On the way towards a generalized entropy maximization procedure

We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Rényi and Tsallis entropies. The generalized entropy maximization procedure for Rényi entropies results in the exponential stationary distribution asymptotically for q∈(0,1] in contrast to the stationary distribution of the inverse power law obtained through the ordinary entropy maximization procedure. Another result of the generalized entropy maximization procedure is that one can naturally obtain all the possible stationary distributions associated with the Tsallis entropies by employing either ordinary or q-generalized Fourier transforms in the averaging procedure.     

Lynden-Bell and Tsallis distributions for the HMF model

Systems with long-range interactions can reach a Quasi Stationary State (QSS) as a result of a violent collisionless relaxation. If the system mixes well (ergodicity), the QSS can be predicted by the statistical theory of Lynden-Bell (1967) based on the Vlasov equation. When the initial condition takes only two values, the Lynden-Bell distribution is similar to the Fermi-Dirac statistics. Such distributions have recently been observed in direct numerical simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we determine the caloric curve corresponding to the Lynden-Bell statistics in relation with the HMF model and analyze the dynamical and thermodynamical stability of spatially homogeneous solutions by using two general criteria previously introduced in the literature. We express the critical energy and the critical temperature as a function of a degeneracy parameter fixed by the initial condition. Below these critical values, the homogeneous Lynden-Bell distribution is not a maximum entropy state but an unstable saddle point. Known stability criteria corresponding to the Maxwellian distribution and the water-bag distribution are recovered as particular limits of our study. In addition, we find a critical point below which the homogeneous Lynden-Bell distribution is always stable. We apply these results to the situation considered in Antoniazzi et al. For a given energy, we find a critical initial magnetization above which the homogeneous Lynden-Bell distribution ceases to be a maximum entropy state. For an energy U=0.69, this transition occurs above an initial magnetization M_x=0.897. In that case, the system should reach an inhomogeneous Lynden-Bell distribution (most mixed) or an incompletely mixed state (possibly fitted by a Tsallis distribution). Thus, our theoretical study proves that the dynamics is different for small and large initial magnetizations, in agreement with numerical results of Pluchino et al. (2004). This new dynamical phase transition may reconcile the two communities by showing that they study different regimes.     

Weighted-density approaches for polymer structure at solid–polymer interfaces

Weighted-density approximations (WDAs), which are based on the weighting function for the second-order direct correlation functions (DCFs) of the uniform polymeric fluids, have been developed to investigate the structural and thermodynamic properties of polymer melts at interfaces. The advantage is the simplicity of calculation of the weighting functions and their accuracies in the applications. They were applied to study the local density distributions and adsorption isotherms of the freely jointed tangent hard-sphere chain, Yukawa chain, and hard-sphere chain mixture in slit pores. The polymer reference interaction model (PRISM) integral equation with the Percus–Yevick (PY) closure has been used to calculate the second-order DCF of the polymeric fluids required as inputs. The mean-field approximation (MFA) has been used to calculate the weighting function for the attractive contribution of a freely jointed tangent Yukawa chain fluid, having attraction among the beads. The calculated results show that (i) for the freely jointed tangent hard-sphere chain, the present theory is in excellent agreement with the computer simulations over a wide range of chain lengths and bulk densities, (ii) the WDA approach for the attraction provides an accurate method for the local density distributions of a freely jointed tangent Yukawa chain fluid, and that (iii) the present theory also yields a reasonably good result for the structural properties of the freely jointed hard-sphere chain mixtures composed of the chain and monomer.     

Generalized Pearson distributions for charged particles interacting with an electric and/or a magnetic field

The linear Boltzmann equation for elastic and/or inelastic scattering is applied to derive the distribution function of a spatially homogeneous system of charged particles spreading in a host medium of two-level atoms and subjected to external electric and/or magnetic fields. We construct a Fokker-Planck approximation to the kinetic equations and derive the most general class of distributions for the given problem by discussing in detail some physically meaningful cases. The equivalence with the transport theory of electrons in a phonon background is also discussed.     

Tsallis distribution as a standard maximum entropy solution with ‘tail’ constraint

We show that Tsallis' distributions can be derived from the standard (Shannon) maximum entropy setting, by incorporating a constraint on the divergence between the distribution and another distribution imagined as its tail. In this setting, we find an underlying entropy which is the Rényi entropy. Furthermore, escort distributions and generalized means appear as a direct consequence of the construction. Finally, the “maximum entropy tail distribution” is identified as a Generalized Pareto Distribution.     

Effective interaction between two giant spheres suspended in a size polydisperse hard-sphere fluid

Analytic expressions for the Laplace transform of the interaction energy and force between two exceedingly large hard spheres at infinite dilution in a polydisperse hard-sphere suspending fluid are presented. The equations are based on the Percus–Yevick approximation for the many-component suspending fluid, supplemented by the hypernetted chain approximation for the correlation function of the suspended spheres. By applying the Derjaguin approximation, the energy and force results for two spheres are related to the energy per unit area and the disjoining pressure between two flat walls suspended in a polydisperse fluid. Numerical results for the representative Schultz distributions of the diameters of the species comprising the suspending fluid are presented and discussed.     

