Dynamical ensembles in nonequilibrium statistical mechanics and their representations

The stationary states of driven systems of particles are considered from the point of view of the invariant probability distributions in the phase space which characterize them. The main features of various representations of such distributions are reviewed, and a brief derivation of the one based on orbital measures is given. We mention the limits of the mathematical derivations, and discuss the expected range of applicability beyond such limits. (c) 1998 American Institute of Physics.     

Positivity of entropy production in nonequilibrium statistical mechanics

We analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee() in a state . When is steady state describing the long term behavior of a system we show thate()0, and sometimes we can provee()>0.     

Unified projection operator formalism in nonequilibrium statistical mechanics

The method of projection operators, which plays an important role in the field of nonequilibrium statistical mechanics, has been established with the use of the Liouville-von Neumann equation for a density matrix to eliminate irrelevant information from a whole system. We formulate a unified and general projection operator method for dynamical variables. The main features of our formalism parallel those for the Liouville-von Neumann equation. (1) Two types of basic equations, time-convolution and time-convolutionless decompositions, are systematically obtained without specifying a projection operator. (2) Expansion formulas for both decompositions are also obtained. (3) Problems incorporating a time-dependent Liouville operator can be flexibly treated. We apply the formulas to problems in random frequency modulation and low field resonance. In conclusion, our formalism yields a more direct and easier means of determining the average time evolution of an operator than the one for the Liouville-von Neumann equation.     

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

Spontaneous synchronisation is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronisation. The model comprises limit-cycle oscillators with distributed natural frequencies interacting through a mean-field coupling. Although more than forty years have passed since its introduction, the model continues to occupy the centre stage of research in the field of non-linear dynamics and is also widely applied to model diverse physical situations. In this brief review, starting with a derivation of the Kuramoto model and the synchronisation phenomenon it exhibits, we summarise recent results on the study of a generalised Kuramoto model that includes inertial effects and stochastic noise. We describe the dynamics of the generalised model from a different yet a rather useful perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. A system is said to be long-range interacting if the inter-particle potential decays slowly as a function of distance. Using tools of statistical physics, we highlight the equilibrium and nonequilibrium aspects of the dynamics of the generalised Kuramoto model, and uncover a rather rich and complex phase diagram that it exhibits, which underlines the basic theme of intriguing emergent phenomena that are exhibited by many-body complex systems.     

Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics

Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.     

Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.     

Thermal mechanics: A quantum mechanical analogue of nonequilibrium statistical thermodynamics

A formal but not conventional equivalence between stochastic processes in nonequilibrium statistical thermodynamics and Schrödinger dynamics in quantum mechanics is shown. It is found, for each stochastic process described by a stochastic differential equation of Itô type, there exists a Schrödinger-like dynamics in which the absolute square of a wavefunction gives us the same probability distribution as the original stochastic process. In utilizing this equivalence between them, that is, rewriting the stochastic differential equation by an equivalent Schrödinger equation, it is possible to obtain the notion of deterministic limit of the stochastic process as a semi-classical limit of the “Schrödinger” equation. The deterministic limit thus obtained improves the conventional deterministic approximation in the sense of Onsager-Machlup. The present approach is valid for a general class of stochastic equations where local drifts and diffusion coefficients depend on the position. Two concrete examples are given. It should be noticed that the approach in the present form has nothing to do with the conventional one where only a formal similarity between the Fokker-Planck equation and the Schrödinger equation is considered.     

人类行为的动力学与统计力学研究

对人类行为的统计特性的研究,涉及研究经济、心理、交通、社会学等学科的根本问题.在最近一个时期以来,针对人类行为的统计揭示出了大量的出乎人们意料的规律,吸引了众多研究者的注意.这些规律不仅对认识人类自身和对大量的社会学科的基础研究有着重要价值,而且在实际应用中有着广阔的前景.文章作者回顾了这一新近发展起来的领域的主要研究成果,并重点对中国科学技术大学的相关研究进行了介绍.     

