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**International Journal of Statistical Mechanics**

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**Open Access journal**

ISSN (Print) 2356-7112 - ISSN (Online) 2314-6850

Published by

**Hindawi Publishing Corporation**[403 journals]

**Fractional Diffusion Equations for Lattice and Continuum:**

Grünwald-Letnikov Differences and Derivatives Approach**Abstract:**Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.**PubDate:**Mon, 08 Dec 2014 13:16:25 +000

**A Fractional Entropy in Fractal Phase Space: Properties and**

Characterization**Abstract:**A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin axioms corresponding to the two-parameter entropy is proposed and verified. We present the relative entropy, Jensen-Shannon divergence measure and check their properties. The Fisher information measure, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are also derived. Also the Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system.**PubDate:**Wed, 24 Sep 2014 00:00:00 +000

**The Statistical Mechanics of Random Set Packing and a Generalization of**

the Karp-Sipser Algorithm**Abstract:**We analyse the asymptotic behaviour of random instances of the maximum set packing (MSP) optimization problem, also known as maximum matching or maximum strong independent set on hypergraphs. We give an analytic prediction of the MSPs size using the 1RSB cavity method from statistical mechanics of disordered systems. We also propose a heuristic algorithm, a generalization of the celebrated Karp-Sipser one, which allows us to rigorously prove that the replica symmetric cavity method prediction is exact for certain problem ensembles and breaks down when a core survives the leaf removal process. The -phenomena threshold discovered by Karp and Sipser, marking the onset of core emergence and of replica symmetry breaking, is elegantly generalized to for one of the ensembles considered, where is the size of the sets.**PubDate:**Mon, 10 Mar 2014 09:07:19 +000

**An Independence Test Based on Symbolic Time Series****Abstract:**An independence test based on symbolic time series analysis (STSA) is developed. Considering an independent symbolic time series there is a statistic asymptotically distributed as a CHI-2 with degrees of freedom. Size and power experiments for small samples were conducted applying Monte Carlo simulations and comparing the results with BDS and runs test. The introduced test shows a good performance detecting independence in nonlinear and chaotic systems.**PubDate:**Mon, 24 Feb 2014 07:49:50 +000

**Exact Solution to the Extended Zwanzig Model for Quasi-Sigmoidal**

Chemically Induced Denaturation Profiles: Specific Heat and

Configurational Entropy**Abstract:**Temperature and chemically induced denaturation comprise two of the most characteristic mechanisms to achieve the passage from the native state to any of the unstructured states in the denatured ensemble in proteins and peptides. In this work we present a full analytical solution for the configurational partition function of a homopolymer chain poly-X in the extended Zwanzig model (EZM) for a quasisigmoidal denaturation profile. This solution is built up from an EZM exact solution in the case where the fraction of native contacts follows exact linear dependence on denaturant’s concentration ; thus an analytical solution for in the case of an exact linear denaturation profile is also provided. A recently established connection between the number of potential nonnative conformations per residue and temperature-independent helical propensity complements the model in order to identify specific proteinogenic poly-X chains, where X represents any of the twenty naturally occurring aminoacid residues. From , equilibrium thermodynamic potentials like entropy and average internal energy and thermodynamic susceptibilities like specific heat are calculated for poly-valine (poly-V) and poly-alanine (poly-A) chains. The influence of the rate at which native contacts denature as function of on thermodynamic stability is also discussed.**PubDate:**Thu, 23 Jan 2014 07:10:18 +000

**Stochastic Regularization and Eigenvalue Concentration Bounds for Singular**

Ensembles of Random Operators**Abstract:**We propose a simple approach allowing reducing the eigenvalue concentration analysis of a class of random operator ensembles with singular probability distribution to the analysis of an auxiliary ensemble with bounded probability density. Our results apply to the Wegner- and Minami-type estimates for single- and multiparticle operators.**PubDate:**Wed, 27 Nov 2013 08:42:11 +000

**Solution and Analysis of a One-Dimensional First-Passage Problem with a**

Nonzero Halting Probability**Abstract:**This paper treats a kind of a one-dimensional first-passage problem, which seeks the probability that a random walker first hits the origin at a specified time. In addition to a usual random walk which hops either rightwards or leftwards, the present paper introduces the “halt” that the walker does not hop with a nonzero probability. The solution to the problem is expressed using a Gauss hypergeometric function. The moment generating function of the hitting time is also calculated, and a calculation technique of the moments is developed. The author derives the long-time behavior of the hitting-time distribution, which exhibits power-law behavior if the walker hops to the right and left with equal probability.**PubDate:**Sun, 27 Oct 2013 13:19:34 +000

**Thermalization of Lévy Flights: Path-Wise Picture in 2D****Abstract:**We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which in the presence of confining potentials may asymptotically set down at Boltzmann-type thermal equilibria. In view of the Eliazar-Klafter no-go statement, such dynamical behavior is plainly incompatible with the standard Langevin modeling of Lévy flights. No explicit path-wise description has been so far devised for the thermally equilibrating random motion we address, and its formulation is the principal goal of the present work. To this end we prescribe a priori the target pdf ρ∗ in the Boltzmann form ~exp[] and next select the Lévy noise (e.g., its Lévy measure) of interest. To reconstruct random paths of the underlying stochastic process we resort to numerical methods. We create a suitably modified version of the time honored Gillespie algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation.**PubDate:**Thu, 03 Oct 2013 15:22:33 +000

**Spectral Functions and Properties of Nuclear Matter****Abstract:**The Green’s function method in the Kadanoff-Baym version provides a basic theory for nuclear dynamics which is applicable also to nonzero temperature and to nonequilibrium systems. At the same time, it maintains the basic many-body techniques of the Brueckner theory that makes reasonable a comparison of the numerical results of the two methods for equilibrium systems. The correct approximation to the spectral function which takes into account the widths of energy levels is offered and discussed, and the comparison of the values of binding energy in the two methods is produced.**PubDate:**Mon, 15 Jul 2013 12:28:04 +000

**An Especial Fractional Oscillator****Abstract:**We propose a peculiar fractional oscillator. By assuming that the motion takes place in a complex media where the level of fractionality is low, we find that the time rate of change of the energy of this system has an oscillatory behavior.**PubDate:**Sun, 23 Jun 2013 13:56:12 +000