Extremal entropy for a constrained probability density

The state of extremal entropy for a one-dimensional probability density is considered. This density is constrained by fixed values of the first and second moment. The grandcanonical distribution yields the extremum of the entropy within a certain range of values of the moments. A different type of density corresponds to an extremum of entropy when the moments are outside this range. The shape of this density is approximated with the Ritz variation method. The results are applied to the formation of breathers in the discrete nonlinear Schrödinger equation.     

Nonextensive statistical approach to non-coding human DNA

We use q-exponential distributions, which maximize the nonextensive entropy S_q (defined as ), to study the size distributions of non-coding DNA (including introns and intergenic regions) in all human chromosomes. We show that the value of the exponent q describing the non-coding size distributions is similar for all chromosomes and varies between 2≤q≤2.3 with the exception of chromosomes X and Y.     

Dynamics in dense suspensions of charge-stabilized colloidal particles

The dynamic behavior of charge-stabilized colloidal particles in suspension was studied by photon correlation spectroscopy with coherent X-rays (XPCS). The short-time diffusion coefficient, D(Q) , was measured for volume concentrations φ ⩽ 0.18 and compared to the free particle diffusion constant D₀ and the static structure factor S(Q) . The data show that indirect, hydrodynamic interactions are relevant for the system and hydrodynamic functions were derived. The results are in striking contrast to the predictions of the PA (pairwise-additive approximation) model, but show features typical for a hard-sphere system. The observed mobility is however considerably smaller than the one of a respective hard-sphere system. The hydrodynamic functions can be modelled quantitatively if one allows for an increased effective viscosity relative to the hard-sphere case.     

Two alternative methods for solving ion transport equation with anisotropic approximation

By means of two alternative methods namely, the maximum entropy and Chapman-Enskog, flux limited approach the ion transport equation for slowing-down problem of low-energy light ions in solid has been solved explicitly. Maximum entropy technique yields approximate solutions in the form of locally Maxwellian distribution function, based on moments expansion truncated upon entropy maximization.The behavior of the approximate maximum entropy and the flux limited solutions have the same tendency. Knowing the distribution function obtained by flux limited approach, allows us to calculate directly the path length distributions of backscattered ions, and compared with that found by other theories such as Laplace-transform and double Legendre polynomial approximation. One can see that the flux limited approach is better than the previous method namely, (DPN) Laplace-transform.The results reported in this article provide further evidence of the usefulness of both maximum entropy and flux limited for obtaining the solution of ion transport equation in compact form.     

Depletion interactions between two spherocylinders

The depletion interactions between two spherocylinders as functions of their separation and their relative orientation, induced by a small hard-sphere fluid, are calculated by Monte Carlo simulations using the acceptance ratio method (ARM). The torque on the spherocylinders is determined from the resulting potential. The calculation shows that the ARM is an effective way to obtain depletion interactions of spherocylinders. The depletion interaction under the Asakura-Oosawa (also excluded-volume) approximation is also calculated numerically.     

Thermodynamics predicts how confinement modifies the dynamics of the equilibrium hard-sphere fluid

We study how confining the equilibrium hard-sphere fluid to restrictive one- and two-dimensional channels with smooth interacting walls modifies its structure, dynamics, and entropy using molecular dynamics and transition-matrix Monte Carlo simulations. Although confinement strongly affects local structuring, the relationships between self-diffusivity, excess entropy, and average fluid density are, to an excellent approximation, independent of channel width or particle-wall interactions. Thus, thermodynamics can be used to predict how confinement impacts dynamics.     

Microcanonical ensemble study of a classical fluid of hard rods under gravity

The study of a system of hard rods in a box of finite length in the presence of a uniform gravitational field is made by means of the microcanonical ensemble. Explicit expressions are derived for the phase volume and the density of states, the primary functions of this ensemble. Related statistical quantities are reported, such as the entropy, the temperature, the heat capacity and the forces exerted on the fluid by the bottom and top walls. The microcanonical number density and higher order molecular distribution functions are also derived. Received: 7 April 1998 / Received in final form: 20 July 1998 / Accepted: 28 July 1998     

Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings

Recently we presented a new technique for numerical simulations of colloidal hard-sphere systems and showed its high efficiency. Here, we extend our calculations to the treatment of both 2- and 3-dimensional monodisperse and 3-dimensional polydisperse systems (with sampled finite Gaussian size distribution of particle radii), focusing on equilibrium pair distribution functions and structure factors as well as volume fractions of random close packing (RCP). The latter were determined using in principle the same technique as Woodcock or Stillinger had used. Results for the monodisperse 3-dimensional system show very good agreement compared to both pair distribution and structure factor predicted by the Percus-Yevick approximation for the fluid state (volume fractions up to 0.50). We were not able to find crystalline 3d systems at volume fractions 0.50–0.58 as shown by former simulations of Reeet al. or experiments of Pusey and van Megen, due to the fact that we used random start configurations and no constraints of particle positions as in the cell model of Hoover and Ree, and effects of the overall entropy of the system, responsible for the melting and freezing phase transitions, are neglected in our calculations. Nevertheless, we obtained reasonable results concerning concentration-dependent long-time selfdiffusion coefficients (as shown before) and equilibrium structure of samples in the fluid state, and the determination of the volume fraction of random close packing (RCP, glassy state). As expected, polydispersity increases the respective volume fraction of RCP due to the decrease in free volume by the fraction of the smaller spheres which fill gaps between the larger particles.     

