Quantum analog of Boltzmann entropy?

[Demystifier]

In classical statistical physics, entropy can be defined either as Boltzmann entropy or Gibbs entropy. In quantum statistical physics we have von Neumann entropy, which is a quantum analog of Gibbs entropy. Is there a quantum analog of Boltzmann entropy?

 

[vanhees71]

I'd say the "Gibbs entropy", in connection with information theory also known as "Shannon entropy", in the context of information-theoretical applications in statistical physics as "Shannon-Jaynes entropy" and specifically in connection with QT as "von Neumann entropy", is the general concept, and boltzmann entropy the special application to the microcanonical ensemble in thermal equilibrium.

 

[Demystifier]

boltzmann entropy the special application to the microcanonical ensemble in thermal equilibrium.

But Botzmann entropy can also be defined out of equilibrium. More importantly, out of equilibrium, Boltzmann and Gibbs entropy are different.

 

[Demystifier]

In the meantime, I found interesting papers, which I'm reading now:

https://arxiv.org/abs/1903.11870

https://www.math.uni-tuebingen.de/de/forschung/maphy/personen/roderichtumulka/talks/laquila20
21.pdf

https://arxiv.org/abs/1909.00961

 

[vanhees71]

But Botzmann entropy can also be defined out of equilibrium. More importantly, out of equilibrium, Boltzmann and Gibbs entropy are different.

It depends what you understand under "Boltzmann entropy". I thought it's just for the microcanonical ensemble $$S=-k_{\text{B}} \ln \Omega,$$
where $\Omega$ is the dimension of the subspace with fixed additive conserved quantities, which is an equilibrium state.

The Shannen-Jaynes entropy,
$$S=-k_{\text{B}} \mathrm{Tr} (\hat{\rho} \ln \hat{\rho}),$$
is general, and applies to all kinds of states, $\hat{\rho}$, be they equilibrium or off-equilibrium.

 

[Demystifier]

It depends what you understand under "Boltzmann entropy".

See the first paper in #4.

 

[vanhees71]

I see, but still with this definition the Boltzmann entropy is not the most general case. It uses only the one-particle distribution function, and the the H-theorem is derived from the Boltzmann equation, which already is "coarse grained" by "cutting" the BBGKY hierarchy by throwing away correlations, and that's where the increase of Boltzmann entropy comes from. Imho, the Gibbs entropy is the more general notion.

In this sense the "quantum analogy" of Boltzmann entropy is just given by "counting" a la Planck, Einstein, Fermi, and Dirac states in with single-particle occupation, neglecting two-particle and higher correlations, which leads to
$$S=\pm \frac{g}{(2 \pi \hbar)^3} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \mathrm{d}^3 x \mathrm{d}^3 p [f \ln f \pm (1\pm f) \ln(1 \pm f)],$$
where the upper sign stands for bosons the lower for fermions.

For this the H-theorem follows for the BUU collision term (including Bose enhancement/Pauli blocking factors). For details, see Sect. 1.8 in

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf

 

[Demystifier]

It uses only the one-particle distribution function

I don't know why do you say that. Before Eq. (1) it is said that the phase space of $N$ particles is considered. The Boltzmann entropy is defined in Eq. (2), where $X$ is the point in this phase space.

 

[vanhees71]

I don't know why do you say that. Before Eq. (1) it is said that the phase space of $N$ particles is considered. The Boltzmann entropy is defined in Eq. (2), where $X$ is the point in this phase space.

Yes, but then I read further, how the authors finally apply it.

 

[Demystifier]

Yes, but then I read further, how the authors finally apply it.

Can you be more specific? Which page or equation applies it to one particle only?

 

[vanhees71]

Sec. 5.4. The title of the section is telling!

 

[Demystifier]

Sec. 5.4. The title of the section is telling!

But this is just one example (as the title is telling). From one example it does not follow that they study only one-particle distributions. In most of the paper they study $N$-particle phase space.

 

[vanhees71]

I have to read this book chapter in detail, but I don't see, why one should prefer a less general concept over a general one. Of course, you can extend the study of (semi-classical) transport including higher-order correlations. Sometimes that's even mandatory, e.g., when there are a long-ranged interactions like electromagnetic interactions (leading to the Vlasov-BUU equations). The most general approach, using the Schwinger-Keldysh formalism (which is the most convenient one; another is the quantum-Liouville approach) can be found here:

K. Chou, Z. Su, B. Hao and L. Yu, Equilibrium and
Nonequilibrium Formalisms made unified, Phys. Rept. 118, 1
(1985), https://doi.org/10.1016/0370-1573(85)90136-X

 

[Demystifier]

but I don't see, why one should prefer a less general concept over a general one.

