Network Theory (Part 32)

20 October, 2014

Okay, today we will look at the ‘black box functor’ for circuits made of resistors. Very roughly, this takes a circuit made of resistors with some inputs and outputs:

and puts a ‘black box’ around it:

forgetting the internal details of the circuit and remembering only how the it behaves as viewed from outside. As viewed from outside, all the circuit does is define a relation between the potentials and currents at the inputs and outputs. We call this relation the circuit’s behavior. Lots of different choices of the resistances $R_1, \dots, R_6$ would give the same behavior. In fact, we could even replace the whole fancy circuit by a single edge with a single resistor on it, and get a circuit with the same behavior!

The idea is that when we use a circuit to do something, all we care about is its behavior: what it does as viewed from outside, not what it’s made of.

Furthermore, we’d like the behavior of a system made of parts to depend in a simple way on the external behaviors of its parts. We don’t want to have to ‘peek inside’ the parts to figure out what the whole will do! Of course, in some situations we do need to peek inside the parts to see what the whole will do. But in this particular case we don’t—at least in the idealization we are considering. And this fact is described mathematically by saying that black boxing is a functor.

So, how do circuits made of resistors behave? To answer this we first need to remember what they are!

Review

Remember that for us, a circuit made of resistors is a mathematical structure like this:

It’s a cospan where:

$\Gamma$ is a graph labelled by resistances. So, it consists of a finite set $N$ of nodes, a finite set $E$ of edges, two functions

$s, t : E \to N$

sending each edge to its source and target nodes, and a function

$r : E \to (0,\infty)$

that labels each edge with its resistance.

$i: I \to \Gamma$ is a map of graphs labelled by resistances, where $I$ has no edges. A labelled graph with no edges has nothing but nodes! So, the map $i$ is just a trick for specifying a finite set of nodes called inputs and mapping them to $N.$ Thus $i$ picks out some nodes of $\Gamma$ and declares them to be inputs. (However, $i$ may not be one-to-one! We’ll take advantage of that subtlety later.)

$o: O \to \Gamma$ is another map of graphs labelled by resistances, where $O$ again has no edges, and we call its nodes outputs.

The principle of minimum power

So what does a circuit made of resistors do? This is described by the principle of minimum power.

Recall from Part 27 that when we put it to work, our circuit has a current $I_e$ flowing along each edge $e \in E.$ This is described by a function

$I: E \to \mathbb{R}$

It also has a voltage across each edge. The word ‘across’ is standard here, but don’t worry about it too much; what matters is that we have another function

$V: E \to \mathbb{R}$

describing the voltage $V_e$ across each edge $e.$

Resistors heat up when current flows through them, so they eat up electrical power and turn this power into heat. How much? The power is given by

$\displaystyle{ P = \sum_{e \in E} I_e V_e }$

So far, so good. But what does it mean to minimize power?

To understand this, we need to manipulate the formula for power using the laws of electrical circuits described in Part 27. First, Ohm’s law says that for linear resistors, the current is proportional to the voltage. More precisely, for each edge $e \in E,$

$\displaystyle{ I_e = \frac{V_e}{r_e} }$

where $r_e$ is the resistance of that edge. So, the bigger the resistance, the less current flows: that makes sense. Using Ohm’s law we get

$\displaystyle{ P = \sum_{e \in E} \frac{V_e^2}{r_e} }$

Now we see that power is always nonnegative! Now it makes more sense to minimize it. Of course we could minimize it simply by setting all the voltages equal to zero. That would work, but that would be boring: it gives a circuit with no current flowing through it. The fun starts when we minimize power subject to some constraints.

For this we need to remember another law of electrical circuits: a spinoff of Kirchhoff’s voltage law. This says that we can find a function called the potential

$\phi: N \to \mathbb{R}$

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.

Using this, we can rewrite the power as

$\displaystyle{ P = \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)})^2 }$

Now we’re really ready to minimize power! Our circuit made of resistors has certain nodes called terminals:

$T \subseteq N$

These are the nodes that are either inputs or outputs. More precisely, they’re the nodes in the image of

$i: I \to \Gamma$

or

$o: O \to \Gamma$

The principle of minimum power says that:

If we fix the potential $\phi$ on all terminals, the potential at other nodes will minimize the power

$\displaystyle{ P(\phi) = \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)})^2 }$

subject to this constraint.

This should remind you of all the other minimum or maximum principles you know, like the principle of least action, or the way a system in thermodynamic equilibrium maximizes its entropy. All these principles—or at least, most of them—are connected. I could talk about this endlessly. But not now!

Now let’s just use the principle of minimum power. Let’s see what it tells us about the behavior of an electrical circuit.

Let’s imagine changing the potential $\phi$ by adding some multiple of a function

$\psi: N \to \mathbb{R}$

If this other function vanishes at the terminals:

$\forall n \in T \; \; \psi(n) = 0$

then $\phi + x \psi$ doesn’t change at the terminals as we change the number $x.$

Now suppose $\phi$ obeys the principle of minimum power. In other words, supposes it minimizes power subject to the constraint of taking the values it does at the terminals. Then we must have

$\displaystyle{ \frac{d}{d x} P(\phi + x \psi)\Big|_{x = 0} }$

whenever

$\forall n \in T \; \; \psi(n) = 0$

This is just the first derivative test for a minimum. But the converse is true, too! The reason is that our power function is a sum of nonnegative quadratic terms. Its graph will look like a paraboloid. So, the power has no points where its derivative vanishes except minima, even when we constrain $\phi$ by making it lie on a linear subspace.

We can go ahead and start working out the derivative:

$\displaystyle{ \frac{d}{d x} P(\phi + x \psi)! = ! \frac{d}{d x} \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)} + x(\psi_{s(e)} -\psi_{t(e)}))^2 }$

To work out the derivative of these quadratic terms at $x = 0,$ we only need to keep the part that’s proportional to $x.$ The rest gives zero. So:

$\begin{array}{ccl} \displaystyle{ \frac{d}{d t} P(\phi + x \psi)\Big|_{x = 0} } &=& \displaystyle{ \frac{d}{d x} \sum_{e \in E} \frac{x}{r_e} (\phi_{s(e)} - \phi_{t(e)}) (\psi_{s(e)} - \psi_{t(e)}) \Big|_{x = 0} } \\ \\ &=& \displaystyle{ \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) (\psi_{s(e)} - \psi_{t(e)}) } \end{array}$

The principle of minimum power says this is zero whenever $\psi : N \to \mathbb{R}$ is a function that vanishes at terminals. By linearity, it’s enough to consider functions $\psi$ that are zero at every node except one node $n$ that is not a terminal. By linearity we can also assume $\psi(n) = 1.$

Given this, the only nonzero terms in the sum

$\displaystyle{ \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) (\psi_{s(e)} - \psi_{t(e)}) }$

will be those involving edges whose source or target is $n.$ We get

$\begin{array}{ccc} \displaystyle{ \frac{d}{d x} P(\phi + x \psi)\Big|_{x = 0} } &=& \displaystyle{ \sum_{e: \; s(e) = n} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)})} \\ \\ && -\displaystyle{ \sum_{e: \; t(e) = n} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) } \end{array}$

So, the principle of minimum power says precisely

$\displaystyle{ \sum_{e: \; s(e) = n} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) = \sum_{e: \; t(e) = n} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }$

for all nodes $n$ that aren’t terminals.

What does this mean? You could just say it’s a set of linear equations that must be obeyed by the potential $\phi.$ So, the principle of minimum power says that fixing the potential at terminals, the potential at other nodes must be chosen in a way that obeys a set of linear equations.

But what do these equations mean? They have a nice meaning. Remember, Kirchhoff’s voltage law says

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

and Ohm’s law says

$\displaystyle{ I_e = \frac{V_e}{r_e} }$

Putting these together,

$\displaystyle{ I_e = \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }$

so the principle of minimum power merely says that

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

for any node $n$ that is not a terminal.

