Gibbs Canonical Distribution Research Articles

Physical and Mathematical Models of Quantum Dielectric Relaxation in Electrical and Optoelectric Elements Based on Hydrogen-Bonded Crystals

The quantum statistical properties of the proton subsystem in hydrogen-bonded crystals (HBC) are investigated. Based on the non-stationary Liouville operator equation (taking into account a number of assumptions established in the experiment), a quantum kinetic equation is constructed for the ensemble of non-interacting protons (an ideal proton gas) moving in the crystal potential image perturbed by the external electric field. The balanced density matrix for the unperturbed proton subsystem is constructed using the quantum canonical Gibbs distribution, and the non-balanced density matrix is calculated from the solutions of the nonlinear quantum kinetic equation by methods in linear approximation of perturbation theory for the blocking electrode model. Full quantum mechanical averaging of the polarization operator makes it possible to study the theoretical frequency-temperature spectra of the complex dielectric permittivity (CDP) calculated using quantum relaxation parameters that differ significantly from their semiclassical counterparts. A scheme is presented for an analytical study of the dielectric loss tangent in the region of quantum nonlinear relaxation in HBC. The results obtained in the given paper are of scientific interest in developing the theoretical foundations of proton conduction processes in energy-independent memory elements (with anomalously high residual polarization) based on thin films of ferroelectric materials in the ultralow temperature range (1–10 K). The theoretical results obtained have a direct application to the study of the tunneling mechanisms of spontaneous polarization in ferroelectric HBC with a rectangular hysteresis loop, in particular in crystals of potassium dideutrophosphate (KDP), widely used in nonlinear optics and laser technology. The quantum properties of proton relaxation in HBC can be applied in the future to the study of solid-state electrolytes with high proton conductivity for hydrogen energy, capacitor technology (superionics, varicodes), and elements of MIS and MSM structures in the development of resonant tunnel diodes for microelectronics and computer technology.

Temperature distribution in finite systems: application to the one-dimensional Ising chain

The theory of superstatistics, originally proposed for the study of complex nonequilibrium systems, has recently been extended to studies of small systems interacting with a finite environment, because such systems display interestingly similar statistical behavior. In both situations there are several applicable definitions of inverse temperature, either intrinsic or dependent of the statistical ensemble. In this work we develop these concepts focusing our attention on a region of an isolated, one-dimensional Ising chain as an example of a subsystem that does not follow the canonical Gibbs distribution. For this example, we explicitly show that superstatistics cannot describe the behavior of the subsystem, and verify a recently reported relation between the fundamental and microcanonical inverse temperatures. Our results hint at a new framework for dealing with regions of microcanonical systems with positive heat capacity, which should be described by some new class of statistical ensembles outside superstatistics but still preserving the notion of temperature fluctuations.

Unified theory of thermodynamics and stochastic thermodynamics for nonlinear Langevin systems driven by non-conservative forces

We construct a unified theory of thermodynamics and stochastic thermodynamics for classical nonequilibrium systems driven by non-conservative forces, using the recently developed covariant Ito-Langevin theory. The thermodynamic forces are split into a conservative part and a non-conservative part. Thermodynamic functions are defined using the reference conservative system. Work and heat are partitioned into excess parts and house-keeping parts, which are due to, respectively, conservative forces and non-conservative forces. Excess entropy production (EP) and house-keeping EP are analogously defined. The splitting of thermodynamic forces is subjected to an arbitrariness resembling a gauge symmetry, with each gauge defining a reference conservative Langevin system. In the special Gibbs gauge, the nonequilibrium steady state (NESS) is characterized by Gibbs canonical distribution, the excess heat agrees with that defined by Hatano and Sasa, and the excess EP agrees with that of Glansdorff and Prigogine, i.e., it is the time rate of the second-order variation of system entropy near the NESS. Adopting the Gibbs gauge, and focusing on the excess parts of thermodynamic quantities, a complete analogy between thermodynamics of non-conservative systems and that of conservative systems is established. One important consequence of this analogy is that both the free energy and excess EP are minimized at NESS. Our theory therefore constitutes a statistical foundation both for the steady-state thermodynamics theory due to Sasa and Tasaki and for the stability theory of NESS due to Glansdorff and Prigogine. These results are valid even if the system is far from equilibrium. By studying detailed fluctuation theorem, we find striking differences between systems with symmetric kinetic matrices and those with asymmetric kinetic matrices. For systems with asymmetric kinetic matrices, the total EP is the sum of house-keeping EP, excess EP, and pumped entropy. Entropy pumping is an exchange of entropy between the system and environment without necessarily involving dissipation. In the presence of entropy pumping, the system may behave as either a demon or an antidemon. Fluctuation theorems and work relations are derived both for total work and for excess work. For systems with symmetric kinetic matrices, there is no entropy pumping, yet in the Gibbs gauge, the excess work and house-keeping work each satisfies a separate fluctuation theorem. We illustrate our theory using many concrete examples.