Their argument is that, in classical physics, Boltzmann is better because Boltzmann entropy is a function of the actual position in the phase space, while Gibbs entropy is a functional of the probability distribution. This also explains why Gibbs entropy has a natural analog in quantum physics (because quantum physics is intrinsically about probability distributions), while it is not so obvious what is a quantum analog of Boltzmann entropy (because it's not clear what's the "actual" state in quantum physics). Indeed, their quantum Boltzmann entropy is clearly defined only in some interpretations, such as the Bohmian one.

They also point out that von Neumann (i.e. quantum Gibbs) entropy is unphysical in some cases. For example for a Schrödinger cat, which has equal probabilities of being dead (i.e. cold) and alive (i.e. warm), the von Neumann entropy is
$$S=\frac{1}{2}S_{\rm cold}+ \frac{1}{2}S_{\rm warm}$$
while the actual entropy is either $S_{\rm cold}$ or $S_{\rm warm}$.

 

[vanhees71]

Of course, the conceptional difficulties of classical statistical mechanics are greater than in quantum statistical physics. Any only-classical theory of matter is conceptionally difficult, and I don't see the merit of developing an ab-initio classical statistical mechanics, when it can be derived much better from the appropriate approximations of quantum statistical mechanics, and there what's most natural indeed is the Shannon-Jaynes-von Neumann entropy as the starting point.

Their argument that for a closed system entropy is constant due to unitary time evolution in QT is also not convincing, because the same holds for Boltzmann entropy, when looking at a closed system due to Liouville's theorem, according to which phase-space volume is conserved.

 

[Demystifier]

Their argument that for a closed system entropy is constant due to unitary time evolution in QT is also not convincing, because the same holds for Boltzmann entropy, when looking at a closed system due to Liouville's theorem, according to which phase-space volume is conserved.

The Boltzmann entropy as they define it is not constant, because they don't define it as you think they do. They consider a single microscopic trajectory $X(t)$ in the phase space, not an ensemble of trajectories. At each time $t$ the state is a single point in the phase space, so its volume is zero and the Liouville's theorem is irrelevant. The finite phase-space volumes are introduced in a totally different way: they divide the whole phase space into finite cells, so that each cell corresponds to one possible mAcroscopic state. A possible mAcroscopic state becomes an actual mAcroscopic state when the microscopic $X$ actually arrives into that mAcroscopic cell. The actual mAcroscopic state is changed when the microscopic state $X$ arrives from one mAcroscopic cell to another. The Boltzmann entropy of macroscopic state is defined as (Boltzmann constant times the logarithm of) volume of the cell, so the entropy increases when the $X$ arrives from a smaller cell to a larger one. The size of $X$ is always the same (and is zero), but what changes is which cell is filled with $X$.(*)

(*)A nice analogy is a car in the city. The size of the car is constant and negligible compared to the size of the city. But the car travels through the city, so often arrives from a smaller city district to a larger one.

 

[Fra]

Interesting question. I'll add my oddball perspective, although I've not sure if it's what you were after.

Their argument is that, in classical physics, Boltzmann is better because Boltzmann entropy is a function of the actual position in the phase space, while Gibbs entropy is a functional of the probability distribution. This also explains why Gibbs entropy has a natural analog in quantum physics (because quantum physics is intrinsically about probability distributions), while it is not so obvious what is a quantum analog of Boltzmann entropy (because it's not clear what's the "actual" state in quantum physics). Indeed, their quantum Boltzmann entropy is clearly defined only in some interpretations, such as the Bohmian one.

FWIW, I also find the boltzmann version more general if I most generally associates the possible complexions with the agents microstates - given it's inferred expectation defined as (in simplest possible case) for example historical frequencies of distinguishable events as received from the environment. So conceptually this is the "boundary" or communication channel ot other agents in the environment. The unknown microstate correspondens then to the HIDDEN microstate of the agent. (MAYBE this will take the role of your solipsist HV, even if our abstractions/interpretations are quite different?).

This is something one can toy around it, and it's a key idea also in my preferred interpretation. We know already there are many attempts to explain some interactions, as "entropic" in nature. For example Ted Jacobson's entropic gravity. Also Ariel Caticha has had such ambitions to derive the laws of physics and GR in particular from the rules of inference (entropic methods). But most of the attemps I am aware of have a fatal flaw, that somewhere (depending on the specific model) there are some arbitrary - hard to justify empirically - prior distributions or ergodic assumptions etc. I think these essentially become sa fine tuning problem conceptually. But what if one tries a similar thing, but uses agent/observer defined, measures? I find this quite interesting and promising. It gets more complicated, and objectionaly to some but is interesting.

To associate to Bohmian mechanics, one can ask, how can a theory about something hidden or unobsevable possible add explanatory power? I think the answer lies not from the ambigous choices, but what happens when ambigous choices are allowed to interact and spontaneous evolutionar selections maybe takes place? After all in quantum gravity we have the black hole information paradoxes, so something strange likely has to be going on anway. Can we expect a simple explanation to weird things?