This is Kirchhoff’s current law: for any node except a terminal, the total current flowing into that node must equal the total current flowing out! That makes a lot of sense. We allow current to flow in or out of our circuit at terminals, but ‘inside’ the circuit charge is conserved, so if current flows into some other node, an equal amount has to flow out.

In short: the principle of minimum power implies Kirchoff’s current law! Conversely, we can run the whole argument backward and derive the principle of minimum power from Kirchhoff’s current law. (In both the forwards and backwards versions of this argument, we use Kirchhoff’s voltage law and Ohm’s law.)

When the node $n$ is a terminal, the quantity

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

need not be zero. But it has an important meaning: it’s the amount of current flowing into that terminal!

We’ll call this $I_n,$ the current at the terminal $n \in T.$ This is something we can measure even when our circuit has a black box around it:

So is the potential $\phi_n$ at the terminal $n.$ It’s these currents and potentials at terminals that matter when we try to describe the behavior of a circuit while ignoring its inner workings.

Black boxing

Now let me quickly sketch how black boxing becomes a functor.

A circuit made of resistors gives a linear relation between the potentials and currents at terminals. A relation is something that can hold or fail to hold. A ‘linear’ relation is one defined using linear equations.

A bit more precisely, suppose we choose potentials and currents at the terminals:

$\psi : T \to \mathbb{R}$

$J : T \to \mathbb{R}$

Then we seek potentials and currents at all the nodes and edges of our circuit:

$\phi: N \to \mathbb{R}$

$I : E \to \mathbb{R}$

that are compatible with our choice of $\psi$ and $J.$ Here compatible means that

$\psi_n = \phi_n$

and

$J_n = \displaystyle{ \sum_{e: \; s(e) = n} I_e \; - \; \sum_{e: \; t(e) = n} I_e }$

whenever $n \in T,$ but also

$\displaystyle{ I_e = \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }$

for every $e \in E,$ and

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

whenever $n \in N - T.$ (The last two equations combine Kirchoff’s laws and Ohm’s law.)

There either exist $I$ and $\phi$ making all these equations true, in which case we say our potentials and currents at the terminals obey the relation… or they don’t exist, in which case we say the potentials and currents at the terminals don’t obey the relation.

The relation is clearly linear, since it’s defined by a bunch of linear equations. With a little work, we can make it into a linear relation between potentials and currents in

$\mathbb{R}^I \oplus \mathbb{R}^I$

and potentials and currents in

$\mathbb{R}^O \oplus \mathbb{R}^O$

Remember, $I$ is our set of inputs and $O$ is our set of outputs.

In fact, this process of getting a linear relation from a circuit made of resistors defines a functor:

$\blacksquare : \mathrm{ResCirc} \to \mathrm{LinRel}$

Here $\mathrm{ResCirc}$ is the category where morphisms are circuits made of resistors, while $\mathrm{LinRel}$ is the category where morphisms are linear relations.

More precisely, here is the category $\mathrm{ResCirc}:$

• an object of $\mathrm{ResCirc}$ is a finite set;

• a morphism from $I$ to $O$ is an isomorphism class of circuits made of resistors:

having $I$ as its set of inputs and $O$ as its set of outputs;

• we compose morphisms in $\mathrm{ResCirc}$ by composing isomorphism classes of cospans.

(Remember, circuits made of resistors are cospans. This lets us talk about isomorphisms between them. If you forget the how isomorphism between cospans work, you can review it in Part 31.)

And here is the category $\mathrm{LinRel}:$

• an object of $\mathrm{LinRel}$ is a finite-dimensional real vector space;

• a morphism from $U$ to $V$ is a linear relation $R \subseteq U \times V,$ meaning a linear subspace of the vector space $U \times V;$

• we compose a linear relation $R \subseteq U \times V$ and a linear relation $S \subseteq V \times W$ in the usual way we compose relations, getting:

$SR = \{(u,w) \in U \times W : \; \exists v \in V \; (u,v) \in R \mathrm{\; and \;} (v,w) \in S \}$

Next steps

So far I’ve set up most of the necessary background but not precisely defined the black boxing functor

$\blacksquare : \mathrm{ResCirc} \to \mathrm{LinRel}$

There are some nuances I’ve glossed over, like the difference between inputs and outputs as elements of $I$ and $O$ and their images in $N.$ If you want to see the precise definition and the proof that it’s a functor, read our paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

The proof is fairly long: there may be a much quicker one, but at least this one has the virtue of introducing a lot of nice ideas that will be useful elsewhere.

Next time I’ll define the black box functor more carefully.

Network Theory Seminar (Part 2)

16 October, 2014

This time I explain more about how ‘cospans’ represent gadgets with two ends, an input end and an output end:

I describe how to glue such gadgets together by composing cospans. We compose cospans using a category-theoretic construction called a ‘pushout’, so I also explain pushouts. At the end, I explain how this gives us a category where the morphisms are electrical circuits made of resistors, and sketch what we’ll do next: study the behavior of these circuits.

These lecture notes provide extra details:

El Niño Project (Part 8)

14 October, 2014

So far we’ve rather exhaustively studied a paper by Ludescher et al which uses climate networks for El Niño prediction. This time I’d like to compare another paper:

• Y. Berezin, Avi Gozolchiani, O. Guez and Shlomo Havlin, Stability of climate networks with time, Scientific Reports 2 (2012).

Some of the authors are the same, and the way they define climate networks is very similar. But their goal here is different: they want to see see how stable climate networks are over time. This is important, since the other paper wants to predict El Niños by changes in climate networks.

They divide the world into 9 zones:

For each zone they construct several climate networks. Each one is an array of numbers $W_{l r}^y$, one for each year $y$ and each pair of grid points $l, r$ in that zone. They call $W_{l r}^y$ a link strength: it’s a measure of how how correlated the weather is at those two grid points during that year.

I’ll say more later about how they compute these link strengths. In Part 3 we explained one method for doing it. This paper uses a similar but subtly different method.

The paper’s first big claim is that $W_{l r}^y$ doesn’t change much from year to year, “in complete contrast” to the pattern of local daily air temperature and pressure fluctuations. In simple terms: the strength of the correlation between weather at two different points tends to be quite stable.

Moreover, the definition of link strength involves an adjustable time delay, $\tau$. We can measure the correlation between the weather at point $l$ at any given time and point $r$ at a time $\tau$ days later. The link strength is computed by taking a maximum over time delays $\tau$. Naively speaking, the value of $\tau$ that gives the maximum correlation is “how long it typically takes for weather at point $l$ to affect weather at point $r$”. Or the other way around, if $\tau$ is negative.

This is a naive way of explaining the idea, because I’m mixing up correlation with causation. But you get the idea, I hope.

Their second big claim is that when the link strength between two points $l$ and $r$ is big, the value of $\tau$ that gives the maximum correlation doesn’t change much from year to year. In simple terms: if the weather at two locations is strongly correlated, the amount of time it takes for weather at one point to reach the other point doesn’t change very much.

The data

How do Berezin et al define their climate network?

They use data obtained from here:

This is not exactly the same data set that Ludescher et al use, namely:

“Reanalysis 2″ is a newer attempt to reanalyze and fix up the same pile of data. That’s a very interesting issue, but never mind that now!

Berezin et al use data for:

• the geopotential height for six different pressures

and

• the air temperature at those different heights

The geopotential height for some pressure says roughly how high you have to go for air to have that pressure. Click the link if you want a more precise definition! Here’s the geopotential height field for the pressure of 500 millibars on some particular day of some particular year:

The height is in meters.

Berezin et al use daily values for this data for:

• locations world-wide on a grid with a resolution of 5° × 5°,

during:

• the years from 1948 to 2006.

They divide the globe into 9 zones, and separately study each zone:

So, they’ve got twelve different functions of space and time, where space is a rectangle discretized using a 5° × 5° grid, and time is discretized in days. From each such function they build a ‘climate network’.