A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation

We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the one-dimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass.

Модифицированный в рамках негауссовской каппа-статистики интегральный критерий устойчивости Чандрасекхара для равновесного сферического облака протозвезды

Within the framework of the non-extensive statistical mechanics of Kanyadakis, a generalization of the integral stability theorem of Chandrasekhar for the spherically symmetric distribution of matter and black radiation in an exoplanetary cloud in a state of gravitational equilibrium is obtained. For this purpose, the elements of deformed thermodynamics for an ideal gas, deformed canonical Gibbs distribution, as well as the effective gravitational constant, calculated in the formalisms of Kanyadakis and Verlinde, are used. In this, the deformation parameter κ (kappa) measures the so-called degree of nonextensiveness of the cloud system. In addition, the modified thermodynamic properties of blackbody radiation, in particular, the analogue of Stefan's law for radiation energy and generalized expressions for the entropy, heat capacity and radiation pressure, are discussed in the context of κ -statistics. The presented method of combining the indicated anomalous physical processes provides an alternative to the classical procedure of Chandrasekhar's derivation of the well-known integral theorems for gas configurations in gravitational equilibrium, and restores all standard expressions in the limit κ → 0. The results obtained will be able, according to the author, to explain some astrophysical problems of stellar-planetary cosmogony, associated, in particular, with modeling the processes of joint formation and evolution of a protosun and an exoplanetary cloud from a single nebula.

Realization of the Landau definitions of effective Hamiltonian and nonequilibrium free energy in microscopic theory

Equilibrium fluctuations of some set of parameters in the states described by the canonical Gibbs distribution are investigated. In the theory of phase transitions of the second kind, these parameters are components of the order parameter. The microscopic realization of the Landau definition of the effective Hamiltonian of the system for studying the equilibrium fluctuations of the specified system of parameters is discussed in the terms of the probability density of their values. A general formula for this function is obtained and it is expressed through the equilibrium correlation functions of these parameters. An expression for the effective Hamiltonian in terms of deviations of the parameters from their equilibrium values is obtained. The deviations are considered small for conducting the calculations. The possibility of calculating the exact free energy of the system using the found effective Hamiltonian is discussed.
 In the microscopic theory, the implementation of the Landau definition of nonequilibrium thermodynamic potentials introduced in his phenomenological theory of phase transitions of the second kind is investigated. Nonequilibrium states of a fluctuating system described with some sets of parameters are considered. A general formula for nonequilibrium free energy expressed through the correlation functions of these parameters is obtained as for the effective Hamiltonian above. Like the previous case, the free energy expression via parameter deviations from the equilibrium values is obtained and small deviations are considered for calculations. The idea of the identity of the effective Hamiltonian of the system and its nonequilibrium free energy is discussed in connection with the Boltzmann distribution. The Gaussian approximation of both developed formalisms is considered.
 A generalization of the constructed theory for the case of spatially inhomogeneous states and the study of long-wave fluctuations are developed.

Living on the edge of instability

Statistical description of stochastic dynamics in highly unstable potentials is strongly affected by properties of divergent trajectories, that quickly leave meta-stable regions of the potential landscape and never return. Using ideas from theory of Q-processes and quasi-stationary distributions, we analyze position statistics of non-diverging trajectories. We discuss two limit distributions which can be considered as (formal) generalizations of the Gibbs canonical distribution to highly unstable systems. Even though the associated effective potentials differ only slightly, properties of the two distributions are fundamentally different for all highly unstable system. The distribution for trajectories conditioned to diverge in an infinitely distant future is localized and light-tailed. The other distribution, describing trajectories surviving in the meta-stable region at the instant of conditioning, is heavy-tailed. The exponent of the corresponding power-law tail is determined by the leading divergent term of the unstable potential. We discuss different equivalent forms of the two distributions and derive properties of the effective statistical force arising in the ensemble of non-diverging trajectories after the Doob h-transform. The obtained explicit results generically apply to non-linear dynamical models with meta-stable states and fast kinetic transitions.

