The second entropy: a general theory for non-equilibrium thermodynamics and statistical mechanics

Abstract

The second entropy is introduced, which is a new type of entropy that provides a basis for the non-equilibrium thermodynamics of time-dependent systems. Whereas the first or ordinary entropy counts the molecular configurations associated with a given structure, the second entropy counts the molecular configurations associated with a transition between two given structures in a specified time. Maximization of the second entropy gives the optimum rate of change or flux, and as such it provides a quantitative principle for non-equilibrium systems. In contrast, the second law of thermodynamics only provides a direction for change, not a rate of change. The probability distribution function for time-dependent systems is also given, which is the focal point of non-equilibrium statistical mechanics. This and the second entropy are used, for example, to derive the Langevin equation, the Green-Kubo relations, the transition and path probability, the fluctuation and work theorems, and a generalised fluctuation-dissipation theorem. They are also used to develop computer simulation algorithms suited for time-dependent systems, specifically non-equilibrium Monte Carlo and stochastic Molecular Dynamics. The analysis is illustrated and quantitatively tested for the case of steady heat flow, and for the case of time-varying, driven, Brownian motion.

Entropy Production from the Master Equation for Driven Lattice Gases

“…An alternative principle for selecting transitions, the principle of maximum second entropy, [13] will be explored in future work. The second or dynamical entropy is the number of configurations connected by a transition between macrostates in a specified time [44] For the driven lattice gas, a macrostate is a set of configurations that share a macroscopic property such as internal entropy.…”

“…The hypothetical principle of maximum entropy production [4][5][6][7][8][9][10] or the converse principle of minimum entropy production would suggest that entropy production may be not just a property but an organizing principle of driven systems such as the DLG. Recent discussions of the role of entropy production in non-equilibrium systems include those by Ross, Vellela and Qian and Attard [11][12][13] A critical review of entropy-production ideas, giving a thorough historical perspective from Carnot's work through the present, was written by Velasco, García-Colín and Uribe [14] The present work is a largely numerical study of the meaning and possible extrema of entropy production in the simple, well-defined DLG model. If DLG steady states are characterized by extreme entropy production then entropy production will vary monotonically, either steadily rising or steadily falling on the way to steady state.…”

“…Dividing by g i is equivalent to subtracting S i =k B ln(g i ) from each term in the sum [13,38] Equivalently, dividing by g i makes the sum over representations equal to the more fundamental sum over configurations, which have equal a priori probability.…”

Thermodynamics and Fluctuations Far From Equilibrium

“…Keizer formulated a stochastic approach for the relaxation to stationary states and fluctuations around a single stationary state [32]; he assumed Gaussian fluctuations, limited to small fluctuations related to linearized kinetics (for chemical kinetics). There are several approaches to the statistical mechanics of stationary states and fluctuating hydrodynamics (see [33] and references therein); some consist of the addition of Gaussian fluctuations to the linearized Navier-Stokes equations [34,35]. Here the thermodynamics may be sufficient for systems approaching equilibrium but not for stationary states far from equilibrium.…”

Entropy vs. Majorization: What Determines Complexity?

“…Attard [18,19] proposed the concept of second entropy to introduce time specifically. His work developed a variational principle, based on maximizing the so-called transition entropy, which is "the number of molecular configurations associated with a transition between macrostates in a specified time".…”