How do they do it?

The climate networks

Berezin’s method of defining a climate network is similar to Ludescher et al‘s, but different. Compare Part 3 if you want to think about this.

Let $\tilde{S}^y_l(t)$ be any one of their functions, evaluated at the grid point $l$ on day $t$ of year $y$.

Let $S_l^y(t)$ be $\tilde{S}^y_l(t)$ minus its climatological average. For example, if $t$ is June 1st and $y$ is 1970, we average the temperature at location $l$ over all June 1sts from 1948 to 2006, and subtract that from $\tilde{S}^y_l(t)$ to get $S^y_l(t)$. In other words:

$\displaystyle{ \tilde{S}^y_l(t) = S^y_l(t) - \frac{1}{N} \sum_y S^y_l(t) }$

where $N$ is the number of years considered.

For any function of time $f$, let $\langle f^y(t) \rangle$ be the average of the function over all days in year $y$. This is different than the ‘running average’ used by Ludescher et al, and I can’t even be 100% sure that Berezin mean what I just said: they use the notation $\langle f^y(t) \rangle$.

Let $l$ and $r$ be two grid points, and $\tau$ any number of days in the interval $[-\tau_{\mathrm{max}}, \tau_{\mathrm{max}}]$. Define the cross-covariance function at time $t$ by:

$\Big(f_l(t) - \langle f_l(t) \rangle\Big) \; \Big( f_r(t + \tau) - \langle f_r(t + \tau) \rangle \Big)$

I believe Berezin mean to consider this quantity, because they mention two grid points $l$ and $r$. Their notation omits the subscripts $l$ and $r$ so it is impossible to be completely sure what they mean! But what I wrote is the reasonable quantity to consider here, so I’ll assume this is what they meant.

They normalize this quantity and take its absolute value, forming:

$\displaystyle{ X_{l r}^y(\tau) = \frac{\Big|\Big(f_l(t) - \langle f_l(t) \rangle\Big) \; \Big( f_r(t + \tau) - \langle f_r(t + \tau) \rangle \Big)\Big|} {\sqrt{\Big\langle \Big(f_l(t) - \langle f_l(t)\rangle \Big)^2 \Big\rangle } \; \sqrt{\Big\langle \Big(f_r(t+\tau) - \langle f_r(t+\tau)\rangle\Big)^2 \Big\rangle } } }$

They then take the maximum value of $X_{l r}^y(\tau)$ over delays $\tau \in [-\tau_{\mathrm{max}}, \tau_{\mathrm{max}}]$, subtract its mean over delays in this range, and divide by the standard deviation. They write something like this:

$\displaystyle{ W_{l r}^y = \frac{\mathrm{MAX}\Big( X_{l r}^y - \langle X_{l r}^y\rangle \Big) }{\mathrm{STD} X_{l r}^y} }$

and say that the maximum, mean and standard deviation are taken over the (not written) variable $\tau \in [-\tau_{\mathrm{max}}, \tau_{\mathrm{max}}]$.

Each number $W_{l r}^y$ is called a link strength. For each year, the matrix of numbers $W_{l r}^y$ where $l$ and $r$ range over all grid points in our zone is called a climate network.

We can think of a climate network as a weighted complete graph with the grid points $l$ as nodes. Remember, an undirected graph is one without arrows on the edges. A complete graph is an undirected graph with one edge between any pair of nodes:

A weighted graph is an undirected graph where each edge is labelled by a number called its weight. But right now we’re also calling the weight the ‘link strength’.

A lot of what’s usually called ‘network theory’ is the study of weighted graphs. You can learn about it here:

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford U. Press, Oxford, 2011.

Suffice it to say that given a weighted graph, there are lot of quantities you can compute from it, which are believed to tell us interesting things!

The conclusions

I will not delve into the real meat of the paper, namely what they actually do with their climate networks! The paper is free online, so you can read this yourself.

I will just quote their conclusions and show you a couple of graphs.

The conclusions touch on an issue that’s important for the network-based approach to El Niño prediction. If climate networks are ‘stable’, not changing much in time, why would we use them to predict a time-dependent phenomenon like the El Niño Southern Oscillation?

We have established the stability of the network of connections between the dynamics of climate variables (e.g. temperatures and geopotential heights) in different geographical regions. This stability stands in fierce contrast to the observed instability of the original climatological field pattern. Thus the coupling between different regions is, to a large extent, constant and predictable. The links in the climate network seem to encapsulate information that is missed in analysis of the original field.

The strength of the physical connection, $W_{l r}$, that each link in this network represents, changes only between 5% to 30% over time. A clear boundary between links that represent real physical dependence and links that emerge due to noise is shown to exist. The distinction is based on both the high link average strength $\overline{W_{l r}}$ and on the low variability of time delays $\mathrm{STD}(T_{l r})$.

Recent studies indicate that the strength of the links in the climate network changes during the El Niño Southern Oscillation and the North Atlantic Oscillation cycles. These changes are within the standard deviation of the strength of the links found here. Indeed in Fig. 3 it is clearly seen that the coefficient of variation of links in the El Niño basin (zone 9) is larger than other regions such as zone 1. Note that even in the El Niño basin the coefficient of variation is relatively small (less than 30%).

Beside the stability of single links, also the hierarchy of the link strengths in the climate network is preserved to a large extent. We have shown that this hierarchy is partially due to the two dimensional space in which the network is embedded, and partially due to pure physical coupling processes. Moreover the contribution of each of these effects, and the level of noise was explicitly estimated. The spatial effect is typically around 50% of the observed stability, and the noise reduces the stability value by typically 5%–10%.

The network structure was further shown to be consistent across different altitudes, and a monotonic relation between the altitude distance and the correspondence between the network structures is shown to exist. This yields another indication that the observed network structure represents effects of physical coupling.

The stability of the network and the contributions of different effects were summarized in specific relation to different geographical areas, and a clear distinction between equatorial and off–equatorial areas was observed. Generally, the network structure of equatorial regions is less stable and more fluctuative.

The stability and consistence of the network structure during time and across different altitudes stands in contrast to the known unstable variability of the daily anomalies of climate variables. This contrast indicates an analogy between the behavior of nodes in the climate network and the behavior of coupled chaotic oscillators. While the fluctuations of each coupled oscillators are highly erratic and unpredictable, the interactions between the oscillators is stable and can be predicted. The possible outreach of such an analogy lies in the search for known behavior patterns of coupled chaotic oscillators in the climate system. For example, existence of phase slips in coupled chaotic oscillators is one of the fingerprints for their cooperated behavior, which is evident in each of the individual oscillators. Some abrupt changes in climate variables, for example, might be related to phase slips, and can be understood better in this context.

On the basis of our measured coefficient of variation of single links (around 15%), and the significant overall network stability of 20–40%, one may speculatively assess the extent of climate change. However, for this assessment our current available data is too short and does not include enough time from periods before the temperature trends. An assessment of the relation between the network stability and climate change might be possible mainly through launching of global climate model “experiments” realizing other climate conditions, which we indeed intend to perform.

A further future outreach of our work can be a mapping between network features (such as network motifs) and known physical processes. Such a mapping was previously shown to exist between an autonomous cluster in the climate network and El Niño. Further structures without such a climate interpretation might point towards physical coupling processes which were not observed earlier.

(I have expanded some acronyms and deleted some reference numbers.)

Finally, here two nice graphs showing the average link strength as a function of distance. The first is based on four climate networks for Zone 1, the southern half of South America:

The second is based on four climate networks for Zone 9, a big patch of the Pacific north of the Equator which roughly corresponds to the ‘El Niño basin':

As we expect, temperatures and geopotential heights get less correlated at points further away. But the rate at which the correlation drops off conveys interesting information! Graham Jones has made some interesting charts of this for the rectangle of the Pacific that Ludescher et al use for El Niño prediction, and I’ll show you those next time.