Quantum perturbations to the radial distribution function of a hard-sphere fluid

Expressions derived in the previous paper for quantum corrections to the radial distribution function of a fluid are applied to the hard-sphere fluid. It is found that the perturbation theory given in the paper is valid only at very high values of temperatures when applied to calculate the correction to the distribution function of the hard-sphere fluid and is not valid at the temperatures at which Gibson and others have obtained numerical results for a given value of ?a³, where ? is the number density and a the hard-sphere diameter.     

Local structure of the high-temperature thermoelectric material PbTe using the maximum entropy method (MEM) and pair distribution function (PDF)

The electron density distribution and the local structure of the high temperature thermoelectric material PbTe has been studied. Powder X-ray data set of PbTe is analyzed in terms of cell parameter, thermal vibration parameters, 1D, 2D and 3-dimensional electron density distributions. The bonding between the atoms using the maximum entropy method (MEM) and bond-length distribution using pair distribution function (PDF) have been analyzed. Both the pictorial and the numerical results of electron density and PDF studies show mixed ionic and covalent characters in PbTe.     

Probability distribution of inherent states in models of granular media and glasses

The present paper develops a Statistical Mechanics approach to the inherent states of glassy systems and granular materials by following the original ideas proposed by Edwards for granular media. We consider three lattice models (a diluted spin glass, a system of hard spheres under gravity and a hard-spheres binary mixture under gravity) introduced to describe glassy and granular systems. They are evolved using a “tap dynamics” analogous to that of experiments on granular media. We show that the asymptotic states reached in such a dynamics are not dependent on the particular sample history and are characterized by a few thermodynamical parameters. We assume that under stationarity these systems are distributed in their inherent states satisfying the principle of maximum entropy. This leads to a generalized Gibbs distribution characterized by new “thermodynamical” parameters, called “configurational temperatures” (related to Edwards compactivity for granular materials). Finally, we show by Monte Carlo calculations that the average of macroscopic quantities over the tap dynamics and over such distribution indeed coincide. In particular, in the diluted spin glass and in the system of hard spheres under gravity, the asymptotic states reached by the system are found to be described by a single “configurational temperature”. Whereas in the hard-spheres binary mixture under gravity the asymptotic states reached by the system are found to be described by two thermodynamic parameters, coinciding with the two configurational temperatures which characterize the distribution among the inherent states when the principle of maximum entropy is satisfied under the constraint that the energies of the two species are independently fixed. Received 19 March 2002 and Received in final form 14 June 2002     

Limitations of Boltzmann's principle

The usual form of Boltzmann's principle assures that maximum entropy, or entropy reduction, occurs with maximum probability, implying a unimodal distribution. Boltzmann's principle cannot be applied to nonunimodal distributions, like the arcsine law, because the entropy may be concave only over a limited portion of the interval. The method of subordination shows that the arcsine distribution corresponds to a process with a single degree of freedom, thereby confirming the invalidation of Boltzmann's principle. The fractalization of time leads to a new distribution in which arcsine and Cauchy distributions can coexist simultaneously for nonintegral degrees of freedom between and 2.     

Entropy of a two-component fluid in a square-well model

An analytical expression valid for two approximations, notably, the random phase approximation (RPA) and the mean spherical approximation (MSA), is derived for entropy of a two-component system with the pair potential of a square well (SW). This expression is applied to calculate the excess entropy of mixing of the Na-K and Na-Cs equiatomic compositions. It is shown that the use of the SW model leads to better results than the application of the hard-sphere model. The MSA gives better convergence with the experiment compared with the RPA.     

Thermodynamics of pure and multicomponent dipolar hard-sphere fluids

The thermodynamics of two recent models of hard-sphere polar fluids is investigated. The two models are the Onsager model developed by Nienhuis and Deutch and the mean spherical approximation results obtained by Wertheim. The equation of state, vapour pressure curve, and other pertinent thermodynamic properties are determined. These results for the pure hard-sphere polar fluid complement the results recently reported by Rushbrooke, Stell and Høye. The thermodynamics of simple multicomponent hard-sphere polar mixtures is examined for the case of equal hard-sphere radii. Results for the multicomponent MSM are based on the exact solution of this model by Adelman and Deutch. The Onsager model is generalized to the case of many components. For both multicomponent models it is demonstrated that a variety of regions of instability exist for the one-phase fluid. Finally a discussion is included of how well these models agree with experiment.