Thermostats: Analysis and application

Gaussian isokinetic and isoenergetic deterministic thermostats are reviewed in the correct historical context with their later justification using Gauss' principle of least constraint. The Nose-Hoover thermostat for simulating the canonical ensemble is also developed. For some model systems the Lyapunov exponents satisfy the conjugate pairing rule and a Hamiltonian formulation is obtained. We prove the conjugate pairing rule for nonequilibrium systems where the force is derivable from a potential. The generalized symplectic structure and Hamiltonian formulation is discussed. The application of such thermostats to the Lorentz gas is considered in some detail. The periodic orbit expansion methods are used to calculate averages and to categorize the generic transitions in the structure of the attractor. We prove that the conductivity in the nonequilibrium Lorentz gas is non-negative. (c) 1998 American Institute of Physics.     

Non-Equilibrium Projection-Operator for a Quenched Thermostatted System

A projection operator formalism is presented which allows to derive an exact set of equations for correlation functions and susceptibilities in out of equilibrium situations of many particle systems. Explicitely considered is the case of an initial temperature quench in a simple liquid stabilized by a Gaussian thermostat. Implications for the violation of the fluctuation dissipation theorem in simple structural glass formers like Lennard–Jones glasses and colloidal glasses and the differences to the Kawasaki–Gunton projection operator are discussed.     

Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.     

Entropy Production in Nonequilibrium Statistical Mechanics

We consider systems of nonequilibrium statistical mechanics, driven by nonconservative forces and in contact with an ideal thermostat. These are smooth dynamical systems for which one can define natural stationary states μ (SRB in the simplest case) and entropy production e(μ) (minus the sum of the Lyapunov exponents in the simplest case). We give exact and explicit definitions of the entropy production e(μ) for the various situations of physical interest. We prove that e(μ)≥0 and indicate cases where e(μ)>0. The novelty of the approach is that we do not try to compute entropy production directly, but make it depend on the identification of a natural stationary state for the system. Received: 15 July 1996 / Accepted: 30 October 1996     

Thermostatted kinetic equations as models for complex systems in physics and life sciences

Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modeling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager–Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics.     

On the Fluctuation Relation for Nosé-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nosé-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in (D.J. Searles, et al., J. Stat. Phys. 128:1337, 2007), for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate Λ and of the dissipation function Ω, a similar relaxation regime at shorter averaging times is found. The quantity Ω satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity Λ appears to begin a monotonic convergence after such times. This is consistent with the fact that Ω and Λ differ by a total time derivative, and that the tails of the probability distribution function of Λ are Gaussian.     

On the Fluctuations about the Vlasov Limit for <Emphasis Type="Italic">N</Emphasis>-particle Systems with Mean-Field Interactions

Whereas the Vlasov (a.k.a. “mean-field”) limit for N-particle systems with sufficiently smooth potentials has been the subject of many studies, the literature on the dynamics of the fluctuations around the limit is sparse and somewhat incomplete. The present work fulfills two goals: 1) to provide a complete, simple proof of a general theorem describing the evolution of a given initial fluctuation field for the particle density in phase space, and 2) to characterize the most general class of initial symmetric probability measures that lead (in the infinite-particle limit) to the same Gaussian random field that arises when the initial phase space coordinates of the particles are assumed to be i.i.d. random variables (so that the standard central limit theorem applies). The strategy of the proof of the fluctuation evolution result is to show first that the deviations from mean-field converge for each individual system, in a purely deterministic context. Then, one obtains the corresponding probabilistic result by a modification of the continuous mapping theorem. The characterization of the initial probability measures is in terms of a higher-order chaoticity condition (a.k.a. “Boltzmann property”).     

