Gibbs Canonical Distribution Research Articles

Renyi entropy as a statistical entropy for complex systems

To describe a complex system, we propose using the Renyi entropy depending on the parameter q (0 < q ≤ 1) and passing into the Gibbs-Shannon entropy at q = 1. The maximum principle for the Renyi entropy yields a Renyi distribution that passes into the Gibbs canonical distribution at q = 1. The thermodynamic entropy of the complex system is defined as the Renyi entropy for the Renyi distribution. In contrast to the usual entropy based on the Gibbs-Shannon entropy, the Renyi entropy increases as the distribution deviates from the Gibbs distribution (the deviation is estimated by the parameter η = 1 − q) and reaches its maximum at the maximum possible value ηmax. As this occurs, the Renyi distribution becomes a power-law distribution. The parameter η can be regarded as an order parameter. At η = 0, the derivative of the thermodynamic entropy with respect to η exhibits a jump, which indicates a kind of phase transition into a more ordered state. The evolution of the system toward further order in this phase state is accompanied by an entropy gain. This means that in accordance with the second law of thermodynamics, a natural evolution in the direction of self-organization is preferable.

A phonon heat bath approach for the atomistic and multiscale simulation of solids

AbstractWe present a novel approach to numerical modelling of the crystalline solid as a heat bath. The approach allows bringing together a small and a large crystalline domain, and model accurately the resultant interface, using harmonic assumptions for the larger domain, which is excluded from the explicit model and viewed only as a hypothetic heat bath. Such an interface is non‐reflective for the elastic waves, as well as providing thermostatting conditions for the small domain. The small domain can be modelled with a standard molecular dynamics approach, and its interior may accommodate arbitrary non‐linearities. The formulation involves a normal decomposition for the random thermal motion term R(t) in the generalized Langevin equation for solid–solid interfaces. Heat bath temperature serves as a parameter for the distribution of the normal mode amplitudes found from the Gibbs canonical distribution for the phonon gas. Spectral properties of the normal modes (polarization vectors and normal frequencies) are derived from the interatomic potential. Approach results in a physically motivated random force term R(t) derived consistently to represent the correlated thermal motion of lattice atoms. We describe the method in detail, and demonstrate applications to one‐ and two‐dimensional lattice structures. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Non-Ergodicity of the Nosé–Hoover Thermostatted Harmonic Oscillator

The Nose–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nose–Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nose–Hoover dynamics are non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nose–Hoover thermostat does not give an ergodynamical system for the one- dimensional harmonic oscillator when the “mass” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nose–Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass.

Theoretical study of characteristics of a molecular single-electron transistor

A technique for the calculation of current-to-voltage curves and control curves of a molecular single-electron transistor with a discrete energy spectrum has been developed. The effective recursive methods for quick computation of the Gibbs canonical distribution of electrons between energy levels, as well as techniques for the fast calculation of the distribution function for a slow relaxation process have been found. Characteristics of the single-electron transistor in the cases of different types of molecule's energy spectrum, fast and slow energy relaxations have been compared.

DEFORMED DENSITY MATRIX AND GENERALIZED UNCERTAINTY RELATION IN THERMODYNAMICS

A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. In our opinion the approach proposed may lead to the proofs of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Planck scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Planck's. The associated deformation of a canonical Gibbs distribution is given explicitly.

Thermodynamics and hydrodynamics (statistical foundations): 2. Thermodynamic equilibrium

We show that the BBGKY hierarchy in thermodynamic equilibrium degenerates into an equilibrium hierarchy. In turn, the latter can be used to obtain both the fundamental system of equations describing the state of matter inside a correlation sphere of the radius R ≈ 10-7 cm and the canonical Gibbs distribution, which, according to the current concepts, describes the state of the entire macroscopic system. We show that the Gibbs distribution is indeed also local, i.e., describes the state of matter only inside the correlation sphere; in the thermostat surrounding this sphere, the Gibbs distribution degenerates into a constant. We state that these two approaches are equivalent to each other.

