arXiv subjects

Robert K. Niven

Publications and source records attributed to Robert K. Niven.

27 records · Page 2Linked to original sources

arXiv2008

Generalized Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac and Acharya-Swamy Statistics and the Polya Urn Model

Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source probabilities $q_i$ for each level $i$, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the Pólya urn model, involving neither independent nor identically distributed ("ninid") sampling. The most probable Pólya distribution contains the Acharya-Swamy intermediate statistic.

cond-mat.stat-mech↗

arXiv2007

Combinatorial Entropy for Distinguishable Entities in Indistinguishable States

The combinatorial basis of entropy by Boltzmann can be written $H= {N}^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy of a system, per unit entity, $N$ is the number of entities and $\mathbb{W}$ is the number of ways in which a given realization of the system can occur, known as its statistical weight. Maximizing the entropy (``MaxEnt'') of a system, subject to its constraints, is then equivalent to choosing its most probable (``MaxProb'') realization. For a system of distinguishable entities and states, $\mathbb{W}$ is given by the multinomial weight, and $H$ asymptotically approaches the Shannon entropy. In general, however, $\mathbb{W}$ need not be multinomial, leading to different entropy measures. This work examines the allocation of distinguishable entities to non-degenerate or equally degenerate, indistinguishable states. The non-degenerate form converges to the Shannon entropy in some circumstances, whilst the degenerate case gives a new entropy measure, a function of a multinomial coefficient, coding parameters, and Stirling numbers of the second kind.

math-ph↗

arXiv2007

Origins of the Combinatorial Basis of Entropy

The combinatorial basis of entropy, given by Boltzmann, can be written $H = N^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy, $N$ is the number of entities and $\mathbb{W}$ is number of ways in which a given realization of a system can occur (its statistical weight). This can be broadened to give generalized combinatorial (or probabilistic) definitions of entropy and cross-entropy: $H=κ(ϕ(\mathbb{W}) +C)$ and $D=-κ(ϕ(\mathbb{P}) +C)$, where $\mathbb{P}$ is the probability of a given realization, $ϕ$ is a convenient transformation function, $κ$ is a scaling parameter and $C$ an arbitrary constant. If $\mathbb{W}$ or $\mathbb{P}$ satisfy the multinomial weight or distribution, then using $ϕ(\cdot)=\ln(\cdot)$ and $κ=N^{-1}$, $H$ and $D$ asymptotically converge to the Shannon and Kullback-Leibler functions. In general, however, $\mathbb{W}$ or $\mathbb{P}$ need not be multinomial, nor may they approach an asymptotic limit. In such cases, the entropy or cross-entropy function can be {\it defined} so that its extremization ("MaxEnt'' or "MinXEnt"), subject to the constraints, gives the ``most probable'' (``MaxProb'') realization of the system. This gives a probabilistic basis for MaxEnt and MinXEnt, independent of any information-theoretic justification. This work examines the origins of the governing distribution $\mathbb{P}$.... (truncated)

physics.class-ph↗

arXiv2007

Combinatorial Information Theory: I. Philosophical Basis of Cross-Entropy and Entropy

This study critically analyses the information-theoretic, axiomatic and combinatorial philosophical bases of the entropy and cross-entropy concepts. The combinatorial basis is shown to be the most fundamental (most primitive) of these three bases, since it gives (i) a derivation for the Kullback-Leibler cross-entropy and Shannon entropy functions, as simplified forms of the multinomial distribution subject to the Stirling approximation; (ii) an explanation for the need to maximize entropy (or minimize cross-entropy) to find the most probable realization; and (iii) new, generalized definitions of entropy and cross-entropy - supersets of the Boltzmann principle - applicable to non-multinomial systems. The combinatorial basis is therefore of much broader scope, with far greater power of application, than the information-theoretic and axiomatic bases. The generalized definitions underpin a new discipline of ``{\it combinatorial information theory}'', for the analysis of probabilistic systems of any type. Jaynes' generic formulation of statistical mechanics for multinomial systems is re-examined in light of the combinatorial approach. (abbreviated abstract)

cond-mat.stat-mech↗

arXiv2006

The Polya Urn: Limit Theorems, Polya Divergence, Maximum Entropy and Maximum Probability

Sanov's Theorem and the Conditional Limit Theorem (CoLT) are established for a multicolor Polya Eggenberger urn sampling scheme, giving the Polya divergence and the Polya extension to the Maximum Relative Entropy (MaxEnt) method. Polya MaxEnt includes the standard MaxEnt as a special case. The universality of standard MaxEnt - advocated by an axiomatic approach to inference for inverse problems - is challenged, in favor of a probabilistic approach based on CoLT and the Maximum Probability principle.

cond-mat.stat-mech↗

arXiv2005

Cost of s-fold Decisions in Exact Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac Statistics

The exact forms of the degenerate Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropy functions, derived by Boltzmann's principle without the Stirling approximation (Niven, Physics Letters A, 342(4) (2005) 286), are further examined. Firstly, an apparent paradox in quantisation effects is resolved using the Laplace-Jaynes interpretation of probability. The energy cost of learning that a system, distributed over s equiprobable states, is in one such state (an s-fold decision) is then calculated for each statistic. The analysis confirms that the cost depends on one's knowledge of the number of entities N and (for BE and FD statistics) the degeneracy, extending the findings of Niven (2005).

cond-mat.stat-mech↗

arXiv2005

q-Exponential Structure of Arbitrary-Order Reaction Kinetics

The rate equation for an arbitrary mth order growth or decay reaction can be expressed in terms of the q-exponential function, with q equal to m. The analysis suggests that a wide variety of reaction rate (kinetic) processes and models, in chemistry, biology/ecology, unit operations and contaminant transport, are amenable to analysis by Tsallis' non-extensive statistical mechanics. A number of q-analogues of common kinetic mathematical models are proposed, and illustrated by several examples.

cond-mat.stat-mech↗

arXiv2005

Exact Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac Statistics

The exact Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirling's approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a "binary decision", exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.

cond-mat.stat-mech↗

arXiv2005

Constrained Forms of the Tsallis Entropy Function and Local Equilibria

The Lagrangian technique of Niven (2004, Physica A, 334(3-4): 444) is used to determine the constrained forms of the Tsallis entropy function - i.e. Lagrangian functions in which the probabilities of each state are independent - for each constraint type reported in the literature (here termed the Mark I, II and III forms). In each case, a constrained form of the Tsallis entropy function exists, which at q=1 reduces to its Shannon equivalent. Since they are fully constrained, each constrained Tsallis function can be "dismembered" to give its partial or local form, providing the means to independently examine each state i relative to its local stationary (maximum entropy) position. The Mark II and III functions depend on q, the probability, the stationary probability, and the respective q-partition function; in contrast the Mark I form depends only on the first three parameters. The Mark II and III forms therefore depend on the structure of the system. The utility of the dismemberment method is illustrated for a system with equispaced energy levels.

cond-mat.stat-mech↗