The series so far

El Niño project (part 1): basic introduction to El Niño and our project here.

El Niño project (part 2): introduction to the physics of El Niño.

El Niño project (part 3): summary of the work of Ludescher et al.

El Niño project (part 4): how Graham Jones replicated the work by Ludescher et al, using software written in R.

El Niño project (part 5): how to download R and use it to get files of climate data.

El Niño project (part 6): Steve Wenner’s statistical analysis of the work of Ludescher et al.

El Niño project (part 7): the definition of El Niño.

El Niño project (part 8): Berezin et al on the stability of climate networks.

Network Theory (Part 31)

13 October, 2014

Last time we came up with a category of labelled graphs and described circuits as ‘cospans’ in this category.

Cospans may sound scary, but they’re not. A cospan is just a diagram consisting of an object with two morphisms going into it:

We can talk about cospans in any category. A cospan is an abstract way of thinking about a ‘chunk of stuff’ $\Gamma$ with two ‘ends’ $I$ and $O.$ It could be any sort of stuff: a set, a graph, an electrical circuit, a network of any kind, or even a piece of matter (in some mathematical theory of matter).

We call the object $\Gamma$ the apex of the cospan and call the morphisms $i: I \to \Gamma, o : O \to \Gamma$ the legs of the cospan. We sometimes call the objects $I$ and $O$ the feet of the cospan. We call $I$ the input and $O$ the output. We say the cospan goes from $I$ to $O,$ though the direction is just a convention: we can flip a cospan and get a cospan going the other way!

If you’re wondering about the name ‘cospan’, it’s because a span is a diagram like this:

Since a ‘span’ is another name for a bridge, and this looks like a bridge from $I$ to $O,$ category theorists called it a span! And category theorists use the prefix ‘co-‘ when they turn all the arrows around. Spans came first historically, and we will use those too at times. But now let’s think about how to compose cospans.

Composing cospans is supposed to be like gluing together chunks of stuff by attaching the output of the first to the input of the second. So, we say two cospans are composable if the output of the first equals the input of the second, like this:

We then compose them by forming a new cospan going all the way from $X$ to $Z$:

The new object $\Gamma +_Y \Gamma'$ and the new morphisms $i'', o''$ are built using a process called a ‘pushout’ which I’ll explain in a minute. The result is cospan from $X$ to $Z,$ called the composite of the cospans we started with. Here it is:

So how does a pushout work? It’s a general construction that you can define in any category, though it only exists if the category is somewhat nice. (Ours always will be.) You start with a diagram like this:

and you want to get a commuting diamond like this:

which is in some sense ‘the best’ given the diagram we started with. For example, suppose we’re in the category of sets and $Y$ is a set included in both $\Gamma$ and $\Gamma'.$ Then we’d like $A$ to be the union of $\Gamma$ and $\Gamma.$ There are other choices of $A$ that would give a commuting diamond, but the union is the best. Something similar is happening when we compose circuits, but instead of the category of sets we’re using the category of labelled graphs we discussed last time.

How do we make precise the idea that $A$ is ‘the best’? We consider any other potential solution to this problem, that is, some other commuting diamond:

Then $A$ is ‘the best’ if there exists a unique morphism $q$ from $A$ to the ‘competitor’ $Q$ making the whole combined diagram commute:

This property is called a universal property: instead of saying that $A$ is the ‘best’, grownups say it is universal.

When $A$ has this universal property we call it the pushout of the original diagram, and we may write it as $\Gamma +_Y \Gamma'.$ Actually we should call the whole diagram

the pushout, or a pushout square, because the morphisms $i'', o''$ matter too. The universal property is not really a property just of $A,$ but of the whole pushout square. But often we’ll be sloppy and call just the object $A$ the pushout.

Puzzle 1. Suppose we have a diagram in the category of sets

where $Y = \Gamma \cap \Gamma'$ and the maps $i, o'$ are the inclusions of this intersection in the sets $\Gamma$ and $\Gamma'.$ Prove that $A = \Gamma \cup \Gamma'$ is the pushout, or more precisely the diagram

is a pushout square, where $i'', o''$ are the inclusions of $\Gamma$ and $\Gamma$ in the union $A = \Gamma \cup \Gamma'.$

More generally, a pushout in the category of sets is a way of gluing together sets $\Gamma$ and $\Gamma'$ with some ‘overlap’ given by the maps

And this works for labelled graphs, too!

Puzzle 2. Suppose we have two circuits of resistors that are composable, like this:

and this:

These give cospans in the category $L\mathrm{Graph}$ where

$L = (0,\infty)$

(Remember from last time that $L\mathrm{Graph}$ is the category of graphs with edges labelled by elements of some set $L.$) Show that if we compose these cospans we get a cospan corresponding to this circuit:

If you’re a mathematician you might find it easier to solve this kind of problem in general, which requires pondering how pushouts work in $L\mathrm{Graph}.$ Alternatively, you might find it easier to think about this particular example: then you can just check that the answer we want has the desired property of a pushout!

If this stuff seems complicated, well, just know that category theory is a very general, powerful tool and I’m teaching you just the microscopic fragment of it that we need right now. Category theory ultimately seems very simple: I can’t really think of any math that’s simpler! It only seem complicated when it’s unfamiliar and you have a fragmentary view of it.

So where are we? We know that circuits made of resistors are a special case of cospans. We know how to compose cospans. So, we know how to compose circuits… and in the last puzzle, we saw this does just what we want.

The advantage of this rather highbrow approach is that a huge amount is known about composing cospans! In particular, suppose we have any category $C$ where pushouts exist: that is, where we can always complete any diagram like this:

to a pushout square. Then we can form a category $\mathrm{Cospan}(C)$ where:

• an object is an object of $C$

• a morphism from an object $I \in C$ to an object $O \in C$ is an equivalence classes of cospans from $I$ to $O:$

• we compose cospans in the manner just described.

Why did I say ‘equivalence class’? It’s because the pushout is not usually unique. It’s unique only up to isomorphism. So, composing cospans would be ill-defined unless we work with some kind of equivalence class of cospans.

To be precise, suppose we have two cospans from $I$ to $O$:

Then a map of cospans from one to the other is a commuting diagram like this:

We say that this is an isomorphism of cospans if $f$ is an isomorphism.

This gives our equivalence relation on cospans! It’s an old famous theorem in category theory—so famous that it’s hard to find a reference for the proof—that whenever $C$ is a category with pushouts, there’s a category $\mathrm{Cospan}(C)$ where:

• an object is an object of $C$

• a morphism from an object $I \in C$ to an object $O \in C$ is an isomorphism class of cospans from $I$ to $O.$

• we compose isomorphism classes of cospans by picking representatives, composing them and then taking the isomorphism class.

This takes some work to prove, but it’s true, so this is how we get our category of circuits!

Next time we’ll do something with this category. Namely, we’ll cook up a category of ‘behaviors’. The behavior of a circuit made of resistors just says which currents and potentials its terminals can have. If we put a circuit in a metaphorical ‘black box’ and refuse to peek inside, all we can see is its behavior.

Then we’ll cook up a functor from the category of circuits to the category of behaviors. We’ll call this the ‘black box functor’. Saying that it’s a functor mainly means that

$\blacksquare(f g) = \blacksquare(f) \blacksquare(g)$

Here $f$ and $g$ are circuits that we can compose, and $f g$ is their composite. The black square is the black box functor, so $\blacksquare(fg)$ is the behavior of the circuit $f g.$ There’s a way to compose behaviors, too, and the equation above says that the behavior of the composite circuit is the composite of their behaviors!

This is very important, because it says we can figure out what a big circuit does if we know what its pieces do. And this is one of the grand themes of network theory: understanding big complicated networks by understanding their pieces. We may not always be able to do this, in practice! But it’s something we’re always concerned with.

