Quantum Techniques for Studying Equilibrium in Reaction Networks

The summer before last, I invited Brendan Fong to Singapore to work with me on my new ‘network theory’ project. He quickly came up with a nice new proof of a result about mathematical chemistry. We blogged about it, and I added it to my book, but then he became a grad student at Oxford and got distracted by other kinds of networks—namely, Bayesian networks.

So, we’ve just now finally written up this result as a self-contained paper:

Check it out and let us know if you spot mistakes or stuff that’s not clear!

The idea, in brief, is to use math from quantum field theory to give a somewhat new proof of the Anderson–Craciun–Kurtz theorem.

This remarkable result says that in many cases, we can start with an equilibrium solution of the ‘rate equation’ which describes the behavior of chemical reactions in a deterministic way in the limit of a large numbers of molecules, and get an equilibrium solution of the ‘master equation’ which describes chemical reactions probabilistically for any number of molecules.

The trick, in our approach, is to start with a chemical reaction network, which is something like this:

and use it to write down a Hamiltonian describing the time evolution of the probability that you have various numbers of each kind of molecule: A, B, C, D, E, … Using ideas from quantum mechanics, we can write this Hamiltonian in terms of annihilation and creation operators—even though our problem involves probability theory, not quantum mechanics! Then we can write down the equilibrium solution as a ‘coherent state’. In quantum mechanics, that’s a quantum state that approximates a classical one as well as possible.

All this is part of a larger plan to take tricks from quantum mechanics and apply them to ‘stochastic mechanics’, simply by working with real numbers representing probabilities instead of complex numbers representing amplitudes!

I should add that Brendan’s work on Bayesian networks is also very cool, and I plan to talk about it here and even work it into the grand network theory project I have in mind. But this may take quite a long time, so for now you should read his paper:

Very interesting! I will look at this more carefully.

Meanwhile, do you know of any literature in the other direction? That is, how can I take a Markov chain, and “quantumize” it by replacing the probabilities by amplitudes, and the stochastic evolution by a quantum evolution? I see on Wikipedia that there are these things called quantum Markov chains, would you say that is the thing to read?

I guess I’m asking if there is a functor from non-equilibrium classical statistical mechanics to non-equilibrium quantum statistical mechanics. I suppose people know how to do this at the level of classical mechanics, by taking the Lagrangian, and treating the variables as operators. My question is how to treat the Laplacian terms in the evolution.

While we can use quantum techniques to study Markov chains, chemical reaction networks and the like, there’s a difference between what counts as a suitable Hamiltonian for a quantum problem and what counts as a suitable Hamiltonian for a stochastic one. The former kind needs to be self-adjoint. The latter kind needs to be infinitesimal stochastic. For more on this, see Part 12 of the network theory series.

So, we can’t in general take any particular quantum system and ‘stochasticize’ it, or take any particular stochastic system and ‘quantumize’ it.

However, there’s a large and interesting class of Hamiltonians that are both self-adjoint and infinitesimal stochastic. These are called ‘Dirichlet operators’. Examples include Laplacians on manifolds and also graph Laplacians. For systems with finitely many states, Dirichlet operators are equivalent to electrical circuits made of resistors. I became very happy when I noticed this, in part because it connects two aspects of the network theory program: Markov processes and electrical circuits. I have ideas about how to exploit this.

You can read more about Dirichlet operators in Part 16, and also here:

• M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980.

(There are later editions but I don’t like them as much.)

So, while I believe that in general there are no functors of the sort you seek, systematically turning random walks into quantum random walks or vice versa, there is one if we restrict attention to systems whose Hamiltonian resembles a Laplacian… or more precisely, is a Dirichlet operator. The most obvious kind of random walk on an undirected graph is of this sort.

I read, superficially, of the quantum Zeno paradox: it seem an interesting topic.
I think that if there is a slowing unstable decay in a stable product, using a measure of the unstable state (is there a coesistence of the quantum states, with a right chemical rate constant?), then can be possible an acceleration of the quantum chemical decay using a measure of the stable state, using a measure of a energy level of a stable molecule with a laser.
If this is true can be possible a cost reduction of the nuclear power plants, reducing the nuclear waste, using laser with the right frequency.
If this is true can be possible the generic toxic chemical waste using a laser inducted decomposition in inert molecules.
If this is true can be obtained a chemical synthesis using laser projection of the molecules (measure of the synthetized molecules).

