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Stream: deprecated: thermodynamics

Topic: statistical mechanical Fock space

Owen Lynch (Nov 17 2021 at 15:30):

I was just learning about Fock space in quantum mechanics, and I feel like there should be some sort of analogue in statistical mechanics.

Suppose that $X$ is a measure space, and let $\mathcal{P}(X)$ be the convex space of probability measures on it. Then consider the convex subset of $\mathcal{P}(X^n)$ consisting of $\mu$ such that $\mu(E_1 \times \cdots \times E_n) = \mu(E_{\sigma(1)} \times \cdots E_{\sigma(n)})$ for all $\sigma \in S_n$, let's call this $\mathcal{SP}^n(X)$.

Then I propose that the correct convex space for a many-particle system is $\prod_{n \in \mathbb{N}} \mathcal{SP}^n(X)$, which I see as some analogue of Fock space.

Now here is the question. If $X$ is a finite set, does the operational definition of entropy for convex spaces end up giving $S(p) = \frac{1}{N!} S_{\mathrm{sh}}(p)$ for $p \in \mathcal{SP}^n(X)$?

John Baez (Nov 17 2021 at 15:56):

Owen Lynch said:

Suppose that $X$ is a measure space, and let $\mathcal{P}(X)$ be the convex space of probability measures on it. Then consider the convex subset of $\mathcal{P}(X^n)$ consisting of $\mu$ such that $\mu(E_1 \times \cdots \times E_n) = \mu(E_{\sigma(1)} \times \cdots E_{\sigma(n)})$ for all $\sigma \in S_n$, let's call this $\mathcal{SP}^n(X)$.

Then I propose that the correct convex space for a many-particle system is $\prod_{n \in \mathbb{N}} \mathcal{SP}^n(X)$, which I see as some analogue of Fock space.

I think you want $\sum_{n \in \mathbb{N}} \mathcal{SP}^n(X)$, right? The system has one particle or two particles or three, etc., not and.

But you are right, there should be a construction like this.

If you think of Fock space as just a vector space and forget the Hilbert structure for a minute, the Fock space on a vector space $V$ is just the free commutative monoid object on $V$:

$\displaystyle{ \sum_{n \in \mathbb{N}} \frac{V^{\otimes n}}{S_n} }$

where the symmetric group $S_n$ acts on the n-fold tensor product $V^{\otimes n}$ by permuting factors.

Similarly given a convex space $X$ we should have a free commutative monoid on $X$ in the category of convex spaces, given by

$\displaystyle{ \sum_{n \in \mathbb{N}} \frac{X^n}{S_n} }$

John Baez (Nov 17 2021 at 16:07):

By the way, this construction is quite general: whenever you have a symmetric monoidal category with countable colimits, and the tensor product distributes over those colimits, the free commutative monoid on an object $x$ is

$\displaystyle{ \sum_{n \in \mathbb{N}} \frac{x^{\otimes n}}{S_n} }$

John Baez (Nov 17 2021 at 16:08):

where we are modding by the action of $S_n$ on $x^{\otimes n}$, using coequalizers.

John Baez (Nov 17 2021 at 16:09):

For obvious reasons people write

$\displaystyle{ \sum_{n \in \mathbb{N}} \frac{x^{\otimes n}}{S_n} } = e^x$

and

and you can show that

$e^{x + y} \cong e^x \otimes e^y$

Owen Lynch (Nov 17 2021 at 16:09):

John Baez said:

I think you want $\sum_{n \in \mathbb{N}} \mathcal{SP}^n(X)$, right? The system has one particle or two particles or three, etc., not and.

Ah, but the system could have a 1/2 probability of having one particle and a 1/2 probability of having two particles!

But I think that there is something wrong here. Really, it should be $\mathcal{P}(N) \times \prod_{n \in \mathbb{N}} \mathcal{SP}^n(X)$.

John Baez (Nov 17 2021 at 16:11):

Ah, but the system could have a 1/2 probability of having one particle and a 1/2 probability of having two particles!

I think I'm right: you have to remember that the coproduct of convex spaces is not their disjoint union. So you get those convex linear combinations automatically. I was just using the mnemonic: "or means coproduct, and means product".

Owen Lynch (Nov 17 2021 at 16:11):

Perhaps by $\sum_{n \in \mathbb{N}} \mathcal{SP}^n(X)$ you mean the categorical sum!!

John Baez (Nov 17 2021 at 16:11):

Of course! What other sum is there?

