In physics, maximum entropy thermodynamics (colloquially, MaxEnt thermodynamics) views equilibrium thermodynamics and statistical mechanics as inference processes. More specifically, MaxEnt applies inference techniques rooted in Shannon information theory, Bayesian probability, and the principle of maximum entropy. These techniques are relevant to any situation requiring prediction from incomplete or insufficient data (e.g., image reconstruction, signal processing, spectral analysis, and inverse problems). MaxEnt thermodynamics began with two papers by Edwin T. Jaynes published in the 1957 Physical Review.
Maximum Shannon entropy Central to the MaxEnt thesis is the principle of maximum entropy. It demands as given some partly specified model and some specified data related to the model. It selects a preferred probability distribution to represent the model. The given data state "testable information"Jaynes, E.T. (1968), p. 229.Jaynes, E.T. (1979), pp. 30, 31, 40. about the probability distribution, for example particular expectation values, but are not in themselves sufficient to uniquely determine it. The principle states that one should prefer the distribution which maximizes the Shannon information entropy,
This is known as the Gibbs algorithm, having been introduced by J. Willard Gibbs in 1878, to set up statistical ensembles to predict the properties of thermodynamic systems at equilibrium. It is the cornerstone of the statistical mechanical analysis of the thermodynamic properties of equilibrium systems (see partition function).
A direct connection is thus made between the equilibrium thermodynamic entropy STh, a state function of pressure, volume, temperature, etc., and the information entropy for the predicted distribution with maximum uncertainty conditioned only on the expectation values of those variables:
kB, the Boltzmann constant, has no fundamental physical significance here, but is necessary to retain consistency with the previous historical definition of entropy by Clausius (1865) (see Boltzmann constant).
However, the MaxEnt school argue that the MaxEnt approach is a general technique of statistical inference, with applications far beyond this. It can therefore also be used to predict a distribution for "trajectories" Γ "over a period of time" by maximising:
This "information entropy" does not necessarily have a simple correspondence with thermodynamic entropy. But it can be used to predict features of nonequilibrium thermodynamic systems as they evolve over time.
For non-equilibrium scenarios, in an approximation that assumes local thermodynamic equilibrium, with the maximum entropy approach, the Onsager reciprocal relations and the Green–Kubo relations fall out directly. The approach also creates a theoretical framework for the study of some very special cases of far-from-equilibrium scenarios, making the derivation of the entropy production fluctuation theorem straightforward. For non-equilibrium processes, as is so for macroscopic descriptions, a general definition of entropy for microscopic statistical mechanical accounts is also lacking.
Technical note: For the reasons discussed in the article differential entropy, the simple definition of Shannon entropy ceases to be directly applicable for random variables with continuous probability distribution functions. Instead the appropriate quantity to maximize is the "relative information entropy",
Hc is the negative of the Kullback–Leibler divergence, or discrimination information, of m(x) from p(x), where m(x) is a prior invariant measure for the variable(s). The relative entropy Hc is always less than zero, and can be thought of as (the negative of) the number of bits of uncertainty lost by fixing on p(x) rather than m(x). Unlike the Shannon entropy, the relative entropy Hc has the advantage of remaining finite and well-defined for continuous x, and invariant under 1-to-1 coordinate transformations. The two expressions coincide for discrete probability distributions, if one can make the assumption that m(xi) is uniform – i.e. the principle of equal a-priori probability, which underlies statistical thermodynamics.
Philosophical implications
Adherents to the MaxEnt viewpoint take a clear position on some of the conceptual/philosophical questions in thermodynamics. This position is sketched below.