/Fredrik

Edit: for some reason i assumed this was in interpretation subforum but now i see its not, i probably took the wrong spin on this.

 

[Demystifier]

To associate to Bohmian mechanics, one can ask, how can a theory about something hidden or unobsevable possible add explanatory power?

Just look at the history of statistical mechanics. How can Boltzmann hypothesis that thermodynamics arises from motion of invisible atoms possibly add explanatory power? At that time (late 19th century) there was no direct evidence for the existence of atoms, and Mach indeed argued that Boltzmann statistical mechanics based on assumption of atoms was a nonsense. It was Einstein in early 20th century (1905) who finally convinced physicists that atoms actually exist, by explaining the Brownian motion from the assumption of atoms.

 

[Demystifier]

Edit: for some reason i assumed this was in interpretation subforum but now i see its not, i probably took the wrong spin on this.

Initially, I was mainly interested in a technical and practical definition of quantum Boltzmann entropy. But of course, the question also evokes some foundational issues.

 

[Fra]

The Boltzmann entropy as they define it is not constant, because they don't define it as you think they do. They consider a single microscopic trajectory $X(t)$ in the phase space, not an ensemble of trajectories. At each time $t$ the state is a single point in the phase space, so its volume is zero and the Liouville's theorem is irrelevant. The finite phase-space volumes are introduced in a totally different way: they divide the whole phase space into finite cells, so that each cell corresponds to one possible mAcroscopic state. A possible mAcroscopic state becomes an actual mAcroscopic state when the microscopic $X$ actually arrives into that mAcroscopic cell. The actual mAcroscopic state is changed when the microscopic state $X$ arrives from one mAcroscopic cell to another. The Boltzmann entropy of macroscopic state is defined as (Boltzmann constant times the logarithm of) volume of the cell, so the entropy increases when the $X$ arrives from a smaller cell to a larger one. The size of $X$ is always the same (and is zero), but what changes is which cell is filled with $X$.(*)

I lost track of what you referenced to, which paper did this come from.
It also sounds closer to what may make sense for an agent view; where in the simplest of cases one can consider that the angent can hold say a string of inputs at any given time. Loosely speaking we can call it X(t) except t would likely be discrete. From this, internal processing can take place and the agent can compute from the one data it has, expectations of future strings or substrings, and assign macrostates as "relative frequences". This way allows a rational way for the statistics witohut ensembles, the effective scores can be defined with the one data set in X(i) - the observers memory essentially. This is how I picture the basal starting points of agents microstates. I was curious to see if this is more than a conincidence.

So which paper did you get this from?

/Fredrik

 

[Demystifier]

So which paper did you get this from?

The first one in #4.

 

[vanhees71]

I find this is a lot of talk but no clear exposition of what the authors criticize about the standard definition of entropy a la Shannon, Jaynes, and von Neumann. That the H-theorem holds for closed systems only when a macroscopic coarse-grained description is used, follows from this more general concept of entropy. If you keep full information about the state of the closed system, its time evolution is described by unitary time evolution and thus entropy is conserved. If you refer to the time evolution of an open system, usually you get the H-theorem, i.e., that the corresponding entropy increases, and equilibrium is thus characterized by maximum entropy (under the given constraints about additive conserved quantities).

 

[renormalize]

I find this is a lot of talk but no clear exposition of what the authors criticize about the standard definition of entropy a la von Shannon, Jaynes, and von Neumann.

Did vanhees71 just promote Claude Shannon from commoner to German nobleman?

 

[vanhees71]

Of the three said scientists, only von Neuman was a (Hungarian) noble man.

 

[Fra]

The first one in #4.

Many interesting problems are raised and not unexpectedly the questions of which is the best IMO is rooted again in the interpreration of probability. After all, entropy is conceptually a way to make the factorization of statistical independence, instead additive, and sometimes to quantity how much memory is required to encode information etc. But to understnad the details, I find that one is led back to the foundations and interpretation of probability, and when it comes to these things I don't share the authors apparent critique against the "subjective entropy" interpretation. In chapter for they raise some valid issues with this, such as cases of "wrong entropy", or the lack of explanatory value as the premise appears abitrary dependng on the choice of obsevers.

From my perspective, the problems they raise must be taken seriously, but I see that they are potentially conceptually solved by a factor that they miss - the choice of observer is not completely arbitrary, because while any observer is "allowed", an observer that is flat out and grossly wrong, are not likely to be stable. This idea is completely LOST when one considers a fixed macroscopic background, which is the standard way. So I think the root of this entropy issue, is pretty much the same as other fountational questions on QM like the observer problem.

