Formulation Of Statistical Mechanics Research Articles

Transport theory of interacting mesoscopic systems: A memory-function approach to charge-counting statistics

We address the subtle problem of formulating mesoscopic transport phenomena in the language of many-body physics. We propose a microscopic description in which the whole system (sample, leads, and detectors) is given fully-quantum-mechanical treatments. The dynamics of the system is obtained from the projection, or memory-function, formalism of nonequilibrium statistical mechanics combined with recent prescriptions for measurements that are extended in the time domain. The associated irreversible quantum dynamics contains an intrinsic doubling of the degrees of freedom identical to the real-time path-integral representation of the Keldysh formalism. We derive a simple formula relating the generating function of the charge-counting statistics to the single-particle matrix Green's function of the interacting system, thereby generalizing the Levitov-Lesovik functional determinant formula. We report an interesting sample-leads duality in our description, which has a simple interpretation in the regime of noninteracting particles, thus establishing the equivalence between the Green's-function technique and the random scattering-matrix approach. We discuss the physical conditions, within the present scheme, for the validity of the Landauer-Buttiker description of mesoscopic transport. We conclude by providing an exact solution for the problem of charge transfer into a noninteracting ballistic cavity in the presence of a periodic time-dependent voltage, using the supersymmetry method.

The spacetime associated with galactic dark matter halos

We show how an adequate post--Newtonian generalization can be obtained for Newtonian dark matter halos associated with an empiric density profile. Applying this approach to halos that follow from the well known numerical simulations of Navarro, Frenk and White (NFW), we derive all dynamical variables and show that NFW halos approximatelly follow an ideal gas type of equation of state which fits very well to a polytropic relation in the region outside the core. This fact suggests that ``outer'' regions of NFW halos might be related to equilibrium states in the non--extensive Statistical Mechanics formalism proposed by Tsallis.

Theoretical and computational models of biological ion channels

The goal of this review is to establish a broad and rigorous theoretical framework to describe ion permeation through biological channels. This framework is developed in the context of atomic models on the basis of the statistical mechanical projection-operator formalism of Mori and Zwanzig. The review is divided into two main parts. The first part introduces the fundamental concepts needed to construct a hierarchy of dynamical models at different level of approximation. In particular, the potential of mean force (PMF) as a configuration-dependent free energy is introduced, and its significance concerning equilibrium and non-equilibrium phenomena is discussed. In addition, fundamental aspects of membrane electrostatics, with a particular emphasis on the influence of the transmembrane potential, as well as important computational techniques for extracting essential information from all-atom molecular dynamics (MD) simulations are described and discussed. The first part of the review provides a theoretical formalism to 'translate' the information from the atomic structure into the familiar language of phenomenological models of ion permeation. The second part is aimed at reviewing and contrasting results obtained in recent computational studies of three very different channels: the gramicidin A (gA) channel, which is a narrow one-ion pore (at moderate concentration), the KcsA channel from Streptomyces lividans, which is a narrow multi-ion pore, and the outer membrane matrix porin F (OmpF) from Escherichia coli, which is a trimer of three beta-barrel subunits each forming wide aqueous multi-ion pores. Comparison with experiments demonstrates that current computational models are approaching semi-quantitative accuracy and are able to provide significant insight into the microscopic mechanisms of ion conduction and selectivity. We conclude that all-atom MD with explicit water molecules can represent important structural features of complex biological channels accurately, including such features as the location of ion-binding sites along the permeation pathway. We finally discuss the broader issue of the validity of ion permeation models and an outlook to the future.

Wigner approach to quantum dynamics simulations of the interacting carriers in disordered systems

AbstractThe new method for solving Wigner–Liouville's type equations and studying dynamics of quantum particles has been developed within the Wigner formulation of quantum statistical mechanics. This approach combines both molecular dynamics and Monte Carlo methods and computes traces and spectra of the relevant dynamical quantities. Considering, as an application, the quantum dynamics of an ensemble of interacting electrons in an array of random scatterers clearly demonstrates that the many‐particle interaction between the electrons can lead to an enhancement of the electrical conductivity. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Statistical mechanics in the extended Gaussian ensemble.

The extended Gaussian ensemble (EGE) is introduced as a generalization of the canonical ensemble. This ensemble is a further extension of the Gaussian ensemble introduced by Hetherington [J. Low Temp. Phys. 66, 145 (1987)]. The statistical mechanical formalism is derived both from the analysis of the system attached to a finite reservoir and from the maximum statistical entropy principle. The probability of each microstate depends on two parameters beta and gamma which allow one to fix, independently, the mean energy of the system and the energy fluctuations, respectively. We establish the Legendre transform structure for the generalized thermodynamic potential and propose a stability criterion. We also compare the EGE probability distribution with the q-exponential distribution. As an example, an application to a system with few independent spins is presented.

