Abstract: Thermodynamic laws have been found applicable to many systems within the universe. But are they applicable to the universe itself as a whole system? Based on the assumption that they are, we are able to propose a modality rooted in quantum physics which can permit astronomical increases in the entropy, phase-space volume and thus the number of position and momentum coordinates that are available in a system, in spite of any prevailing adiabatic conditions. We conclude that a study of the thermodynamic consequences of energy introduction into a state at low or absolute zero temperature may increase our understanding of any possible cosmic genesis.

Abstract: Recently Bruers proposed a simple setup, illustrating Dewar’s Maximum Entropy Production (MaxEP). The setup is used as a framework for discussing Dewar dice problem – and analogue of the well-known Jaynes dice, – from a probabilistic point view, which rests on Conditional Law of Large Numbers and Maximum Probability/Maximum Entropy asymptotic correspondence. A couple of examples is worked out. It is noted that in Bruers’ setup, MaxEP distribution can be obtained without solving constrained optimization problem, utilizing its independence property.

Abstract: We define an entropy based on a chosen governing probability distribution. If a certain kind of measurements follow such a distribution it also gives us a suitable scale to study it with. This scale will appear as a link function that is applied to the measurements. A link function can also be used to define an alternative structure on a set. We will see that generalized entropies are equivalent to using a different scale for the phenomenon that is studied compared to the scale the measurements arrive on. An extensive measurement scale is here a scale for which measurements fulfill a memoryless property. We conclude that the alternative algebraic structure defined by the link function must be used if we continue to work on the original scale. We derive Tsallis entropy by using a generalized log-logistic governing distribution. Typical applications of Tsallis entropy are related to phenomena with power-law behaviour.

Abstract: A rate equation for a discrete probability distribution is discussed as a route to describe smooth relaxation towards the maximum entropy distribution compatible at all times with one or more linear constraints. The resulting dynamics follows the path of steepest entropy ascent compatible with the constraints. The rate equation is consistent with the Onsanger theorem of reciprocity and the fluctuation-dissipation theorem. The mathematical formalism was originally developed to obtain a quantum theoretical unification of mechanics and thermodinamics. It is presented here in a general, non-quantal formulation as a part of an effort to develop tools for the phenomenological treatment of non-equilibrium problems with applications in engineering, biology, sociology, and economics. The rate equation is also extended to include the case of assigned time-dependences of the constraints and the entropy, such as for modeling non-equilibrium energy and entropy exchanges.
 
Manuscript ID: Entropy-12-05
Type of the Paper: Full Research Paper
Title: Generalised exponential families and associated entropy functions
Author: Jan Naudts
Abstract: A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential families. The well-known result that the inequality of Cram´er and Rao becomes an equality in the case of an exponential family can be generalised. However, this requires the introduction of escort probabilities.
 
Manuscript ID: Entropy-12-06
Type of the Paper: Full Research Paper
Title: Differential entropy relation as an alternative to MaxEnt
Authors:  A. Plastino, A. R. Plastino, E. M. S. Curado and M. Casas
Abstract: We show that, to generate the statistical operator appropriate for a given system, and as an alternative to Jaynes’ MaxEnt approach, that refers to the entropy S, one can use instead the differential dS. To such an effect, one uses the macroscopic thermodynamic relation that links dS to changes in i) the internal energy E and ii) the remaining M relevant extensive quantities A_i, i = 1, ..., M, that characterize the context one is working with.
 
Manuscript ID: Entropy-12-07
Type of the Paper: Full Research Paper
Title: Quesne-like generalization of the semiclassical entropy
Authors: G.L. Ferri, F. Olivares, F. Pennini, A. Plastino, and A. R. Plastino *
Abstract: We explicitly obtain here a novel expression for the semiclassical Wehrl’s entropy using deformed algebras built up with the q--coherent states of Quesne’s [J. Phys. A 2002, 35, 9213]. The generalization is investigated with emphasis on i) its behavior as a function of temperature and ii) the results obtained when the deformation-parameter tends to unity.
 
