Related papers: Gibbs Entropy and Irreversibility
A generalized Kullback-Leibler relative entropy is introduced starting with the symmetric Jackson derivative of the generalized overlap between two probability distributions. The generalization retains much of the structure possessed by the…
There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time-reversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations…
Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits…
The possible distinction between inanimate and living matter has been of interest to humanity since thousands of years. Clearly, such a rich question can not be answered in a single manner, and a plethora of approaches naturally do exist.…
The Boltzmann--Gibbs entropy is a functional on the space of probability measures. When a state space is countable, one characterization of the Boltzmann--Gibbs entropy is given by the Shannon--Khinchin axioms, which consist of continuity,…
We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for $L^1$ measures is obtained as a corollary.…
Entropic cosmology assumes several forms of entropy on the horizon of the universe, where the entropy can be considered to behave as if it were related to the exchange (the transfer) of energy. To discuss this exchangeability, the…
Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to…
The Boltzmann and Gibbs approaches to statistical mechanics have very different definitions of equilibrium and entropy. The problems associated with this are discussed and it is suggested that they can be resolved, to produce a version of…
A dynamical estimate is given for the Boltzmann entropy of the Universe, under the simplifying assumptions provided by Newtonian cosmology. We first model the cosmological fluid as the probability fluid of a quantum-mechanical system. Next,…
Within the Tsallis thermodynamics' framework, and using scaling properties of the entropy, we derive a generalization of the Gibbs-Duhem equation. The analysis suggests a transformation of variables that allows standard thermodynamics to be…
We give a closed-form solution of von Neumann entropy as a function of geometric phase modulated by visibility and average distinguishability in Hilbert spaces of two and three dimensions. We show that the same type of dependence also…
For a Markovian dynamics on discrete states, the logarithmic ratio of waiting-time distributions between two successive, instantaneous transitions in forward and backward direction is a measure of time-irreversibility. It thus serves as an…
A complex interplay between the academic issue about generalization of the thermodynamics and the practical matter about setting standards for a sustainable evolution of both tailored devices and natural systems is considered. It is…
Ordinary Boltzmann-Gibbs entropy is inadequate to be used in systems depending on a control parameter that yield different mean energy values. Such systems fail to give the correct comparison between the off-equilibrium and equilibrium…
Gibbs random fields play an important role in statistics. However they are complicated to work with due to an intractability of the likelihood function and there has been much work devoted to finding computational algorithms to allow…
The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. Independent or weakly interacting variables render the number of configurations scale exponentially with the…
We define a general notion of entropy in elementary, algebraic terms. Based on that, weak forms of a scalar product and a distance measure are derived. We give basic properties of these quantities, generalize the Cauchy-Schwarz inequality,…
We prove the invariance of the Gibbs measure for the defocusing quintic nonlinear Schr\"odinger equation on the real line. This builds on earlier work by Bourgain, who treated the cubic nonlinearity. The key new ingredient is a growth…
The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to…