The phenomena of dielectric saturation in the water domains of polymer electrolyte membranes

A statistical mechanical formalism is presented for the calculation of the thermodynamic property of the permittivity of the water in the water and ion containing domains of polymer electrolyte membranes, with particular emphasis on the polarization. The present work differs from previous work in that the electric field due to the dissociated acidic groups (i.e. –SO 3H) of the polymer is included in the expression of the total Hamiltonian of the system. With this important consideration we compute the dielectric constant of the water in a fully hydrated Nafion membrane and show that there exists a central region within the pores where the water is similar to bulk water (i.e. ε=80) but as the pore walls are approached the water becomes increasingly more bound.

Self-organization from structural refrigeration.

The self-organization of a classical current is studied, in an exactly solvable model where both the quantum statistics over microhistories of particles, and the macroscopic phenomenology, can be computed in closed form. It is shown that for thermodynamically reversible systems, the Jaynes formulation of statistical mechanics naturally extends to include explicit macroscopic dynamics and heterogeneities in temperature, while preserving the structure of partition functions, effective potentials, and ground states of the equilibrium theory. Self-organization in such reversible systems is constrained by entropy transport through engine and refrigeration cycles, rather than by diffusion in gradients. Limitations in the ability to decompose such systems sensibly into components with additive entropies, and in the extrapolation of entropy functions from equilibrium forms, are discussed with examples.

Nonextensivity in turbulence in rotating two-dimensional and three-dimensional flows

Our experiments on turbulent flow in a rotating annulus yield probability distribution functions (PDFs) for velocity increments δv(ℓ), where ℓ is the separation between points. We fit these PDFs to a form derived for turbulent flows by Beck, who used the Tsallis nonextensive statistical mechanics formalism. For slow rotation rates, we find that the fit parameter q is 1.25 for small ℓ. At large ℓ, q decreases to unity, the value corresponding to the usual Boltzmann–Gibbs statistics. These results agree with those previously measured in experiments on Couette–Taylor turbulence. However, with rapid rotation of the annulus, the turbulent flow becomes strongly two-dimensional (2D) rather than three-dimensional (3D), and we find q=1.32±0.04, independent of ℓ. This suggests that the coherent structures (vortices), which are a source of intermittency, are important at all length scales in the 2D case.

Complexation in dense gases: concentrations of argon monomers, dimers and trimers

A classical statistical mechanical cluster formalism is presented for computing the thermodynamic properties and equilibrium state of aggregation of a fluid composed of structureless monomeric units which are capable of forming weakly bound van der Waals complexes. The general procedure is not restricted to any particular definition of the clusters, provided that the definition used satisfies certain reasonable criteria. The formalism is illustrated by applying it to two different cluster definitions. Calculations based on these definitions have been performed for the concentrations of argon monomers, dimers and trimers. It is found that gas imperfections due to cluster-cluster interactions can significantly affect the calculated values of these concentrations.

Geometry of escort distributions.

Given an original distribution, its statistical and probabilistic attributes may be scanned using the associated escort distribution introduced by Beck and Schlögl and employed in the formulation of nonextensive statistical mechanics. Here, the geometric structure of the one-parameter family of the escort distributions is studied based on the Kullback-Leibler divergence and the relevant Fisher metric. It is shown that the Fisher metric is given in terms of the generalized bit variance, which measures fluctuations of the crowding index of a multifractal. The Cramér-Rao inequality leads to a fundamental limit for the precision of the statistical estimate of the order of the escort distribution. We also show quantitatively that it is inappropriate to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics.

Keldysh-Green function formalism for current profiles in mesoscopic systems

We provide a general procedure for the description and evaluation of the current distribution in mesoscopic quantum wires. Our approach is based on the Keldysh-Green function formalism of nonequilibrium quantum statistical mechanics and exploits the real-space renormalization method within the tight-binding framework. We obtain a detailed spatial description of the microscopic currents between any couple of sites of the system, both in the presence and in the absence of time-reversal symmetry. As an application we present current profiles for a disordered quantum wire in the regime of universal conductance fluctuations, and we illustrate the random path of the current flow also in the presence of a magnetic field.

Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states

Jaynes' information theory formalism of statistical mechanics is applied to the stationary states of open, non-equilibrium systems. The key result is the construction of the probability distribution for the underlying microscopic phase space trajectories. Three consequences of this result are then derived : the fluctuation theorem, the principle of maximum entropy production, and the emergence of self-organized criticality for flux-driven systems in the slowly-driven limit. The accumulating empirical evidence for these results lends support to Jaynes' formalism as a common predictive framework for equilibrium and non-equilibrium statistical mechanics.

New strain measure tensor for entangled polymeric liquids

A novel approach to the description of entangled polymeric liquids has recently been presented [F. Greco, Phys. Rev. Lett. 88, 108301 (2002)], where the mechanical behavior of any subchain connecting two entanglements is determined through the grand canonical formalism of statistical mechanics, thus properly allowing for exchange of particles (Kuhn segments) among subchains. The deduction of a strain measure tensor in this new approach is given and discussed in detail. It is also shown that the new strain measure tensor fulfills the stress-optical law. Predictions obtained with this strain measure are compared with data for step–strain deformations both in shear and elongation, and good agreement is found. The normal stress ratio in step shear is found to be better described here than with the classical “rigorous” Doi–Edwards strain measure.