I am thinking a chemical reactor with laser catalyst.
If the quantum Zeno effect can catalize a reaction, then can be possible to use a reflective mirror that is doped with the product of reaction: the infrared emission of the inner product of reaction is reflected by the shiny molecules layer (light filter for the product of reaction).
This can happen in nature (with long time) in the chimney inside an hydrothermal vent, to obtain molecules of the same chirality: light catalysis of a chiral product for overlap of chiral layer.

[…] But sometimes our state will stay coherent forever! For one case where this happens, see the companion paper, which I blogged about a little while ago: […]

“In the first part of this mini-series, I describe how various ideas important in probability theory arise naturally when you start doing linear algebra using only the nonnegative real numbers.”

Although the kind of numbers that represent probability– real
numbers greater than zero– are compared to each other by the greater-than/less-than relation, in the world of my imagination I’d prefer the kind of numbers representing possibility NOT to be compared to each other by a greater-than/less-than relation.

It’s a choice– and there may be other choices than this one.

The kind of numbers In the world I imagine which represent possibilities also should work logically — or at least it seems to me.

First they convey the logical “OR”:

— Example: (A) the possibility “I exist somewhere” comprises (B) the possibility I am home OR (C) the possibility I am somewhere else.

A = B + C

— Logically, it doesn’t matter which of (A) or (B) is stated first.

And multiplication seems to convey the logical “AND.”

An example:

— I’m having a BBQ and invite Homer. But Homer hates being rained on at BBQs and I am considering the possibilities.

— (C) the possibility it will rain on Homer at the BBQ comprises
(D) the possibility Homer will attend AND (E) the possibility that it will rain during the BBQ.

C = DE

— Again, it logically doesn’t matter which of (D) or (E) is stated
first.

C = DE = ED

— Further, if either (D) or (E) is zeroed, the possibility it will
rain on Homer at the BBQ is also zeroed. That is, if it is impossible
that it will rain during the BBQ, then it’s logically impossible that
(C) it will rain on Homer at the BBQ.

C = DE, E = 0 ==> C = 0

— if it is impossible that Homer will attend the BBQ, then in that
case it is impossible it will rain on Homer at the BBQ.

C = DE, D = 0 ==> C = 0

– Now say that other concerns enter my mind,Nand I become inconsiderate!

Then in my consideration of (C) the possibility it will rain on
Homer at the BBQ, if I factor out (E) the possibility it will rain at
the BBQ, then this is equivalent to considering only (D) the
possibility that Homer will attend the BBQ.

C = DE

In my imagination I factor out E from C:

C/E = D ( I consider only D)

– Similarly in my consideration of (C), the possibility it will
rain on Homer at the BBQ, if I factor out from my consideration (D), the possibility that Homer will attend, then logically this is equivalent
to considering only (E) the possibility that it will rain at the BBQ.

C = DE

In this case I factor out D from C:

C/D = E

(I consider only E)

— In other words In my imagination possibility means commutative numbers which do not support the > or < relation.

But, (see above) they must support division.

So from the four kinds of numbers in the division algebras– in my imagined world the complex numbers represent possibility.

Your analysis is very interesting, but you’re jumping the gun a bit when you introduce division algebras. There are four normed division algebras over the real numbers, and those italicized buzzwords introduce restrictions that don’t show up in your analysis… so you’re throwing out options you might find interesting.

It seems to me that your analysis says you want to represent possibility using a semifield: a structure with an associative and commutative addition and a 0 element such that

an associative and commutative multiplication that distributes over addition, and having the property that every nonzero element has a multiplicative inverse

Actually you didn’t mention zero, and you seemed to allow division by anything. But it seems good to have an element 0 to stand for ‘impossible’ or more precisely ‘probability zero’, and it seems bad to allow division by 0. Also, you didn’t mention the distributive law

but I’d like to include that, since it’s very hard to work with addition and multiplication if they don’t get along in some way like this.