/Fredrik

 

[Demystifier]

Of the three said scientists, only von Neuman was a (Hungarian) noble man.

You wrote "von Shannon" in #22.

 

[vanhees71]

Thanks, I corrected it :-).

 

[Demystifier]

I find this is a lot of talk but no clear exposition of what the authors criticize about the standard definition of entropy a la Shannon, Jaynes, and von Neumann.

Note that the authors are Bohmians, so they have a very different mindset than a typical practical physicist like you, which explains why you have difficulties with understanding their point. It's not about different interpretations of QM (most of the paper is about classical statistical mechanics, anyway), but about general way of thinking in physics. You mainly ask practical epistemic questions, while they ask foundational ontological questions. From their point of view, the most important quantity is the exact position and momentum of every particle, in a system of large number $N$ (typically, $N\sim 10^{23}$) of particles, and everything else should be defined in terms of this. From your point of view, even though you will accept that in classical physics one can talk about exact positions and momenta in principle, it does not make much sense in practice when $N$ is so big. That's why you find the Gibbs point of view much more natural and can't easily understand why they prefer the Boltzmann point of view. Loosely speaking, Boltzmann entropy is better suited for foundational ontological questions, while Gibbs entropy is better suited for practical epistemic questions. And sometimes physicists (including me) are not certain whether a question is foundational or practical, which is a part of the reason why they often get confused by different notions of entropy.

And of course, the Boltzmann vs Gibbs is not the only dilemma that causes a confusion. There is also the fine grained vs coarse grained dilemma; the frequentist vs Bayesian probability dilemma; the choice of ensemble (microcanonical, canonical, grand-canonical, etc.) problem; the real vs imagined ensemble; the difference between notions of ensemble, probability and typicality; the $N$-particle phase space vs 1-particle phase space; the number of particles in the system vs the number of systems in the ensemble; the equilibrium vs non-equilibrium dilemma; the open vs closed system dilemma; and last but not least, the classical vs quantum statistical physics. There are so many levels at which someone can get confused in statistical mechanics.

 

[vanhees71]

Note that the authors are Bohmians, so they have a very different mindset than a typical practical physicist like you, which explains why you have difficulties with understanding their point. It's not about different interpretations of QM (most of the paper is about classical statistical mechanics, anyway), but about general way of thinking in physics. You mainly ask practical epistemic questions, while they ask foundational ontological questions. From their point of view, the most important quantity is the exact position and momentum of every particle, in a system of large number $N$ (typically, $N\sim 10^{23}$) of particles, and everything else should be defined in terms of this. From your point of view, even though you will accept that in classical physics one can talk about exact positions and momenta in principle, it does not make much sense in practice when $N$ is so big. That's why you find the Gibbs point of view much more natural and can't easily understand why they prefer the Boltzmann point of view. Loosely speaking, Boltzmann entropy is better suited for foundational ontological questions, while Gibbs entropy is better suited for practical epistemic questions. And sometimes physicists (including me) are not certain whether a question is foundational or practical, which is a part of the reason why they often get confused by different notions of entropy.

That explains a lot. Bohmians tend to preach their gospel... It's clear that within QT exact phase-space points don't make sense and that description of the state become useful when coarse-graining over phase-space cells much larger than $\hbar^{2N}$ is a sufficient description, i.e., if there's a separation of microscopic and macroscopic scales, and you are interested only in the macroscopic coarse-grained observables.

And of course, the Boltzmann vs Gibbs is not the only dilemma that causes a confusion. There is also the fine grained vs coarse grained dilemma; the frequentist vs Bayesian probability dilemma; the choice of ensemble (microcanonical, canonical, grand-canonical, etc.) problem; the real vs imagined ensemble; the difference between notions of ensemble, probability and typicality; the $N$-particle phase space vs 1-particle phase space; the number of particles in the system vs the number of systems in the ensemble; the equilibrium vs non-equilibrium dilemma; the open vs closed system dilemma; and last but not least, the classical vs quantum statistical physics. There are so many levels at which someone can get confused in statistical mechanics.

At least the applicability of the standard ensembles is pretty clear. If you have rather small systems, you need the microcanonical ensemble. If you have large energies but not so much (net) conseved charge you need the canonical and otherwise the grand-canonical ensemble. It depends on, whether the fluctuations of the macroscopic quantities of interest are large compared to the mean values or not, which ensemble is the better choice.

 

[Fra]

You mainly ask practical epistemic questions, while they ask foundational ontological questions.

I think both ways are complementary and there is no need for conflict. I think the tension arises only because the "observer" is not taken as the active particpant and part of physical reality that is obviously is. In fact I see the agent perspective as aiming to unify this in that the ontology is motivated epistemologically via inference, and simultaneously the set of all "possible inferences" are constrained by the ontology, so both parts evolve in interaction.

/Fredrik