Simulation of characteristics of a molecular single-electron tunneling transistor with a discrete energy spectrum of the central electrode

Current–voltage curves of molecular single-electron tunneling transistors are simulated based on a modified theory of single electronics that accounts for the discreteness of the energy spectrum of the molecule. The simulation was performed including effects of energy relaxation of the electrons in the molecule for two limiting cases of fast and slow relaxation, and for both equidistant and randomly spaced energy levels of the molecule. An efficient recursion method allowing a fast calculation of the Gibbs canonical distribution for electrons in the molecule is suggested and realized. A comparison of the simulated I–V curves with the experimental ones shows that the experimental conditions correspond to the slow relaxation case.

A Quantum Theory of Thermodynamic Relaxation

A new approach to quantum Markov processes is developed and the corresponding Fokker-Planck equation is derived. The latter is examined to reproduce known results from classical and quantum physics. It was also applied to the phase-space description of a mechanical system thus leading to a new treatment of this problem different from the Wigner presentation. The equilibrium probability density obtained in the mixed coordinate-momentum space is a reasonable extension of the Gibbs canonical distribution.

The random-variable canonical distribution

An alternative interpretation to Gibbs' concept of the canonical distribution for an ensemble of systems in statistical equilibrium is proposed. Whereas Gibbs' theory is based upon a consideration of systems subject to dynamical law, the present analysis relies neither on the classical equations of motion nor makes use of any a priori probability of a complexion; rather, it makes avail of the basic algebra of random variables and, specifically, invokes the law of large numbers. Thereby, a canonical distribution is derived which describes a macrosystem in probabilistic, rather than deterministic, terms, and facilitates the understanding of energy fluctuations which occur in macrosystems at an overall constant ensemble temperature. A discussion is given of a modified form of the Gibbs canonical distribution which takes full account of the effects of random energy fluctuations. It is demonstrated that the results from this modified analysis are entirely consonant with those derived from the random-variable approach.

Maximum Marginal Likelihood Estimation and Constrained Optimization in Image Restoration.

In image restorations, two different mathematical frameworks for massive probabilistic models are given in the standpoint of Bayes statistics. One of them is formulated by assuming that a priori probability distribution has a form of Gibbs micro-canonical distribution and can be reduced to a constrained optimization. In this framework, we have to know a quantity estimated from the original image though we do not have to know the con.guration of the original image. We give a new method to estimate the length of boundaries between areas with different grey-levels in the original image only from the degraded image in grey-level image restorations. When the length of boundaries in the original image is estimated from the given degraded image, we can construct a probabilistic model for image restorations by means of Bayes formula. This framework can be regarded as probabilistic information processing at zero temperature. The other framework is formulated by assuming the a priori probability distribution has a form of Gibbs canonical distribution. In this framework, some hyperparameters are determined so as to maximize a marginal likelihood. In this paper, it is assumed that the a priori probability distribution is a Potts model, which is one of familiar statistical-mechanical models. In this assumption, the logarithm of marginal likelihood can be expressed in terms of free energies of some probabilistic models and hence this framework can be regarded as probabilistic information processing at finite temperature. The practical algorithms in the constrained optimization and the maximum marginal likelihood estimation are given by means of cluster variation method, which is a statistical-mechanical approximation with high accuracy for massive probabilistic models. We compare them with each others in some numerical experiments for grey-level image restorations.

From Molecular Dynamics to Dissipative Particle Dynamics

A procedure is introduced for deriving a coarse-grained dissipative particle dynamics from molecular dynamics. The rules of the dissipative particle dynamics are derived from the underlying molecular interactions, and a Langevin equation is obtained that describes the forces experienced by the dissipative particles and specifies the associated canonical Gibbs distribution for the system.

Maslov's complex germ and the asymptotic formula for the Gibbs canonical distribution

The Gibbs canonical distribution for a system of N classical particles is studied under the following conditions: the external potential isO(1), the potential of pairwise interaction isO(1/N), the potential of triple interaction isO(1/N 2), etc. The asymptotics of free energy and of the partition function asN→∞ is found. An asymptotic formula approximating the normalized canonical distribution in theL 1 norm asN→∞ is also constructed. It is proved that the chaos property is satisfied fork-particle distributions,k = const, and is not satisfied for theN-particle distribution.