The nature of the probabilities in statistical mechanics
Jaynes (1985,Jaynes, E.T. (1985). 2003,Jaynes, E.T. (2003). et passim) discussed the concept of probability. According to the MaxEnt viewpoint, the probabilities in statistical mechanics are determined jointly by two factors: by respectively specified particular models for the underlying state space (e.g. Liouvillian phase space); and by respectively specified particular partial descriptions of the system (the macroscopic description of the system used to constrain the MaxEnt probability assignment). The probabilities are objective in the sense that, given these inputs, a uniquely defined probability distribution will result, the same for every rational investigator, independent of the subjectivity or arbitrary opinion of particular persons. The probabilities are epistemic in the sense that they are defined in terms of specified data and derived from those data by definite and objective rules of inference, the same for every rational investigator.Jaynes, E.T. (1979), p. 28. Here the word epistemic, which refers to objective and impersonal scientific knowledge, the same for every rational investigator, is used in the sense that contrasts it with opiniative, which refers to the subjective or arbitrary beliefs of particular persons; this contrast was used by Plato and Aristotle, and stands reliable today.
Jaynes also used the word 'subjective' in this context because others have used it in this context. He accepted that in a sense, a state of knowledge has a subjective aspect, simply because it refers to thought, which is a mental process. But he emphasized that the principle of maximum entropy refers only to thought which is rational and objective, independent of the personality of the thinker. In general, from a philosophical viewpoint, the words 'subjective' and 'objective' are not contradictory; often an entity has both subjective and objective aspects. Jaynes explicitly rejected the criticism of some writers that, just because one can say that thought has a subjective aspect, thought is automatically non-objective. He explicitly rejected subjectivity as a basis for scientific reasoning, the epistemology of science; he required that scientific reasoning have a fully and strictly objective basis.Jaynes, E.T. (1968), p. 228. Nevertheless, critics continue to attack Jaynes, alleging that his ideas are "subjective". One writer even goes so far as to label Jaynes' approach as "ultrasubjectivist",Guttmann, Y.M. (1999), pp. 28, 36, 38, 57, 61. and to mention "the panic that the term subjectivism created amongst physicists".Guttmann, Y.M. (1999), p. 29.
The probabilities represent both the degree of knowledge and lack of information in the data and the model used in the analyst's macroscopic description of the system, and also what those data say about the nature of the underlying reality.
The fitness of the probabilities depends on whether the constraints of the specified macroscopic model are a sufficiently accurate and/or complete description of the system to capture all of the experimentally reproducible behavior. This cannot be guaranteed, a priori. For this reason MaxEnt proponents also call the method predictive statistical mechanics. The predictions can fail. But if they do, this is informative, because it signals the presence of new constraints needed to capture reproducible behavior in the system, which had not been taken into account.
Is entropy "real"?
The thermodynamic entropy (at equilibrium) is a function of the state variables of the model description. It is therefore as "real" as the other variables in the model description. If the model constraints in the probability assignment are a "good" description, containing all the information needed to predict reproducible experimental results, then that includes all of the results one could predict using the formulae involving entropy from classical thermodynamics. To that extent, the MaxEnt STh is as "real" as the entropy in classical thermodynamics.
Of course, in reality there is only one real state of the system. The entropy is not a direct function of that state. It is a function of the real state only through the (subjectively chosen) macroscopic model description.
Is ergodic theory relevant?
The Gibbsian ensemble idealizes the notion of repeating an experiment again and again on different systems, not again and again on the same system. So long-term time averages and the ergodic hypothesis, despite the intense interest in them in the first part of the twentieth century, strictly speaking are not relevant to the probability assignment for the state one might find the system in.
However, this changes if there is additional knowledge that the system is being prepared in a particular way some time before the measurement. One must then consider whether this gives further information which is still relevant at the time of measurement. The question of how 'rapidly mixing' different properties of the system are then becomes very much of interest. 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.
The second law
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 summarizable 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.
Classical thermodynamics is built on the assumption that entropy is a state function of the macroscopic variables—i.e., that none of the history of the system matters, so that it can all be ignored.
The extended, wispy, evolved probability distribution, which still has the initial Shannon entropy STh(1), should reproduce the expectation values of the observed macroscopic variables at time t2. However it will no longer necessarily be a maximum entropy distribution for that new macroscopic description. On the other hand, the new thermodynamic entropy STh(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 6N-dimensional probability distribution, this result represents coarse graining—i.e., information loss by smoothing out very fine-scale detail.
Caveats with the argument
Some caveats should be considered with the above.