Network Theory Seminar (Part 1)

11 October, 2014

Check out this video! I start with a quick overview of network theory, and then begin building a category where the morphisms are electrical circuits. These lecture notes provide extra details:

With luck, this video will be the first of a series. I’m giving a seminar on network theory at U.C. Riverside this fall. I’ll start by sketching the results here:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

But this is a big paper, and I also want to talk about other papers, so I certainly won’t explain everything in here—just enough to help you get started! If you have questions, don’t be shy about asking them.

I thank Blake Pollard for filming this seminar, and Muhammad “Siddiq” Siddiqui-Ali for providing the videocamera and technical support.

The Large-Number Limit for Reaction Networks (Part 3)

8 October, 2014

joint with Arjun Jain

We used to talk about reaction networks quite a lot here. When Arjun Jain was visiting the CQT, we made a lot of progress understanding how the master equation reduces to the rate equation in the limit where there are very large numbers of things of each kind. But we never told you the end of the story, and by now it’s been such a long time that you may need a reminder of some basic facts!

So…

The rate equation treats the number of things of each kind as continuous—a nonnegative real number—and says how it changes in a deterministic way.

The master equation treats the number of things of each kind as discrete—a nonnegative integer—and says how it changes in a probabilistic way.

You can think of the master equation as the ‘true’ description, and the rate equation as an approximation that’s good in some limit where there are large numbers of molecules — or more precisely, where the probability distribution of having some number of molecules of each kind is sharply peaked near some large value.

You may remember that in the master equation, the state of a chemical system is described by a vector $\psi$ in a kind of ‘Fock space’, while time evolution is described with the help of an operator on this space, called the ‘Hamiltonian’ $H$:

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

The Hamiltonian is built from annihilation and creation operators, so all of this looks very much like quantum field theory. The details are here, and we won’t try to review them all:

• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics.

The point is this: the ‘large-number limit’ where the master equation reduces to the rate equation smells a lot like the ‘classical limit’ of quantum field theory, where the description of light in terms of photons reduces to the good old Maxwell equations. So, maybe we can understand the large-number limit by borrowing techniques from quantum field theory!

How do we take the classical limit of quantum electromagnetism and get the classical Maxwell equations? For simplicity let’s ignore charged particles and consider the ‘free electromagnetic field': just photons, described by the quantum version of Maxwell’s equations. When we take the classical limit we let Planck’s constant $\hbar$ go to zero: that much is obvious. However, that’s not all! The energy of each photon is proportional to $\hbar,$ so to take the classical limit and get a solution of the classical Maxwell’s equations with nonzero energy we also need to increase the number of photons. We cleverly do this in such a way that the total energy remains constant as $\hbar \to 0.$

So, in quantum electromagnetism the classical limit is also a large-number limit!

That’s a good sign. It suggests the same math will also apply to our reaction network problem.

But then we hit an apparent roadblock. What’s the analogue of Planck’s constant in chemical reaction networks? What should go to zero?

We told you the answer to that puzzle a while ago: it’s the reciprocal of Avogadro’s number!

You see, chemists measure numbers of molecules in ‘moles’. There’s a large number of molecules in each mole: Avogadro’s number. If we let the reciprocal of Avogadro’s number go to zero, we are taking a limit where chemistry becomes ‘continuous’ and the discreteness of molecules goes away. Of course this is just a mathematical trick, but it’s a very useful one.

So, we got around that roadblock. And then something nice happened.

When taking the classical limit of quantum electromagnetism, we focus attention on certain quantum states that are the ‘best approximations’ to classical states. These are called ‘coherent states’, and it’s very easy to study how the behave as we simultaneously let $\hbar \to 0$ and let the expected number of photons go to infinity.

And the nice thing is that these coherent states are also important in chemistry! But because chemistry involves probabilities rather than amplitudes, they have a different name: ‘Poisson distributions’. On this blog, Brendan Fong used them to give a new proof of a great result in mathematical chemistry, the Anderson–Craciun–Kurtz theorem.

So, we have most of the concepts and tools in place, and we can tackle the large-number limit using quantum techniques.

You can review the details here:

So, after a quick refresher on the notation, we’ll plunge right in.

As you’ll see, we solve the problem except for one important technical detail: passing a derivative through a limit! This means our main result is not a theorem. Rather, it’s an idea for how to prove a theorem. Or if we act like physicists, we can call it a theorem.

Review of notation

The rate equation says

$\displaystyle{\frac{d}{dt}{x}(t) = \sum_{\tau \in T} r(\tau) (t(\tau)-s(\tau)) {x}(t)^{s(\tau)} }$

where:

$x(t) \in \mathbb{R}^k$ is a vector describing concentrations of $k$ different species at time $t.$ In chemistry these species could be different kinds of molecules.

• Each $\tau \in T$ is a transition, or in chemistry, a reaction.

$s(\tau) \in \mathbb{N}^k$ is a vector of natural numbers saying how many items of each species appear as inputs to the reaction $\tau.$ This is called the source of the reaction.

$t(\tau) \in \mathbb{N}^k$ is a vector of natural numbers saying how many items of each species appear as outputs of the reaction $\tau.$ This is called the target of the reaction. So, $t(\tau) - s(\tau)$ says the net change of the number of items of each species in the reaction $\tau.$

• The rate at which the reaction $\tau$ occurs is proportional to the rate constant $r(\tau)$ times the number

$x(t)^{s(\tau)}$

Here we are raising a vector to a vector power and getting a number, using this rule:

$x^r = x_1^{r_1} \cdots x_k^{r_k}$

where $r$ is any vector of natural numbers $(r_1, \dots, r_k),$ and $x$ is any vector of nonnegative real numbers. From now on we’ll call a vector of natural numbers a multi-index.

In this paper:

• John Baez, Quantum techniques for reaction networks.

it was shown that the master equation implies

$\displaystyle{\frac{d}{dt}\langle N_i \psi(t)\rangle = \sum_{\tau \in T} r(\tau) (t_i(\tau)-s_i(\tau)) \; \langle N^{\, \underline{s(\tau)}} \, \psi(t)\rangle }$

Here:

$\psi(t)$ is the stochastic state saying the probability of having any particular number of items of each species at each time $t.$ We won’t review the precise details of how this work; for that reread the relevant bit of Part 8.

$N_i$ is the $i$th number operator, defined using annihilation and creation operators as in quantum mechanics:

$N_i = a_i^\dagger a_i$

For the annihilation and creation operators, see Part 8.

$\langle N_i \psi(t) \rangle$ is the expected number of items of the $i$th species at time $t.$

• Similarly, $\langle N^{\underline{s(\tau)}} \psi(t)\rangle$ is the expected value of a certain product of operators. For any multi-index $r$ we define the falling power

$N_i^{\underline{r}_i} = N_i (N_i - 1) (N_i - 2) \cdots (N_i - r_i +1)$

and then we define

$N^{\underline{r}} = N_1^{\underline{r_1}} \cdots N_k^{\underline{r_k}}$

The large-number limit

Okay. Even if you don’t understand any of what we just said, you’ll see the master and rate equation look similar. The master equation implies this:

$\displaystyle{\frac{d}{dt}\langle N_i \psi(t)\rangle = \sum_{\tau \in T} r(\tau) (t_i(\tau)-s_i(\tau)) \; \langle N^{\, \underline{s(\tau)}} \, \psi(t)\rangle }$

while the rate equation says this:

$\displaystyle{\frac{d}{dt}{x}(t) = \sum_{\tau \in T} r(\tau) (t(\tau)-s(\tau)) \; {x}(t)^{s(\tau)} }$

So, we will try to get from the first to the second second with the help of a ‘large-number limit’.

We start with a few definitions. We introduce an adjustable dimensionless quantity which we call $\hbar.$ This is just a positive number, which has nothing to do with quantum theory except that we’re using a mathematical analogy to quantum mechanics to motivate everything we’re doing.