Weakly Nonergodic Dynamics in the Gross-Pitaevskii Lattice.

The microcanonical Gross-Pitaevskii (also known as the semiclassical Bose-Hubbard) lattice model dynamics is characterized by a pair of energy and norm densities. The grand canonical Gibbs distribution fails to describe a part of the density space, due to the boundedness of its kinetic energy spectrum. We define Poincaré equilibrium manifolds and compute the statistics of microcanonical excursion times off them. The tails of the distribution functions quantify the proximity of the many-body dynamics to a weakly nonergodic phase, which occurs when the average excursion time is infinite. We find that a crossover to weakly nonergodic dynamics takes place inside the non-Gibbs phase, being unnoticed by the largest Lyapunov exponent. In the ergodic part of the non-Gibbs phase, the Gibbs distribution should be replaced by an unknown modified one. We relate our findings to the corresponding integrable limit, close to which the actions are interacting through a short range coupling network.

On the relationship between microcanonical and canonical Gibbs distributions

It is established that the agreement between the use of microcanonical and canonical Gibbs distributions is possible only if the temperature of a closed equilibrium system and its specific energy in a thermostat are mutually inverse functions. The analysis shows that the derivation of the canonical Gibbs distribution from the microcanoncial does not agree with the physical condition for separating a macroscopic part of a closed equilibrium system.

Thermostat algorithm for generating target ensembles.

We present a deterministic algorithm called contact density dynamics that generates any prescribed target distribution in the physical phase space. Akin to the famous model of Nosé and Hoover, our algorithm is based on a non-Hamiltonian system in an extended phase space. However, the equations of motion in our case follow from contact geometry and we show that in general they have a similar form to those of the so-called density dynamics algorithm. As a prototypical example, we apply our algorithm to produce a Gibbs canonical distribution for a one-dimensional harmonic oscillator.

The statistical mechanics of a class of dissipative systems

The statistical mechanics of dynamical systems on which only isotropic viscous friction forces act is developed. A non-stationary analogue of the Gibbs canonical distribution, which enables each such system to be made to correspond to a certain thermodynamic system that satisfies the first and second laws of thermodynamics, is introduced. The evolution of non-Gibbs probability distributions with time is also considered.

Comparing the Efficiencies of Stochastic Isothermal Molecular Dynamics Methods

Molecular dynamics typically incorporates a stochastic-dynamical device, a “thermostat,” in order to drive the system to the Gibbs (canonical) distribution at a prescribed temperature. When molecular dynamics is used to compute time-dependent properties, such as autocorrelation functions or diffusion constants, at a given temperature, there is a conflict between the need for the thermostat to perturb the time evolution of the system as little as possible and the need to establish equilibrium rapidly. In this article we define a quantity called the “efficiency” of a thermostat which relates the perturbation introduced by the thermostat to the rate of convergence of average kinetic energy to its equilibrium value. We show how to estimate this quantity analytically, carrying out the analysis for several thermostats, including the Nosé-Hoover-Langevin thermostat due to Samoletov et al. (J. Stat. Phys. 128:1321–1336, 2007) and a generalization of the “stochastic velocity rescaling” method suggested by Bussi et al. (J. Chem. Phys. 126:014101, 2007). We find efficiency improvements (proportional to the number of degrees of freedom) for the new schemes compared to Langevin Dynamics. Numerical experiments are presented which precisely confirm our theoretical estimates.

A Quantum Statistical Approach to Remediation Effect of Humic Substances

Among various remediation factors, dissolved organic matter including humic substances (HS) has substantial effect on environmental contamination significantly changing the contaminant’s degradation, bioavailability, reactivity, and immobilization. However, the effects strongly depend on HS concentrations and their aromaticity index (AI). To understand underlying phenomena of remediation action of HS, which is revealed to occur within a definite interval of HS concentrations in water solution, a quantum statistical approach is supposed. Developing this approach, a model of protons as Fermi particles in humic substances was advanced for the first time and applied to describe transformations of HS molecules, i.e., multipoles into micelle structures, which in turn provide for mediating effects in water. Sufficiently high concentration of micelle granules in water solution exists if the concentration of HS lies within a definite interval. It was demonstrated applying a grand canonical Gibbs distribution method to a statistical ensemble of HS particles. Our approach allows for understanding and quantifying some biological and physiological processes connected with mediating action of HS, as for example the reversible red cell aggregation influenced by HS, adsorption of HS particles by cancer cells, and effect of HS on human resistibility to inflammatory processes of different kinds. Application of our results to water systems may be helpful to optimize waste processing and disposal.