People often talk about fields, where you also have subtraction, but we may not want negative probabilities, so it’s better not to demand that always exists.

To take some rather silly examples, the rational numbers also have all the properties you list. So do the nonnegative rational numbers. So do the Gaussian numbers, that is, numbers where and are rational. So do the numbers But there are also lots of stranger possibilities. And it might be fun to investigate these.

If we include an ordering with some reasonable properties, we are very easily led to conclude that probabilities should be nonnegative real numbers, or at least lie in some ‘sub-semifield’ of the nonnegative real numbers. (For example, some people argue that only nonnegative rational numbers are truly meaningful probabilities.)

But I don’t mind if someone wants to drop this and explore other options.

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Very interesting! I will look at this more carefully.

Meanwhile, do you know of any literature in the other direction? That is, how can I take a Markov chain, and “quantumize” it by replacing the probabilities by amplitudes, and the stochastic evolution by a quantum evolution? I see on Wikipedia that there are these things called quantum Markov chains, would you say that is the thing to read?

I guess I’m asking if there is a functor from non-equilibrium classical statistical mechanics to non-equilibrium quantum statistical mechanics. I suppose people know how to do this at the level of classical mechanics, by taking the Lagrangian, and treating the variables as operators. My question is how to treat the Laplacian terms in the evolution.

While we can use quantum techniques to study Markov chains, chemical reaction networks and the like, there’s a difference between what counts as a suitable Hamiltonian for a quantum problem and what counts as a suitable Hamiltonian for a stochastic one. The former kind needs to be self-adjoint. The latter kind needs to be infinitesimal stochastic. For more on this, see Part 12 of the network theory series.

So, we can’t in general take any particular quantum system and ‘stochasticize’ it, or take any particular stochastic system and ‘quantumize’ it.

However, there’s a large and interesting class of Hamiltonians that are

bothself-adjointandinfinitesimal stochastic. These are called ‘Dirichlet operators’. Examples include Laplacians on manifolds and also graph Laplacians. For systems with finitely many states, Dirichlet operators are equivalent to electrical circuits made of resistors. I became very happy when I noticed this, in part because it connects two aspects of the network theory program: Markov processes and electrical circuits. I have ideas about how to exploit this.You can read more about Dirichlet operators in Part 16, and also here:

• M. Fukushima,

Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980.(There are later editions but I don’t like them as much.)

So, while I believe that in general there are no functors of the sort you seek, systematically turning random walks into quantum random walks or vice versa, there is one if we restrict attention to systems whose Hamiltonian resembles a Laplacian… or more precisely, is a Dirichlet operator. The most obvious kind of random walk on an undirected graph is of this sort.

PS: A small typo in para 4 of intro: Craciun, not Cracuin.

Thanks – fixed!

I read, superficially, of the quantum Zeno paradox: it seem an interesting topic.

I think that if there is a slowing unstable decay in a stable product, using a measure of the unstable state (is there a coesistence of the quantum states, with a right chemical rate constant?), then can be possible an acceleration of the quantum chemical decay using a measure of the stable state, using a measure of a energy level of a stable molecule with a laser.

If this is true can be possible a cost reduction of the nuclear power plants, reducing the nuclear waste, using laser with the right frequency.

If this is true can be possible the generic toxic chemical waste using a laser inducted decomposition in inert molecules.

If this is true can be obtained a chemical synthesis using laser projection of the molecules (measure of the synthetized molecules).

I am thinking a chemical reactor with laser catalyst.

If the quantum Zeno effect can catalize a reaction, then can be possible to use a reflective mirror that is doped with the product of reaction: the infrared emission of the inner product of reaction is reflected by the shiny molecules layer (light filter for the product of reaction).

This can happen in nature (with long time) in the chimney inside an hydrothermal vent, to obtain molecules of the same chirality: light catalysis of a chiral product for overlap of chiral layer.

