Dynamic Maximum Entropy Reduction

Mixed-Up-Ness or Entropy?

Mixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect "order" to states of total "disorder". Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems.

Entropy 2021 Best Paper Award

On behalf of the Editor-in-Chief, Prof [...].

Multiscale Thermodynamics

Multiscale thermodynamics is a theory of the relations among the levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case, but it is applicable to both static and time evolving processes in externally and internally driven macroscopic systems that are far from equilibrium and are investigated at the microscopic, mesoscopic, and macroscopic levels. In this paper we formulate multiscale thermodynamics, explain its origin, and illustrate it in mesoscopic dynamics that combines levels.

Dynamic and Renormalization-Group Extensions of the Landau Theory of Critical Phenomena

We place the Landau theory of critical phenomena into the larger context of multiscale thermodynamics. The thermodynamic potentials, with which the Landau theory begins, arise as Lyapunov like functions in the investigation of the relations among different levels of description. By seeing the renormalization-group approach to critical phenomena as inseparability of levels in the critical point, we can adopt the renormalization-group viewpoint into the Landau theory and by doing it bring its predictions closer to results of experimental observations.

The fourth law of thermodynamics: steepest entropy ascent

When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose 'greatness' derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The second law of thermodynamics is universally contemplated among the great laws of Nature. In this paper, we show that in the past four decades, an enormous body of scientific research devoted to modelling the essential features of non-equilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature that we propose to call the 'fourth law of thermodynamics'. We state it as follows: every non-equilibrium state of a system or local subsystem for which entropy is well defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. To illustrate the power of the fourth law, we derive (nonlinear) extensions of Onsager reciprocity and fluctuation-dissipation relations to the far-non-equilibrium realm within the framework of the rate-controlled constrained-equilibrium approximation (also known as the quasi-equilibrium approximation). This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Gradient and GENERIC time evolution towards reduced dynamics

Reduction of a mesoscopic dynamical theory to equilibrium thermodynamics brings to the latter theory the fundamental thermodynamic relation (i.e. entropy as a function of the thermodynamic state variables). The reduction is made by following the mesoscopic time evolution to its conclusion, i.e. to fixed points at which the time evolution ceases to continue. The approach to fixed points is driven by entropy, that, if evaluated at the fixed points, becomes the thermodynamic entropy. Since the fixed points are parametrized by the thermodynamic state variables (by constants of motion), the thermodynamic entropy arises as a function of the thermodynamic state variables and thus the final outcome of the reduction is the fundamental thermodynamic relation. This reduction process extends also to reductions in which the reduced theory still involves the time evolution (e.g. reduction of kinetic theory to hydrodynamics). The essence of the extension is the replacement of the mesoscopic time evolution of the state variables with the corresponding mesoscopic time evolution of the vector field (i.e. of the fluxes). The fixed point in this flux time evolution is the vector field generating the reduced mesoscopic time evolution. The flux-entropy driving the flux time evolution becomes, if evaluated at the fixed point, the flux fundamental thermodynamic relation in the reduced dynamical theory. We show that the flux-entropy is a potential related to the entropy production. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.