人类行为的动力学与统计力学研究

对人类行为的统计特性的研究,涉及研究经济、心理、交通、社会学等学科的根本问题.在最近一个时期以来,针对人类行为的统计揭示出了大量的出乎人们意料的规律,吸引了众多研究者的注意.这些规律不仅对认识人类自身和对大量的社会学科的基础研究有着重要价值,而且在实际应用中有着广阔的前景.文章作者回顾了这一新近发展起来的领域的主要研究成果,并重点对中国科学技术大学的相关研究进行了介绍.     

On a heuristic point of view related to quantum nonequilibrium statistical mechanics

In this paper I propose a new way for counting the microstates of a system out of equilibrium. As, according to quantum mechanics, things happen as if a given particle can be found in more than one state at once, I extend this concept to propose the coherent access by a particle to the available states of a system. By coherent access I mean the possibility for the particle to act as if it is populating more than one microstate at once. This hypothesis has experimental implications, since the thermodynamical probability and, as a consequence, the Bose-Einstein distribution as well as the argument of the Boltzmann factor is modified.     

Quantum dynamics and statistical mechanics of sine-gordon ø-particles

We examine quantal corrections to the dynamics and statistical mechanics of classical sine-Gordon ø-particles (solitons) in weakly-pinned quasi-one- dimensional charge-density-wave condensates at low temperatures. We find that (i) the motion of the quantum ø-particle is governed by Schrödinger's equation, and (ii) the sine-Gordon system at low temperatures can be regarded as a low-density “ideal gas” of ø-particles, anti-ø-particles, phase phonons, and bound pairs of ø- and anti-ø-particles.     

The time-local view of nonequilibrium statistical mechanics. II. Generalized Langevin equations

On a semiphenomenological level, generalized Langevin equations are usually obtained by adding a random force (RF) term to macroscopic deterministic equations assumed to be known. Here this procedure is made rigorous by conveniently redefining the RF, which is shown to be colored noise weakly correlated with the observables at earlier times due to the finite lifetime of microscopic events. Corresponding fluctuation-dissipation theorems are derived. Explicit expressions for the spectral density of the fluctuations are obtained in a particularly simple form, with the deviation of the line shape from the Lorentzian being related most explicitly to the spectral density of the RF. Well-known low-frequency expressions and the Einstein relation of (generalized) Brownian motion theory are modified so as to include lifetime effects. New sum rules are obtained relating dissipative quantities to contour integrals (in the complex frequency domain) over spectral densities or corresponding response functions. The Heisenberg dynamics of a complete set of macroobservables is shown to be equivalent to a generalized Orstein-Uhlenbeck stochastic process which is a non-Markovian process due to the lifetime effects.     

Synthetic Hamiltonian mechanics

This paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. Fundamental results concerning Hamiltonian vector fields, Poisson brackets, and momentum mappings are discussed. The significance of the Lie derivative in the synthetic context is also consistently stressed. In particular, the notion of an infinitesimally Euclidean space is introduced, and the Jacobi identity of vector fields with respect to Lie brackets is established naturally for microlinear, infinitesimally Euclidean spaces by using Lie derivatives instead of a highly combinatorial device such as P. Hall's 42-letter identity.     

Universality in solar flare, magnetic storm and earthquake dynamics using Tsallis statistical mechanics

The universal character of the dynamics of various extreme phenomena is an outstanding scientific challenge. We show that X-ray flux and D_st time series during powerful solar flares and intense magnetic storms, respectively, obey a nonextensive energy distribution function for earthquake dynamics with similar values for the Tsallis entropic index q. Thus, evidence for universality in solar flares, magnetic storms and earthquakes arise naturally in the framework of Tsallis statistical mechanics. The observed similarity suggests a common approach to the interpretation of these diverse phenomena in terms of driving physical mechanisms that have the same character.