Maximum entropy thermodynamics

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. 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' 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, Boltzmann's 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's 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.