Definition. The rescaled number operators are defined as $\widetilde{N}_i = \hbar N_i.$ This can be thought of as a rescaling of the number of objects, so that instead of counting objects individually, we count them in bunches of size $1/\hbar.$

Definition. For any multi-index $r$ we define the rescaled falling power of the number operator $N_i$ by:

$\widetilde{N}_i^{\underline{r_i}} = \widetilde{N}_i (\widetilde{N}_i - \hbar)(\widetilde{N}_i-2\hbar)\cdots (\widetilde{N}_i-r_i\hbar+\hbar)$

and also define

$\widetilde{N}^{\underline{r}} = \widetilde{N}_1^{\underline{r_1}} \; \cdots \;\widetilde{N}_k^{\underline{r_k}}$

for any multi-index $r.$

Using these, we get the following equation:

$\displaystyle{\frac{1}{\hbar}\frac{d}{dt} \langle \widetilde{N}_i \psi(t)\rangle = \sum_{\tau \in T} r(\tau) (t_i(\tau)-s_i(\tau)) \; \langle \widetilde{N}^{\underline{s(\tau)}} \psi(t)\rangle \; \frac{1}{\hbar^{|s(\tau)|}} }$

where for any multi-index $r$ we set

$|r| = r_1 + \cdots + r_k$

This suggests a way to rescale the rate constants in the master equation:

Definition. The rescaled rate constants are

$\displaystyle{\widetilde{r}(\tau) = \frac{r(\tau)}{\hbar^{|s(\tau)|-1}}}$

From here onwards, we change our viewpoint. We consider the rescaled rate constants $\widetilde{r}(\tau)$ to be fixed, instead of the original rate constants $r(\tau).$ So, as we decrease $\hbar,$ we are studying situations where the original rate constants change to ensure that the rescaled rate constants stays fixed!

So, we switch to working with a rescaled master equation:

Definition. The rescaled master equation is:

$\displaystyle{\frac{d}{dt} \langle \widetilde{N}_i\widetilde{\psi}(t)\rangle = \sum_{\tau \in T} \widetilde{r}(\tau) (t_i(\tau)-s_i(\tau)) \; \langle \widetilde{N}^{\underline{s(\tau)}} \widetilde{\psi}(t)\rangle }$

This is really a one-parameter family of equations, depending on $\hbar\in (0,\infty).$ We write a solution of the rescaled master equation as $\widetilde{\psi}(t),$ but it is really one solution for each value of $\hbar.$

Following the same procedure as above, we can rescale the rate equation, using the same definition of the rescaled rate constants:

Definition. The rescaled number of objects of the $i\mathrm{th}$ species is defined as $\widetilde{x_i}=\hbar x_i,$ where $x_i$ is the original number of objects of the $i\mathrm{th}$ species. Here again, we are counting in bunches of $1/\hbar.$

Using this to rescale the rate equation, we get

Definition. The rescaled rate equation is

$\displaystyle{\frac{d}{dt}\widetilde{x}(t) = \sum_{\tau \in T} \widetilde{r}(\tau) (t(\tau)-s(\tau))\; \widetilde{x}(t)^{s(\tau)} }$

where

$\widetilde{x}(t)=(\widetilde{x_1}(t),\widetilde{x_2}(t),\dots, \widetilde{x_k}(t))$

Therefore, to go from the rescaled master equation to the rescaled rate equation, we require that

$\langle\widetilde{N}^{\underline{r}}\, \widetilde{\psi}(t)\rangle \to \langle\widetilde{N}\widetilde{\psi}(t)\rangle^r$

as $\hbar \to 0.$ If this holds, we can identify $\langle\widetilde{N}\widetilde{\psi}(t)\rangle$ with $\widetilde{x}(t)$ and get the rate equation from the master equation!

To this end, we introduce the following crucial idea:

Definition. A semiclassical family of states, $\widetilde{\psi},$ is defined as a one-parameter family of states depending on $\hbar \in (0,\infty)$ such that for some $\widetilde{c} \in [0,\infty)^k$ we have

$\langle\widetilde{N}^r\widetilde{\psi}\rangle \to \widetilde{c}^{\, r}$

for every $r\in \mathbb{N}^k$ as $\hbar \to 0.$

In particular, this implies

$\langle\widetilde{N}_i\widetilde{\psi}\rangle \to \widetilde{c}_i$

for every index $i.$

Intuitively, a semiclassical family is a family of probability distributions that becomes more and more sharply peaked with a larger and larger mean as $\hbar$ decreases. We would like to show that in this limit, the rescaled master equation gives the rescaled rate equation.

We make this precise in the following propositions.

Proposition 1. If $\widetilde{\psi}$ is a semiclassical family as defined above, then in the $\hbar \to 0$ limit, we have $\langle\widetilde{N}^{\underline{r}}\widetilde{\psi}\rangle \to \widetilde{c}^{\; r}$ as well.

Proof. For each index $i,$

$\displaystyle{ \langle\widetilde{N}_i^{\; \underline{r_i}}\, \widetilde{\psi}\rangle = \displaystyle{ \langle \widetilde{N}_i (\widetilde{N}_i - \hbar)(\widetilde{N}_i-2\hbar)\cdots(\widetilde{N}_i-r_i\hbar+\hbar)\,\widetilde{\psi}\rangle} }$

$\displaystyle{ = \Big\langle\Big(\widetilde{N}_i^r + \hbar\frac{r_i(r_i-1)}{2}\widetilde{N}_i^{r_i-1}+\cdots + \hbar^{r_i-1}(r_i-1)!\Big)\,\widetilde{\psi}\Big\rangle }$

By the definition of a semiclassical family,

$\displaystyle{ \lim_{\hbar\to 0} \langle\Big(\widetilde{N}_i^{r_i} + \hbar\frac{r_i(r_i-1)}{2}\widetilde{N}_i^{r_i-1}+ \cdots + \hbar^{r_i-1}(r_i-1)!\Big)\;\widetilde{\psi}\rangle} = \widetilde{c}_i^{\; r_i}$

since every term but the first approaches zero. Thus, we have

$\displaystyle{ \lim_{\hbar \to 0} \langle\widetilde{N}_i^{\; \underline{r_i}}\, \widetilde{\psi}\rangle = \widetilde{c}_i^{\; r_i} }$

A similar but more elaborate calculation shows that

$\displaystyle{ \lim_{\hbar \to 0} \langle\widetilde{N}_1^{\, \underline{r_1}} \cdots\widetilde{N}_k^{\, \underline{r_k}} \, \widetilde{\psi}\rangle = \lim_{\hbar \to 0} \langle\widetilde{N}_1^{\, r_1}\cdots \widetilde{N}_k^{\, r_k} \widetilde{\psi}\rangle= \lim_{\hbar \to 0}\langle\widetilde{N}^{\, r} \, \widetilde{\psi}\rangle }$

or in other words

$\langle\widetilde{N}^{\, \underline{r}}\,\widetilde{\psi}\rangle \to \widetilde{c}^{\, r}$

as $\hbar \to 0.$   █

Proposition 2. If $\widetilde{\psi}$ is a semiclassical family of states, then

$\displaystyle{ \langle (\widetilde{N}-\widetilde{c})^{r}\, \widetilde{\psi}\rangle \to 0 }$

for any multi-index $r.$

Proof. Consider the $r_i\mathrm{th}$ centered moment of the $i\mathrm{th}$ number operator:

$\displaystyle{\langle(\widetilde{N}_i-\widetilde{c}_i)^{r_i}\widetilde{\psi}\rangle = \sum_{p =0}^{r_i} {r_i \choose p}\langle\widetilde{N}_i^p\widetilde{\psi}\rangle(-\widetilde{c}_i)^{r_i-p} }$