[…] But sometimes our state will stay coherent forever! For one case where this happens, see the companion paper, which I blogged about a little while ago: […]

“In the first part of this mini-series, I describe how various ideas important in probability theory arise naturally when you start doing linear algebra using only the nonnegative real numbers.”

Although the kind of numbers that represent probability– real

numbers greater than zero– are compared to each other by the greater-than/less-than relation, in the world of my imagination I’d prefer the kind of numbers representing possibility NOT to be compared to each other by a greater-than/less-than relation.

It’s a choice– and there may be other choices than this one.

The kind of numbers In the world I imagine which represent possibilities also should work logically — or at least it seems to me.

First they convey the logical “OR”:

— Example: (A) the possibility “I exist somewhere” comprises (B) the possibility I am home OR (C) the possibility I am somewhere else.

A = B + C

— Logically, it doesn’t matter which of (A) or (B) is stated first.

And multiplication seems to convey the logical “AND.”

An example:

— I’m having a BBQ and invite Homer. But Homer hates being rained on at BBQs and I am considering the possibilities.

— (C) the possibility it will rain on Homer at the BBQ comprises

(D) the possibility Homer will attend AND (E) the possibility that it will rain during the BBQ.

C = DE

— Again, it logically doesn’t matter which of (D) or (E) is stated

first.

C = DE = ED

— Further, if either (D) or (E) is zeroed, the possibility it will

rain on Homer at the BBQ is also zeroed. That is, if it is impossible

that it will rain during the BBQ, then it’s logically impossible that

(C) it will rain on Homer at the BBQ.

C = DE, E = 0 ==> C = 0

— if it is impossible that Homer will attend the BBQ, then in that

case it is impossible it will rain on Homer at the BBQ.

C = DE, D = 0 ==> C = 0

– Now say that other concerns enter my mind,Nand I become inconsiderate!

Then in my consideration of (C) the possibility it will rain on

Homer at the BBQ, if I factor out (E) the possibility it will rain at

the BBQ, then this is equivalent to considering only (D) the

possibility that Homer will attend the BBQ.

C = DE

In my imagination I factor out E from C:

C/E = D ( I consider only D)

– Similarly in my consideration of (C), the possibility it will

rain on Homer at the BBQ, if I factor out from my consideration (D), the possibility that Homer will attend, then logically this is equivalent

to considering only (E) the possibility that it will rain at the BBQ.

C = DE

In this case I factor out D from C:

C/D = E

(I consider only E)

— In other words In my imagination possibility means commutative numbers which do not support the > or < relation.

But, (see above) they must support division.

So from the four kinds of numbers in the division algebras– in my imagined world the complex numbers represent possibility.

Of course I have no authority in these matters.

John what's your position?

Your analysis is very interesting, but you’re jumping the gun a bit when you introduce division algebras. There are four

normeddivision algebrasover the real numbers, and those italicized buzzwords introduce restrictions that don’t show up in your analysis… so you’re throwing out options you might find interesting.It seems to me that your analysis says you want to represent possibility using a

semifield: a structure with an associative and commutative addition and a 0 element such thatan associative and commutative multiplication that distributes over addition, and having the property that every nonzero element has a multiplicative inverse

Actually you didn’t mention zero, and you seemed to allow division by anything. But it seems good to have an element 0 to stand for ‘impossible’ or more precisely ‘probability zero’, and it seems bad to allow division by 0. Also, you didn’t mention the distributive law

but I’d like to include that, since it’s very hard to work with addition and multiplication if they don’t get along in some way like this.

People often talk about fields, where you also have subtraction, but we may not want negative probabilities, so it’s better not to demand that always exists.

To take some rather silly examples, the rational numbers also have all the properties you list. So do the nonnegative rational numbers. So do the Gaussian numbers, that is, numbers where and are rational. So do the numbers But there are also lots of stranger possibilities. And it might be fun to investigate these.

If we include an ordering with some reasonable properties, we are very easily led to conclude that probabilities should be nonnegative real numbers, or at least lie in some ‘sub-semifield’ of the nonnegative real numbers. (For example, some people argue that only nonnegative rational numbers are truly meaningful probabilities.)

But I don’t mind if someone wants to drop this and explore other options.