Fracture of disordered solids in compression as a critical phenomenon. I. Statistical mechanics formalism.

This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in defining the ensemble. Ensembles are obtained by dividing a huge rock sample into many mesoscopic volumes. Because rocks are a disordered collection of grains in cohesive contact, we expect that once shear strain is applied and cracks begin to arrive in the system, the mesoscopic volumes will have a wide distribution of different crack states. These mesoscopic volumes are the members of our ensembles. We determine the probability of observing a mesoscopic volume to be in a given crack state by maximizing Shannon's measure of the emergent-crack disorder subject to constraints coming from the energy balance of brittle fracture. The laws of thermodynamics, the partition function, and the quantification of temperature are obtained for such cracking systems.

Extensive form of equilibrium nonextensive statistics

It is argued that, in nonextensive statistical mechanics with Tsallis entropy, the factorization of compound probability over subsystems is a consequence of the existence of thermodynamic equilibrium in the composite system and should be respected by all exact calculations concerning equilibrium subsystems. Using nonadditive energy satisfying this factorization, we propose an additive formalism of nonextensive statistical mechanics with additive q-deformed physical quantities and exponential distributions. This formalism leads to exact quantum gas distributions different from those given by factorization approximation with additive energy. The fermion distribution of the present work shows similar characteristics to the distribution of strongly correlated electrons given by numerical analysis with the Kondo t-J model.

On the definition of physical temperature and pressure for nonextensive thermostatistics

It is shown that a recent proposal to give physically meaningful definitions of temperature and pressure within Tsallis formalism for non-extensive thermostatistics leads to expressions which coincide with those obtained by using the standard Boltzmann formalism of statistical mechanics.

Complex and non-complex phase structures in models of spin glasses and information processing

The gauge theory of spin glasses and statistical–mechanical formulation of error-correcting codes are reviewed with an emphasis on their similarities. For the gauge theory, we explain the functional identities on dynamical autocorrelation functions and on the distribution functions of order parameters. These functional identities restrict the possibilities of slow dynamics and complex structure of the phase space. An inequality for error-correcting codes is shown to be interpreted to indicate non-monotonicity of spin orientation as a function of the temperature in spin glasses.

Monte Carlo simulations in the isothermal—isobaric ensemble: the requirement of a ‘shell’ molecule and simulations of small systems

A modification of the widely used Monte Carlo method for determining thermophysical properties in the isothermal-isobaric ensemble is presented. The new Monte Carlo method, now consistent with recent derivations describing the proper statistical mechanical formulation of the constant pressure ensemble for small systems, requires a ‘shell’ molecule to uniquely identify the volume of the system, thereby avoiding the redundant counting of configurations. Ensemble averages obtained with the new algorithm differ from averages calculated with the old Monte Carlo method, particularly for small system sizes, although both sets of averages become equal in the thermodynamic limit. Monte Carlo simulations in the constant pressure ensemble applied to the Lennard-Jones fluid demonstrate these differences for small systems. Peculiarities of small systems are also discussed, revealing that ‘shape’ is an important thermodynamic variable. Finally, an extension of the Monte Carlo method to mixtures is presented.

Statistical mechanical models with effective potentials: Definitions, applications, and thermodynamic consequences

Realistic interactions that operate in condensed matter systems can exhibit complicated many-particle characteristics. However, it is often useful to seek a more economical description using at most singlet and pair effective interactions that are density dependent, to take advantage of the theoretical and computational simplifications that result. This paper analyzes the statistical mechanical formalism required to describe thermal equilibrium in that kind of approach. Two distinct interpretations are available for the role of density dependence. Either one can be treated with internal consistency, but generally they lead to differing thermodynamic predictions. One regards the density dependence of effective interactions as merely a passive index for the state at which the optimal choice of those effective interactions was determined (Case I). The other treats the density as an active variable on the same footing as particle coordinates (Case II). Virial pressure and isothermal compressibility expressions in terms of particle distribution functions are displayed for both cases. Under special circumstances it is possible for the two interpretations to yield the same pressure isotherms; the conditions producing this coincidental concordancy have been explored by means of density expansions.

Multi-mode q-oscillator algebras with q2(k + 1) = 1 and related thermo field dynamics

A type of multi-mode q-oscillator algebra with q2(k + 1) = 1 is set up and the associated qk-thermo field dynamics is constructed for all k = 1,2,,∞ in a unified form. It is demonstrated that these qk-thermo field dynamics can all be nicely fitted into the algebraic formulation of statistical mechanics (axiomatized form for statistical physics). This means that we obtain infinitely many realizations of the algebraic scheme, which extend the consideration of Ojima [1981 Ann. Phys. 137 1] and contain the usual thermo field dynamics for the fermionic (k = 1) and bosonic (k = ∞) systems as special cases. As simple applications, the qk-statistical average of some operators are given.