Taking the limit as $\hbar$ goes to zero, this becomes

$\begin{array}{ccl} \displaystyle{ \lim_{\hbar \to 0}\sum_{p =0}^{r_i} {r_i \choose p}\langle\widetilde{N}_i^p\widetilde{\psi}\rangle(-\widetilde{c}_i)^{r_i-p} } &=& \displaystyle{ \sum_{p =0}^{r_i} {r_i \choose p}(\widetilde{c}_i)^p(-\widetilde{c}_i)^{r_i-p} } \\ \\ &=& (\widetilde{c}_i-\widetilde{c}_i)^{r_i} \\ \\ &=& 0 \end{array}$

For a general multi-index $r$ we can prove the same sort of thing with a more elaborate calculation. First note that

$\langle (\widetilde{N}-\widetilde{c})^{r}\widetilde{\psi}\rangle=\langle(\widetilde{N_1}-\widetilde{c_1})^{r_1} \cdots (\widetilde{N_k}-\widetilde{c_k})^{r_k})\widetilde{\psi}\rangle$

The right-hand side can be expanded as

$\displaystyle{ \langle(\sum_{p_1 =0}^{r_1} {r_1 \choose p_1}\widetilde{N}_1^{p_1}(-\widetilde{c}_1)^{r_1-p_1} ) \cdots (\sum_{p_k =0}^{r_k} {r_k \choose p_k}\widetilde{N}_k^{p_k}(-\widetilde{c}_k)^{r_k-p_k} )\widetilde{\psi} }\rangle$

We can write this more tersely as

$\displaystyle{ \sum_{p=0}^r} {r\choose p} \langle \widetilde{N}^p\widetilde{\psi}\rangle (-\widetilde{c})^{r-p}$

where for any multi-index $r$ we define

$\displaystyle{{\sum_{p=0}^{r}}= \sum_{p_1 =0}^{r_1} \cdots \sum_{p_k =0}^{r_k} }$

and for any multi-indices $r, p$ we define

$\displaystyle{ {r \choose p}={r_1 \choose p_1}{r_2 \choose p_2}\cdots {r_k \choose p_k}}$

Now using the definition of a semiclassical state, we see

$\displaystyle{ \lim_{\hbar \to 0} \sum_{p=0}^r} {r\choose p} \langle \widetilde{N}^p\widetilde{\psi}\rangle (-\widetilde{c})^{r-p}= \displaystyle{ \sum_{p=0}^r} {r\choose p} (\widetilde{c})^{p} (-\widetilde{c})^{r-p}$

But this equals zero, as the last expression, expanded, is

$\displaystyle{ (\widetilde{c})^r \left( \sum_{p_1=0}^{r_1} {r_1\choose p_1} (-1)^{r_1-p_1}\right) \cdots \left( \sum_{p_k=0}^{r_k} {r_k\choose p_k} (-1)^{r_k-p_k} \right) }$

where each individual sum is zero.   █

Here is the theorem that would finish the job if we could give a fully rigorous proof:

“Theorem.” If $\widetilde{\psi}(t)$ is a solution of the rescaled master equation and also a semiclassical family for the time interval $[t_0,t_1],$ then $\widetilde{x}(t) = \langle \widetilde{N} \widetilde{\psi}(t) \rangle$ is a solution of the rescaled rate equation for $t \in [t_0,t_1].$

Proof sketch. We sketch a proof that relies on the assumption that we can pass the $\hbar \to 0$ limit through a time derivative. Of course, to make this rigorous, we would need to justify this. Perhaps it is true only in certain cases.

Assuming that we can pass the limit through the derivative:

$\displaystyle{\lim_{\hbar \to 0}\frac{d}{dt} \langle \widetilde{N}\widetilde{\psi}(t)\rangle = \lim_{\hbar \to 0} \sum_{\tau \in T} \widetilde{r}(\tau) (t(\tau)-s(\tau))\langle \widetilde{N}^{\, \underline{s(\tau)}} \, \widetilde{\psi}(t)\rangle }$

and thus

$\displaystyle{\frac{d}{dt}\lim_{\hbar \to 0} \langle \widetilde{N}\widetilde{\psi}(t)\rangle = \sum_{\tau \in T} \widetilde{r}(\tau) (t(\tau)-s(\tau)) \lim_{\hbar \to 0}\langle \widetilde{N}^{\, \underline{s(\tau)}} \, \widetilde{\psi}(t)\rangle }$

and thus

$\displaystyle{\frac{d}{dt}\widetilde{x}(t) = \sum_{\tau \in T} \widetilde{r}(\tau) (t(\tau)-s(\tau))\widetilde{x}^{\, s(\tau)} }.$

As expected, we obtain the rescaled rate equation.   █

Another question is this: if we start with a semiclassical family of states as our initial data, does it remain semiclassical as we evolve it in time? This will probably be true only in certain cases.

An example: rescaled coherent states

The best-behaved semiclassical states are the coherent states.
Consider the family of coherent states

$\displaystyle{\widetilde{\psi}_{\widetilde{c}} = \frac{e^{(\widetilde{c}/\hbar) z}}{e^{\widetilde{c}/\hbar}}}$

using the notation developed in the earlier mentioned paper. In that paper it was shown that for any multi-index $m$ and any coherent state $\Psi$ we have

$\langle N^{\underline{m}}\Psi\rangle = \langle N\Psi \rangle^m$

Using this result for $\widetilde{\psi}_{\widetilde{c}}$ we get

$\displaystyle{\langle \widetilde{N}^{\underline{m}}\widetilde{\psi}_{\widetilde{c}}\rangle~=~\hbar^{|m|}\langle N^{\underline{m}}\widetilde{\psi}_{\widetilde{c}}\rangle~=~\hbar^{|m|}\langle N\widetilde{\psi}_{\widetilde{c}}\rangle^m~=~\hbar^{|m|}\frac{\widetilde{c}^m}{\hbar^{|m|}}~=~\widetilde{c}^m}$

Since $\langle\widetilde{N}^{\underline{m}}\widetilde{\psi}_{\widetilde{c}}\rangle$ equals $\langle \widetilde{N}^{m} \widetilde{\psi}_{\widetilde{c}}\rangle$ plus terms of order $\hbar,$ as $\hbar \to 0$ we have

$\langle\widetilde{N}^{\underline{m}}\widetilde{\psi}_{\widetilde{c}}\rangle~\to~\langle\widetilde{N}^{m}\widetilde{\psi}_{\widetilde{c}}\rangle=\widetilde{c}^{m}$

showing that our chosen $\widetilde{\psi}_{\widetilde{c}}$ is indeed a semiclassical family.

Network Theory (Part 30)

3 October, 2014

The network theory series is back! You may have thought it died out, but in fact it’s just getting started. Over the last year my grad students have made huge strides in working out the math of networks. Now it’s time to explain what they’ve done.

In the last three episodes I explained how electrical circuits made of resistors, inductors and capacitors are a great metaphor for many kinds of complex systems made of interacting parts. And it’s not just a metaphor; there are mathematically rigorous analogies—in other words, isomorphisms—between such electrical circuits and various other kinds of ‘passive linear networks’.

I showed you a chart of these analogies last time:

 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

But what do I mean by a ‘passive linear network’? Let me explain this very roughly at first, since we’ll be painfully precise later on.

Right now by ‘network’ I mean a graph with gizmos called ‘components’ on the edges. For example:

In a network there is some kind of ‘flow’ running along each edge, and also some kind of ‘effort’ across that edge. For example, in electronics the flow is electrical current and the effort is voltage. The chart shows the meaning of flow and effort in other examples.

‘Passivity’ means roughly that none of the components put out energy that didn’t earlier come in. For example, resistors lose energy (which goes into heat, which we’re not counting). Capacitors can store energy and later release it. So, resistors and capacitors are passive—and so are inductors. But batteries and current sources actually put out energy, so we won’t allow them in our networks yet. For now, we’re just studying how passive components respond to a source of flow or effort.

For some subtleties that show up when you try to make the concept of passivity precise, try:

Passivity (engineering), Wikipedia.

