Manual Predictive Statistical Mechanics: A Nonequilibrium Ensemble Formalism

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Information about some degrees of freedom of the combined system may become unusable very quickly; information about other properties of the system may go on being relevant for a considerable time. If nothing else, the medium and long-run time correlation properties of the system are interesting subjects for experimentation in themselves. Failure to accurately predict them is a good indicator that relevant macroscopically determinable physics may be missing from the model.

According to Liouville's theorem for Hamiltonian dynamics , the hyper-volume of a cloud of points in phase space remains constant as the system evolves. Therefore, the information entropy must also remain constant, if we condition on the original information, and then follow each of those microstates forward in time:.

However, as time evolves, that initial information we had becomes less directly accessible. Instead of being easily summarisable in the macroscopic description of the system, it increasingly relates to very subtle correlations between the positions and momenta of individual molecules.

Compare to Boltzmann's H-theorem. Equivalently, it means that the probability distribution for the whole system, in 6N-dimensional phase space, becomes increasingly irregular, spreading out into long thin fingers rather than the initial tightly defined volume of possibilities.

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Classical thermodynamics is built on the assumption that entropy is a state function of the macroscopic variables —i. The extended, wispy, evolved probability distribution, which still has the initial Shannon entropy S Th 1 , should reproduce the expectation values of the observed macroscopic variables at time t 2. However it will no longer necessarily be a maximum entropy distribution for that new macroscopic description. On the other hand, the new thermodynamic entropy S Th 2 assuredly will measure the maximum entropy distribution, by construction. Therefore, we expect:.

At an abstract level, this result implies that some of the information we originally had about the system has become "no longer useful" at a macroscopic level. At the level of the 6 N -dimensional probability distribution, this result represents coarse graining —i. Like all statistical mechanical results according to the MaxEnt school, this increase in thermodynamic entropy is only a prediction. It assumes in particular that the initial macroscopic description contains all of the information relevant to predicting the later macroscopic state.

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This may not be the case, for example if the initial description fails to reflect some aspect of the preparation of the system which later becomes relevant. In that case the "failure" of a MaxEnt prediction tells us that there is something more which is relevant that we may have overlooked in the physics of the system. It is also sometimes suggested that quantum measurement , especially in the decoherence interpretation, may give an apparently unexpected reduction in entropy per this argument, as it appears to involve macroscopic information becoming available which was previously inaccessible.

However, the entropy accounting of quantum measurement is tricky, because to get full decoherence one may be assuming an infinite environment, with an infinite entropy. The argument so far has glossed over the question of fluctuations. It has also implicitly assumed that the uncertainty predicted at time t 1 for the variables at time t 2 will be much smaller than the measurement error.

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But if the measurements do meaningfully update our knowledge of the system, our uncertainty as to its state is reduced, giving a new S I 2 which is less than S I 1. Note that if we allow ourselves the abilities of Laplace's demon , the consequences of this new information can also be mapped backwards, so our uncertainty about the dynamical state at time t 1 is now also reduced from S I 1 to S I 2. This then leaves open the possibility for fluctuations in S Th. The thermodynamic entropy may go "down" as well as up.


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A more sophisticated analysis is given by the entropy Fluctuation Theorem , which can be established as a consequence of the time-dependent MaxEnt picture. As just indicated, the MaxEnt inference runs equally well in reverse. So given a particular final state, we can ask, what can we "retrodict" to improve our knowledge about earlier states?

However the Second Law argument above also runs in reverse: given macroscopic information at time t 2 , we should expect it too to become less useful.

The two procedures are time-symmetric. But now the information will become less and less useful at earlier and earlier times. Compare with Loschmidt's paradox. The MaxEnt inference would predict that the most probable origin of a currently low-entropy state would be as a spontaneous fluctuation from an earlier high entropy state.

But this conflicts with what we know to have happened, namely that entropy has been increasing steadily, even back in the past. The MaxEnt proponents' response to this would be that such a systematic failing in the prediction of a MaxEnt inference is a "good" thing. If it is correct that the dynamics "are" time-symmetric , it appears that we need to put in by hand a prior probability that initial configurations with a low thermodynamic entropy are more likely than initial configurations with a high thermodynamic entropy.

This cannot be explained by the immediate dynamics. Quite possibly, it arises as a reflection of the evident time-asymmetric evolution of the universe on a cosmological scale see arrow of time. The Maximum Entropy thermodynamics has some important opposition, in part because of the relative paucity of published results from the MaxEnt school, especially with regard to new testable predictions far-from-equilibrium. The theory has also been criticized in the grounds of internal consistency.

Balescu states that Jaynes' and coworkers theory is based on a non-transitive evolution law that produces ambiguous results. Although some difficulties of the theory can be cured, the theory "lacks a solid foundation" and "has not led to any new concrete result". Though the maximum entropy approach is based directly on informational entropy, it is applicable to physics only when there is a clear physical definition of entropy.

There is no clear unique general physical definition of entropy for non-equilibrium systems, which are general physical systems considered during a process rather than thermodynamic systems in their own internal states of thermodynamic equilibrium. This problem is related to the fact that heat may be transferred from a hotter to a colder physical system even when local thermodynamic equilibrium does not hold so that neither system has a well defined temperature.

Classical entropy is defined for a system in its own internal state of thermodynamic equilibrium, which is defined by state variables, with no non-zero fluxes, so that flux variables do not appear as state variables. But for a strongly non-equilibrium system, during a process, the state variables must include non-zero flux variables. Classical physical definitions of entropy do not cover this case, especially when the fluxes are large enough to destroy local thermodynamic equilibrium.


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In other words, for entropy for non-equilibrium systems in general, the definition will need at least to involve specification of the process including non-zero fluxes, beyond the classical static thermodynamic state variables. The 'entropy' that is maximized needs to be defined suitably for the problem at hand. If an inappropriate 'entropy' is maximized, a wrong result is likely.

Thierry Bodineau - Nonequilibrium statistical mechanics & large deviation theory

In principle, maximum entropy thermodynamics does not refer narrowly and only to classical thermodynamic entropy. It is about informational entropy applied to physics, explicitly depending on the data used to formulate the problem at hand. According to Attard, for physical problems analyzed by strongly non-equilibrium thermodynamics, several physically distinct kinds of entropy need to be considered, including what he calls second entropy. Attard writes: "Maximizing the second entropy over the microstates in the given initial macrostate gives the most likely target macrostate.

From Wikipedia, the free encyclopedia. Application of information theory to thermodynamics and statistical mechanics. Physical Review. Bibcode : PhRv.. The entropy of classical thermodynamics, Chapter 8 of Greven, A. Bajkova, A. Astronomical and Astrophysical Transactions. Caticha, Ariel Dewar, R. A: Math. Bibcode : JPhA Grinstein, G. Shows invalidity of Dewar's derivations a of maximum entropy production MaxEP from fluctuation theorem for far-from-equilibrium systems, and b of a claimed link between MaxEP and self-organized criticality.

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A Nonequilibrium Ensemble Formalism

William T. Coffey Yuri P. Didier De Fontaine. Felix L. Chernousko I. Ananievski S.