A proof of the canonical Gibbs distribution

We propose a rigorous procedure to obtain the distribution function of a macroscopic system of microstates in contact with a thermal bath, namely the Gibbs canonical distribution, supposing the validity of the microcanonical distribution for the whole system.

Development of the activation process model: Compensation effect

In this article, an attempt was made to develop an activation process model. The average energy of the translational motion of the atoms, taking part in the elementary activation process and being in the thermodynamic equilibrium with thermal radiation, was obtained using the quantum canonical Gibbs distribution and the model principles of elementary activation. The degeneracy and exclusion of some excited vibrational levels were taken into consideration, the result being a strong dependence of the probability of surmounting the activation barrier on the behavior of the excited vibrational states of the quantum subsystems. As an application of the development model, the formulas of the preexponential factor for solid-state atomic diffusivity and first-order chemical reaction rate constants were derived. Quantitative analysis of the atomic diffusion in solids in the framework of our model has made it possible to describe the diffusion processes in metals, covalent semiconductors, as well as diffusion anomalies, connected with the “nonclassical” behavior of the empirical Arrhenius dependence. A possible physical essence of a kinetic compensation effect is discussed. It was shown that compensation may be caused by only changing the degeneracy of the vibrational levels of the quantum subsystems. © 1996 John Wiley & Sons, Inc.

Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns

The likelihood method is developed for the analysis of socalled regular point patterns. Approximating the normalizing factor of Gibbs canonical distribution, we simultaneously estimate two parameters, one for the scale and the other which measures the softness (or hardness), of repulsive interactions between points. The approximations are useful up to a considerably high density. Some real data are analyzed to illustrate the utility of the parameters for characterizing the regular point pattern.

Statistical Operator for Thin Solid Films

AbstractUsing the Schrödinger variational principle for average energy in the case of a system interacting weakly with environment and for a given entropy, the equilibrium statistical operator is obtained. In the case when the interaction with the environment can be neglected this statistical operator is identical with that of the canonical Gibbs distribution. This statistical operator may be useful in describing thermodynamical properties of thin solid films and similar systems.

The role of fluctuations in thermodynamics: a critical answer to Jaworski's paper 'Higher-order moments and the maximum entropy inference'

For original paper see ibid. vol.20, p.915, (1987). The authors show that energy fluctuations, and thus the higher-order moments of energy, contain essential information and cannot be neglected even in the thermodynamical limit. In contrast with some of Jaworski's statements, they point out how all the thermodynamical properties depend in a crucial way on the fluctuations on the basis of realistic physical assumptions. Indeed, phenomenological thermodynamics allows one to conclude that the Gibbs canonical distribution is the only possible probability distribution which does not violate the second principle. It follows that the knowledge of the free energy as a function of temperature beta -1 is equivalent to that of the probability law governing fluctuations. This probability law is therefore a characteristic of a body which can be investigated by measuring either energy moments at fixed beta or the mean values of entropy and energy at varying beta .

Unified formulation of quantum and statistical mechanics

Using one and the same simple variational principle for average energy, a new equation is obtained equivalent to the set of two equations: the time-independent Schrödinger equation and that describing canonical Gibbs distribution.

Likelihood Analysis of Spatial Point Patterns

SUMMARY The likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns. Approximation methods of the normalizing factor of Gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness (or hardness) of repulsive interactions. The approximations are useful up to a considerably high density. The validity of our procedure is demonstrated by some computer experiments. Some real data are analysed.

Continuity of the temperature and derivation of the Gibbs canonical distribution in classical statistical mechanics

For a classical system of interacting particles we prove, in the microcanonical ensemble formalism of statistical mechanics, that the thermodynamic-limit entropy density is a differentiable function of the energy density and that its derivative, the thermodynamic-limit inverse temperature, is a continuous function of the energy density. We also prove that the inverse temperature of a finite system approaches the thermodynamic-limit inverse temperature as the volume of the system increases indefinitely. Finally, we show that the probability distribution for a system of fixed size in thermal contact with a large system approaches the Gibbs canonical distribution as the size of the large system increases indefinitely, if the composite system is distributed microcanonically.