Finally, ‘linearity’ means that the flow along each edge of our network is linearly related to the effort across that edge. Here are the key examples:

• For electrical resistors, linearity is captured by Ohm’s law. If an edge $e$ in our network is labelled by a resistor of resistance $R,$ usually drawn like this:

then Ohm’s law says:

$V = R I$

where $V$ is the voltage across that edge and $I$ is the current along that edge.

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

we have

$\displaystyle{ V = L \frac{d I}{d t} }$

Here we need to think of the voltage and current as functions of time.

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

we write the equation

$\displaystyle{ I = C \frac{d V}{d t} }$

where again we think of the voltage and current as functions of time.

Both linearity and passivity are simplifying assumptions that we eventually want to drop. If we include batteries or current sources, we’re dropping passivity. And if include transistors, we’re dropping linearity. Obviously both these are important!

However, there is a lot to do even with these simplifying assumptions. And now it’s time to get started!

In what follows, I will not use the terms ‘flow’ and ‘effort’, which are chosen to be neutral and subject-independent. Instead, I’ll use the vocabulary from electronics, e.g. ‘current’ and ‘voltage’. The reason is that we’ve all heard of resistors, capacitors, Ohm’s law and Kirchhoff’s laws, and while these have analogues in every row of the chart, it seems pointless to make up weird new ‘neutral’ terms for all these concepts.

But don’t be fooled by the jargon! We’re not merely studying electrical circuits. We’ll be studying passive linear networks in full generality… with the help of category theory.

Linear passive networks as morphisms

To get going, let’s think about circuits made of resistors. We can do this without harm, because we’ll later include capacitors and inductors using a simple effortless trick. Namely, we’ll generalize the ‘resistance’ of a resistor, which is a real number, to something called ‘impedance’, which is an element of some larger field. Everything will be so abstract that replacing resistances with impedances will be as easy as snapping our fingers.

Right now I want to define a category where the morphisms are circuits made of resistors. Any morphism will go from some ‘inputs’ to some ‘outputs’, like this:

So a morphism is roughly a graph with edges labelled by numbers called ‘resistances’, with some special nodes called ‘inputs’ and some special nodes called ‘outputs’.

What can do with morphisms? Compose them! So, suppose we have a second morphism whose inputs match the outputs of the first:

Then we can compose them, attaching the outputs of the first to the inputs of the second. We get this morphism as the result:

So, composing morphisms is a way to build big electrical circuits—or other ‘linear passive networks’—out of little ones.

This seems pretty simple, but let’s try to formalize it and see why we have a category. In fact it takes a bit of thought. To check that we get a category, we need to check that composition is associative:

$(fg)h = f(gh)$

and that each object $x$ has an identity morphism $1_x : x \to x$ that does what an identity should:

$f 1_x = f$

$1_x g = g$

All these equations seem obviously true in our example… until you try to prove them.

You might think an identity morphism should be a bunch of straight pieces of wire—a bunch of edges each with an input node and an output node—but that doesn’t quite work, since sticking an extra edge onto a graph gives a new graph with an extra edge!

Also, we are composing circuits by ‘sticking them together’. This process is formalized in category theory using a pushout, and pushouts are only defined ‘up to canonical isomorphism’. The very simplest example is the disjoint union of two sets. We all know what it means, but if you examine it carefully, you’ll see it’s only defined up to canonical isomorphism, because it involves a choice of how we make the two sets disjoint, and this choice is somewhat arbitrary.

All this means the category we’re after is a bit subtler than you might at first expect; in fact, it’s most naturally thought of as a bicategory, meaning roughly that all the equations above hold only ‘up to canonical isomorphism’.

So, we proceed like this.

First we define a concept of ‘labelled graph’, where (for now) only the edges are labelled. We do this because we want our circuits to have edges labelled by ‘resistances’, which are real numbers. But we do it in greater generality because later we’ll want the edges to be labelled by ‘impedances’, which are elements of some other field. And since we’re studying electrical circuits just as examples of networks, later still we will probably want graphs whose edges are labelled in still other ways.

So:

Definition. A graph consists a finite set $E$ of edges, a finite set $N$ of nodes, and two functions

$s,t : E \to N$

Thus each edge $e$ will have some node $s(e)$ as its source and some node $t(e)$ as its target:

Definition. Given a set $L,$ we define an $L$-labelled graph to be a graph together with a function $r : E \to L.$ This assigns to each edge $e \in E$ its label $r(e) \in L.$ We call $L$ the label set.

We use the letter $r$ because for circuits of resistors we will take the label set to be

$L = (0,\infty) \subset \mathbb{R}$

the positive real numbers, and $r(e)$ will be the resistance of the edge $e.$ For circuits that also contain inductors and capacitors we will take the label set to be the positive elements of some larger field… but more about that later!

Now we want to give our $L$-labelled graph a set of nodes called ‘inputs’ and a set of nodes called ‘outputs’. You might think the set of inputs should be disjoint from the set of outputs, but that’s a tactical error! It turns out an identity morphism in our category should have the inputs being exactly the same as the outputs… and no edges at all:

To handle this nicely, we need to make a category of $L$-labelled graphs. This works in the obvious way, if you’re used to this stuff. A morphism from one $L$-labelled graph to another sends edges to edges, nodes to nodes, and preserves everything in sight:

Definition. Given $L$-graphs $\Gamma = (E,N,s,t,r)$ and $\Gamma' = (E',N',s',t',r'),$ a morphism of $L$-labelled graphs from $\Gamma$ to $\Gamma'$ is a pair of functions

$\epsilon: E \to E'$

$\nu : N \to N'$

such that the following diagrams commute:

There is a category $L\mathrm{Graph}$ where the objects are $L$-labelled graphs and the morphisms are as we’ve just defined them.

Warning: the morphisms in $L\mathrm{Graph}$ are not the morphisms of the kind we really want, the ones that look like this:

They are just a step along the way. A morphism of the kind we really want is a diagram like this in $L\mathrm{Graph}$:

where $\Gamma$ is an $L$-labelled graph and $I, O$ are $L$-labelled graphs with no edges!

You see, if $I$ and $O$ have no edges, all they have is nodes. We call the nodes of $I$ the inputs and those of $O$ the outputs. The morphisms $i: I \to \Gamma$ and $o : O \to \Gamma$ say how these nodes are included in $\Gamma.$ $\Gamma$ is our circuit made of resistors.

In general, any diagram shaped like this is called a cospan:

If we turned the arrows around it would be called a span. Cospans are good whenever you a thing with an ‘input end’ and an ‘output end’, and you want to describe how the ends are included in the thing. So, they’re precisely what we need for describing a circuit made of resistors, like this:

This makes us want to cook up a category $L\mathrm{Circ}$ where:

• an object $I$ is an $L$-labelled graph with no edges. We can alternatively think of it as a finite set: a set of nodes.

• a morphism from $I$ to $O$ is a cospan of $L$-labelled graphs:

We still need to say how to compose these morphisms. We know it will amount to attaching the outputs of one circuit to the inputs of the next—that’s all there is to it! But we need to make this precise and prove we get a category. And as I’ve hinted, we will actually get something bigger and better: a bicategory! This will come as no surprise to if you’re familiar with span and cospan bicategories–but it may induce a heart attack otherwise.

This bicategory can then be ‘watered down’ to give our category $L\mathrm{Circ}.$ And when we take

$L = (0,\infty)$

we’ll get the category where morphisms are circuits made of resistors! We’ll call this $\mathrm{ResCirc}.$

Then I’ll explain what we can do with this category! There’s no end of things we could do with it. But the main thing Brendan does is study the ‘black-boxing’ operation, where we take a circuit, forget its inner details, and only keep track of what it does. This turns out to be quite interesting.

References

I thank Brendan Fong for drawing some of the pictures of circuits here. For the details of what I’m starting to explain here, read our paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.