Manuscript ID: Entropy-12-08
Type of the Paper: Full Research Paper
Title: Estimating the Entropy of Binary Time Series: Methodology, Some Theory and a Simulation Study
Authors: Yun Gao, Ioannis Kontoyiannis, Elie Bienenstock
Abstract: Partly motivated by entropy-estimation problems in neuroscience, we present a detailed and extensive comparison between some of the most popular and effective entropy estimation methods used in practice: The plug-in method, four different estimators based on the Lempel-Ziv (LZ) family of data compression algorithms, an estimator based on the Context-Tree Weighting (CTW) method, and the renewal entropy estimator. METHODOLOGY. Three new entropy estimators are introduced; two new LZ-based estimators, and the “renewal entropy estimator,” which is tailored to data generated by a binary renewal process. For two of the four LZ-based estimators, a bootstrap procedure is described for evaluating their standard error, and a practical rule of thumb is heuristically derived for selecting the values of their parameters in practice. THEORY. We prove that, unlike their earlier versions, the two new LZ-based estimators are universally consistent, that is, they converge to the entropy rate for every finite-valued, stationary and ergodic process. An effective method is derived for the accurate approximation of the entropy rate of a finite-state HMM with known distribution. Heuristic calculations are presented and approximate formulas are derived for evaluating the bias and the standard error of each estimator. SIMULATION. All estimators are applied to a wide range of data generated by numerous different processes with varying degrees of dependence and memory. The main conclusions drawn from these experiments include: (i) For all estimators considered, the main source of error is the bias. (ii) The CTW method is repeatedly and consistently seen to provide the most accurate results. (iii) The performance of the LZ-based estimators is often comparable to that of the plug-in method. (iv) The main drawback of the plug-in method is its computational inefficiency; with small word-lengths it fails to detect longer-range structure in the data, and with longer word-lengths the empirical distribution is severely undersampled, leading to large biases.
 
Manuscript ID: Entropy-12-09
Type of the Paper: Full Research Paper
Title: Additive Composed Quantum Statistical Entropy
Authors: Philipp Dedié, Wolfgang Muschik
Abstract: The incompatibility between the quantum statistical description of isolated undecomposed systems by the subadditive VON NEUMANN entropy and the irreversible behavior of the corresponding subdivided two-part composed system is discussed. An entropy definition for the composed system is offered. This composed quantum statistical entropy is additive and describes in contrast to the VON NEUMANN entropy the irreversibility of the composed system. This entropy definition dissolves another incompatibility, namely the one using the VON NEUMANN entropy for any isolated undecomposed system and the thermodynamic axiom testifying that any reversible composed system can only consist of reversible subsystems. Moreover, the proposed composed entropy definition yields an endoreversible description of the composed system in consideration.
 
Manuscript ID: Entropy-12-10
Type of the Paper: Full Research Paper
Title: Maximum Entropy Parameter Learning at Elevated Training Temperature
Authors: Ronny Melz
Abstract: ForMaximum Entropy (ME) parameter inference, the Improved Iterative Scaling algorithm (IIS) is often preferred over Generalized Iterative Scaling (GIS) due to its better convergence properties. But effectively, IIS requires the feature sum for each training event to be drawn from quite a limited, finite set of discrete values to allow for an efficiently computable parameter update step. Quite some generality of ME models is lost by requiring the feature functions to sum to discrete values. We re-interpret the maximum feature sum (originally determined by the GIS convergence proof) as an inverse “training temperature”, i.e. an additional free hyper parameter of the model. We provide empirical evidence that GIS outperforms IIS for suitable values of the training temperature, especially in the most interesting early iterations, despite its less complex implementation.
 
Manuscript ID: Entropy-12-11
Type of the Paper: Full Research Paper
Title: Graph Entropy and Conditioning
Authors: Arthur Ramer and Marian Grendar
Abstract: How to perform conditioning when certain letters/outcomes are not distinguishable? Distinguishability being specified by a graph, we apply K¨orner’s graph entropy and related information divergence on graphs to address this question.