Thermodynamic constraints on fluctuation phenomena

The relationships among reversible Carnot cycles, the absence of perpetual motion machines, and the existence of a nondecreasing globally unique entropy function form the starting point of many textbook presentations of the foundations of thermodynamics. However, the thermal fluctuation phenomena associated with statistical mechanics has been argued to restrict the domain of validity of this basis of the second law of thermodynamics. Here we demonstrate that fluctuation phenomena can be incorporated into the traditional presentation, extending rather than restricting the domain of validity of the phenomenologically motivated second law. Consistency conditions lead to constraints upon the possible spectrum of thermal fluctuations. In a special case this uniquely selects the Gibbs canonical distribution and more generally incorporates the Tsallis distributions. No particular model of microscopic dynamics need be assumed.

What Is Gibbs's Canonical Distribution?

Although the canonical distribution is one of the central tools of statistical mechanics, the reason for its effectiveness is poorly understood. This is due in part to the fact that there is no clear consensus on what it means to use the canonical distribution to describe a system in equilibrium with a heat bath. I examine some traditional views as to what sort of thing we should take the canonical distribution to represent. I argue that a less explored alternative, according to which the canonical distribution represents a time ensemble of sorts, has a number of advantages that rival interpretations lack.

The thermodynamic characteristics of the rupture of a Morse lattice

We consider the thermodynamic characteristics of the rupture of a one-dimensional lattice with the Morse interaction potential between neighboring particles. The description is given in terms of the Gibbs canonical distribution. Equations for partition functions corresponding to states with equal bond lengths and one broken bond were obtained in the quasi-harmonic low-temperature approximation. The ratios between these partition functions allow lattice lifetimes in both states to be estimated. The approximation used corresponds to a substantially truncated phase space volume, when states other than those in the immediate vicinity of potential energy minima are excluded from consideration. Nevertheless, satisfactory agreement between analytic equations and molecular dynamics simulation results was obtained for the number of particles in the Morse lattice N = 10.

Stationary Density Matrix of a Pumped Polariton System

The density matrix rho of a model polariton system is obtained numerically from a master equation which takes account of pumping and losses. In the stationary limit, the coherences between eigenstates of the Hamiltonian are 3 orders of magnitude smaller than the occupations, meaning that the stationary density matrix is approximately diagonal in the energy representation. A weakly distorted grand canonical Gibbs distribution fits well the occupations.

On the completeness of description of an equilibrium canonical ensemble using a reduceds-particle distribution function

In this paper it is shown that for a classical equilibrium canonical ensemble of molecules with sufficientlysmall s-body interaction, the full Gibbs distribution can be uniquely expressed in terms of a reduceds-particle distribution function. This means that whenever the number of particlesN and the volumeV of such a system arefixed, the reduced s-particle distribution function contains as much information about theequilibrium system as the canonical Gibbs distribution function. The latter isrepresented as an absolutely convergent power series relative to the reduceds-particle distribution function. As an example, a linear term of this expansion iscalculated. It is also shown that reduced distribution functions of order less thans do not possess such a property and, to all appearances, do not contain all of theinformation about the system under consideration.

Two-dimensional Brownian vortices

We introduce a stochastic model of 2D Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (“plasma” case) and at negative temperatures, like-sign vortices attract each other (“gravity” case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker–Planck equation satisfied by the N -body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1 / N , where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N → + ∞ . For spatially homogeneous systems, we study the two-body correlation function, in a Debye–Hückel approximation valid at the order O ( 1 / N ) . The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye–Hückel model of electrolytes.

Trapped slender vortex filaments in statistical equilibrium

AbstractSystems of nearly parallel, slender vortex filaments in which angular momentum is conserved are an important simplification of the Navier‐Stokes equations where turbulence can be studied in statistical equilibrium. We study the canonical Gibbs distribution based on the Klein‐Majda‐Damodaran (KMD) (Klein et al. [1995]) model and find a divergence in the mean square vortex position from that of the point vortex model of Onsager [1949] at high temperature. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)