Entropy: The Markov Ordering Approach

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of the various entropy additivity properties and separation of variables for independent subsystems in MaxEnt problem is added in Section 4.2. Bibliography is extende

Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model

We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume-Emery-Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical entropy is nonconcave, the microcanonical distributions of states are nowhere realized in the canonical ensemble. Moreover, we show that for values of the mean energy where the microcanonical entropy is strictly concave, the equilibrium microcanonical distributions of states can be put in one-to-one correspondence with equivalent canonical equilibrium distributions of states. Our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by Ellis, Haven, and Turkington [J. Stat. Phys. 101, 999 (2000)].Comment: 13 pages, 4 figures, minor typos corrected and one reference adde

The Nonequilibrium Thermodynamics of Small Systems

The interactions of tiny objects with their environment are dominated by thermal fluctuations. Guided by theory and assisted by micromanipulation tools, scientists have begun to study such interactions in detail.Comment: PDF file, 13 pages. Long version of the paper published in Physics Toda

Maxallent: Maximizers of all Entropies and Uncertainty of Uncertainty

The entropy maximum approach (Maxent) was developed as a minimization of the subjective uncertainty measured by the Boltzmann--Gibbs--Shannon entropy. Many new entropies have been invented in the second half of the 20th century. Now there exists a rich choice of entropies for fitting needs. This diversity of entropies gave rise to a Maxent "anarchism". Maxent approach is now the conditional maximization of an appropriate entropy for the evaluation of the probability distribution when our information is partial and incomplete. The rich choice of non-classical entropies causes a new problem: which entropy is better for a given class of applications? We understand entropy as a measure of uncertainty which increases in Markov processes. In this work, we describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). For inference, this approach results in a set of conditionally "most random" distributions. Each distribution from this set is a maximizer of its own entropy. This "uncertainty of uncertainty" is unavoidable in analysis of non-equilibrium systems. Surprisingly, the constructive description of this set of maximizers is possible. Two decomposition theorems for Markov processes provide a tool for this description.Comment: 23 pages, 4 figures, Correction in Conclusion (postprint

A morphological study of cluster dynamics between critical points

We study the geometric properties of a system initially in equilibrium at a critical point that is suddenly quenched to another critical point and subsequently evolves towards the new equilibrium state. We focus on the bidimensional Ising model and we use numerical methods to characterize the morphological and statistical properties of spin and Fortuin-Kasteleyn clusters during the critical evolution. The analysis of the dynamics of an out of equilibrium interface is also performed. We show that the small scale properties, smaller than the target critical growing length $\xi(t) \sim t^{1/z}$ with $z$ the dynamic exponent, are characterized by equilibrium at the working critical point, while the large scale properties, larger than the critical growing length, are those of the initial critical point. These features are similar to what was found for sub-critical quenches. We argue that quenches between critical points could be amenable to a more detailed analytical description.Comment: 26 pages, 13 figure

Creating conditions of anomalous self-diffusion in a liquid with molecular dynamics

We propose a computational method to simulate anomalous self-diffusion in a simple liquid. The method is based on a molecular dynamics simulation on which we impose the following two conditions: firstly, the inter-particle interaction is described by a soft-core potential and secondly, the system is forced out of equilibrium. The latter can be achieved by subjecting the system to changes in the length scale at intermittent times. In many respects, our simulation system bears resemblance to slowly driven sandpile models displaying self-organised criticality. We find non-Gaussian single time step displacement distributions during the out-of-equilibrium time periods of the simulation.Comment: Extended version: 12 pages, 9 figure

Non-Equilibrium Processes in the Solar Corona, Transition Region, Flares, and Solar Wind \textit{(Invited Review)}

We review the presence and signatures of the non-equilibrium processes, both non-Maxwellian distributions and non-equilibrium ionization, in the solar transition region, corona, solar wind, and flares. Basic properties of the non-Maxwellian distributions are described together with their influence on the heat flux as well as on the rates of individual collisional processes and the resulting optically thin synthetic spectra. Constraints on the presence of high-energy electrons from observations are reviewed, including positive detection of non-Maxwellian distributions in the solar corona, transition region, flares, and wind. Occurrence of non-equilibrium ionization is reviewed as well, especially in connection to hydrodynamic and generalized collisional-radiative modelling. Predicted spectroscopic signatures of non-equilibrium ionization depending on the assumed plasma conditions are summarized. Finally, we discuss the future remote-sensing instrumentation that can be used for detection of these non-equilibrium phenomena in various spectral ranges.Comment: Solar Physics, accepte