John Baez (Nov 17 2021 at 16:12):

:upside_down:

Owen Lynch (Nov 17 2021 at 16:12):

Hahaha, good point

John Baez (Nov 17 2021 at 16:13):

If you read what I wrote you'll see I meant the categorical sum:

John Baez said:

By the way, this construction is quite general: whenever you have a symmetric monoidal category with countable colimits, and the tensor product distributes over those colimits, the free commutative monoid on an object $x$ is

$\displaystyle{ \sum_{n \in \mathbb{N}} \frac{x^{\otimes n}}{S_n} }$

Owen Lynch (Nov 17 2021 at 16:13):

Ah, and in the category of Hilbert spaces, (as opposed to vector spaces) an infinite sum allows you to actually take infinite sums, because of completeness

Owen Lynch (Nov 17 2021 at 16:13):

So we need a similar condition for our convex spaces so that this infinite categorical sum allows there to be a non-zero probability for every number of particles

Owen Lynch (Nov 17 2021 at 16:14):

Otherwise you can only take finite convex combinations.

John Baez (Nov 17 2021 at 16:14):

Well, unfortunately the "infinite direct sum of Hilbert spaces" is not their coproduct, unlike the case for vector spaces, where the infinite direct sum really is the coproduct.

Owen Lynch (Nov 17 2021 at 16:14):

Oh, that is unfortunate

Owen Lynch (Nov 17 2021 at 16:15):

Is there a subtlety that fixes this, or is it just something that we have to live with?

Owen Lynch (Nov 17 2021 at 16:16):

I.e., is there a categorical construction of the infinite direct sum of Hilbert spaces?

John Baez (Nov 17 2021 at 16:16):

You can see that if you take bounded linear maps with larger and larger norm from Hilbert spaces $H_i$ to a Hilbert space $H$, you can't put them together and get a bounded linear map from $\bigoplus_i H_i$ to $H$.

Owen Lynch (Nov 17 2021 at 16:16):

Ah right! So maybe we need to think about normed-space enriched categories!

Owen Lynch (Nov 17 2021 at 16:16):

Where you can only take bounded limits!

John Baez (Nov 17 2021 at 16:17):

Owen Lynch said:

Is there a subtlety that fixes this, or is it just something that we have to live with?

So far everyone has decided that analysis is cursed from the viewpoint of category theory, and category theory is cursed from the viewpoint of analysis. Someday someone will solve this problem.

Owen Lynch (Nov 17 2021 at 16:18):

I'm very interested in fixing this problem, but I need to avoid it so I stay focused on thermodynamics :upside_down:

John Baez (Nov 17 2021 at 16:19):

Yes, don't try to solve it before you get your masters diploma on the wall.

Owen Lynch (Nov 17 2021 at 16:20):

:thumbs_up:

John Baez (Nov 17 2021 at 16:20):

This sort of problem is why most work on categorical quantum mechanics studies only "kindergarten quantum mechanics", meaning finite-dimensional Hilbert spaces. They argue that's good enough for quantum computers... and that's probably true as long as you don't actually try to build one.

John Baez (Nov 17 2021 at 16:21):

But some poor engineer out there is gonna have to think about Schrodinger's equation. :upside_down:

Owen Lynch (Nov 17 2021 at 16:21):

You tell me not to try and solve this before I have my masters diploma on the wall, and then dangle that juicy problem in front of me??

John Baez (Nov 17 2021 at 16:22):

You have to have some reason to live after you've solved all the foundational problems of thermodynamics.

Owen Lynch (Nov 17 2021 at 16:22):

You make some good points

John Baez (Nov 17 2021 at 16:24):

Heh. Now I have to fix our paper.

But I think you might want to think about this $\exp(x) = \sum x^n / S_n$ construction for measure spaces, and how that applies to stat mech.

Owen Lynch (Nov 22 2021 at 20:49):

I've been thinking about this a bit, and I wonder if it's connected to the derivation of Shannon entropy from empirical frequencies

Owen Lynch (Nov 22 2021 at 20:50):

I.e., if $X$ is a finite set, then two elements of $X^n/S_n$ are equal if and only if they have the same empirical frequency

Owen Lynch (Nov 22 2021 at 20:51):

And the preimage of any element in the map $X^n \to X^n/S_n$ has a cardinality of a multinomial coefficient

John Baez (Nov 22 2021 at 21:41):

I suspect that when we finally understand what Hong Qian is talking about here, it'll be related to this!

Owen Lynch (Dec 01 2021 at 21:36):

I'm now thinking that the statmech version of Fock space may be the space of point processes.

Owen Lynch (Dec 02 2021 at 18:11):

Specifically, a statistical-mechanical ideal gas should be a point process on the phase space of a single molecule.

Owen Lynch (Dec 02 2021 at 18:12):

The grand canonical ensemble would be a Poisson point process

Owen Lynch (Dec 02 2021 at 18:15):

Collisions with the wall of a container should be a point process on the space of {position on the wall} x {velocity at impact} x {time of impact}.

Owen Lynch (Dec 02 2021 at 18:19):

Microscopically, I think that heat transfer should be mediated by kinetic energy transfer into the wall, and pressure should be mediated by momentum transfer into the wall. These each could be modeled with jump processes, maybe.