The variational principles called maximum entropy (MaxEnt) and maximum caliber (MaxCal) are reviewed. MaxEnt originated in the statistical physics of Boltzmann and Gibbs, as a theoretical tool for predicting the equilibrium states of thermal systems. Later, entropy maximization was also applied to matters of information, signal transmission, and image reconstruction. Recently, since the work of Shore and Johnson, MaxEnt has been regarded as a principle that is broader than either physics or information alone. MaxEnt is a procedure that ensures that inferences drawn from stochastic data satisfy basic self-consistency requirements. The different historical justifications for the entropy $S=\ensuremath{-}\ensuremath{\sum}_{i}{p}_{i}\mathrm{log}﻿{p}_{i}$ and its corresponding variational principles are reviewed. As an illustration of the broadening purview of maximum entropy principles, maximum caliber, which is path entropy maximization applied to the trajectories of dynamical systems, is also reviewed. Examples are given in which maximum caliber is used to interpret dynamical fluctuations in biology and on the nanoscale, in single-molecule and few-particle systems such as molecular motors, chemical reactions, biological feedback circuits, and diffusion in microfluidics devices.

Principles of maximum entropy and maximum caliber in statistical physics

Thermodynamic Theory of Structure, Stability and Fluctuations. By P. Glansdorff and I. Prigogine. Pp. xxiii + 306. (Wiley: New York and London, June 1971.) £5.50.

The Fourth Law of Thermodynamics

'There exists an additive function of state S, called the entropy, and a set of additive functions of state, called components of content, such ,that, for any states a and b, a -b if, and only if S(a) :s;;S(b), and Q(a) = Q(b) for every component of content Q." Somehow this does not sound familiar. In fact, this is the fundamental theorem of a thermodynamics of Giles (1). Only three primitive concepts are necessary: that of a !>tate, tha.t of addition (+), and that of the relation beween two states a and obsuch that there exists a natural process leading from a to b (a --4 b). By using only these th ee notions, a coherent structure of c!assical thermodynamics has been constructed; the results are equivalent to those obtained by other, more complicated approach<. This was done some years ago. Yet, relatively very little aHention has been paid to Giles's construction. This is surprising, as users of thermodynamics are a multitude. Many of these, however, act under a mistaken assumption, that the scope and contents of thermodynamics are time-independent. Thermodynamics is relatively old, compared

Between Laws of Thermodynamics and Coding of Information: There is nothing wrong with Carnot's cycle, but contemporary thermodynamics may be more useful.

Distinctions are drawn between various contrasting approaches to thermodynamics: it appears that thermodynamics is to be regarded as a plural discipline. Different types of thermodynamic theory are distinguished and classified, with particular reference to the distinction between a system of thermodynamics in which mechanics and other branches of macroscopic science are presupposed at the start, and a system of thermodynamics in which this is not the case.

https://google.iopscience.iop.org/article/10.1088/0143-0807/1/2/003/pdf

Conceptually distinct types of thermodynamics

The view is advocated that the basic ideas of thermodynamics are the same for all students, engineers, chemists and physicists, while the applications used should differ. What are these basic ideas? It is suggested that they are the four (phonetic) ‘F’s — phase space, Pfaffians, fluids and fluxes, including some statistical interpretation and a clear understanding of inexact differentials. Sample arguments are given.

One World — One Thermodynamics

A presentation is given of the form taken by the thermodynamic theory of homogeneous chemical equilibrium when transcribed into Landsberg’s (1961) formulation of thermodynamics in terms of set theory. Certain sets of points can usefully be defined, in order to deal separately and explicitly with metastable equilibrium states, and with open systems as such, without having simultaneously to consider non-equilibrium states. ‘Narrower’ and ‘wider’ thermodynamic principles can be distinguished in terms of what is the set for which some proposition is asserted to apply, and Landsberg’s formulation makes possible greater clarity in the enunciation of the special axioms of chemical thermodynamics. In particular, the axiom of the ideal behaviour of dilute systems takes a very clear-cut significance. When arguments of the type introduced by Landsberg are used, the assumptions involved in a formal development of the thermodynamic theory of homogeneous chemical equilibrium are clearly apparent. Metastable equilibrium states, and non-equilibrium states, each have their own distinct place in the theory. Some comments are made in relation to the usual presentation.

Chemical Thermodynamics in Landsberg's Formulation

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Chemical Thermodynamics in Landsberg's Formulation