerful in the strict sense, essentially because of all the decidable properties of Petri Nets. However, they can approximate a Turing machine up to any fixed error bound, so they are ‘almost’ Turing-complete, or ‘Turing-complete-up-to-epsilon’. The error bound can be fixed independently of the length of the computation (which, being a Turing machine, is not going to be known ahead of time); in practice, that means progressively slowing down the computation to make it more accurate over time and to remain below the global error bound.

Note also that polymerization is a chemical operation that goes beyond the power of Stochastic Petri Nets and plain chemical reaction networks: if you can form unbounded polymers (like, e.g., DNA), you can use them as registers or tapes and obtain full Turing completeness, chemically (or, you might say ‘biochemically’ because that’s where the most interesting polymers are found). An unbounded polymer corresponds to an infinite set of reactions (a small set of reactions for each polymer length), i.e. to an ‘actually infinite program’ in the language of simple reaction networks. Infinite programs of course are no good for any notion of Turing computation, so you need to use a more powerful language for describing polymerization, that is, a language that has the equivalent of molecular binding/unbinding as a primitive. That kind of language can be found in Process Algebra.

So, in addition to the

Chemical-Reaction-Networks/
Stochastic-Petri-Nets/
Turing-Completeness-Up-To-Epsilon

connection, there is another connection between

‘Biochemical’-Reaction-Networks/
Stochastic-Process-Algebra/
Full-Turing-Completeness.

And he provided a list of references. The correct answer to my question appeared first here:

• D. Soloveichik, M. Cook, E. Winfree and J. Bruck, Computation with finite stochastic chemical reaction networks, Natural Computing 7 (2008), 615–633.

contradicting an earlier claim here:

• M.O. Magnasco, Chemical kinetics is Turing universal, Phys. Rev. Lett. 78 (1997), 1190–1193.

Further work can be found here:

• Matthew Cook, David Soloveichik, Erik Winfree and Jehoshua Bruck Programmability of chemical reaction networks, in Algorithmic Bioprocesses, ed. Luca Cardelli, Springer, Berlin, 543–584, 2009.

and some of Cardelli’s own work is here:

• Gianluigi Zavattaro and Luca Cardelli, Termination problems in chemical kinetics, in CONCUR 2008 – Concurrency Theory, eds. Gianluigi Zavattaro and Luca Cardelli, Lecture Notes in Computer Science 5201, Springer, Berlin, 2008, pp. 477–491.

• Luca Cardelli and Gianluigi Zavattaro, Turing universality of the biochemical ground form, Math. Struct. Comp. Sci. 20 (2010), 45–73.

You can find lots more interesting work on Cardelli’s homepage.


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