An efficient method of distinguishing chaos from noise

It is an important problem in chaos theory whether an observed irregular signal is deterministic chaotic or stochastic. We propose an efficient method for distinguishing deterministic chaotic from stochastic time series for short scalar time series. We first investigate, with the increase of the embedding dimension, the changing trend of the distance between two points which stay close in phase space. And then, we obtain the differences between Gaussian white noise and deterministic chaotic time series underlying this method. Finally, numerical experiments are presented to testify the validity and robustness of the method. Simulation results indicate that our method can distinguish deterministic chaotic from stochastic time series effectively even when the data are short and contaminated.     

The Nosé–Hoover Thermostated Lorentz Gas

We apply the Nosé–Hoover thermostat and three variations of it, which control different combinations of velocity moments, to the periodic Lorentz gas. Switching on an external electric field leads to nonequilibrium steady states for the four models. By performing computer simulations we study the probability density, the conductivity and the attractor in nonequilibrium. The results are compared to the Gaussian thermostated Lorentz gas and to the Lorentz gas as thermostated by deterministic scattering. We find that slight modifications of the Nosé–Hoover thermostat lead to different dynamical properties of our models. However, in all cases the attractor appears to be multifractal.     

SINGULAR PERTURBATION EXPANSION FOR MARKOVIAN MASTER EQUATIONS

Asymptotic properties of the master equations for chemical reactive systems whose macroacopic rate equations have more than one stationary state are discussed Itsing generating function method. The systematic singular perturbation expansion method for equation of generating function is generalized to include the case of multi-stationary system beyond the bifurcation point and the following conclusions are proved: For such systems, there is a Gaussian fluctuation before next genuine bifurcation point] a critical fluctuation at a genuine bifurcation point; and a macroscopic fluctuation when the system is on the coexistence line which is determined by the master equation. Furthermore, the relation between the cumulant of generat-ing function and the stochastic potentia 1 is also established. Our discussion, however, is limited to homogeneous system of one variable only.     

Does the Gaussian thermostat maximize the phase-space compression factor?

A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat maximizes the average phase-space compression factor in nonequilibrium steady states is analyzed for a dilute gas under uniform shear flow. Three routes have been followed: (i) an exact solution of the Bhatnagar-Gross-Krook kinetic equation for arbitrary shear rate, (ii) an exact solution of the Boltzmann equation through super-Burnett order, and (iii) a numerical solution of the Boltzmann equation for finite shear rates. The results show that the above hypothesis does not exactly hold for arbitrary shear rates, although the thermostat that maximizes is close to the Gaussian one. In addition, the influence of the thermostat considered on the nonlinear shear viscosity is also analyzed.     

Entropy Production in Thermostats II

We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies for every 2-plane σ, where K _w is the sectional curvature of the associated Weyl connection and is the orthogonal projection of E onto σ. Then the entropy production of any SRB measure is positive unless E has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.     

Fractality of the hydrodynamic modes of diffusion

Transport by normal diffusion can be decomposed into hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with 2 degrees of freedom, the fine scale structures of these modes are singular and fractal, characterized by a Hausdorff dimension given in terms of Ruelle's topological pressure. For long-wavelength modes, we relate the Hausdorff dimension to the diffusion coefficient and the Lyapunov exponent. This relationship is tested numerically on two Lorentz gases, one with hard repulsive forces, the other with attractive, Yukawa forces. The agreement with theory is excellent.     

Some comments on the Mindlin-Goodman method of solving vibrations problems with time dependent boundary conditions

Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.     

OPTIMAL TRACKING OF PARAMETER DRIFT IN A CHAOTIC SYSTEM: EXPERIMENT AND THEORY

We present a method of optimal tracking for chaotic dynamical systems with a slowly drifting parameter. The net drift in the parameter is assumed to be small: this makes detecting and tracking the drift more difficult. The method relies on the existence of underlying deterministic behavior in the dynamical system, yet neither requires a system model nor develops one. We begin by describing an experimental study where a heuristic optimality criterion gave good tracking performance: the tracking method there was based on maximizing smoothness and overall variation in the drift observer, which was found by solving an eigenvalue problem. We then develop a theory, based on simplifying assumptions about the chaotic dynamics, to explain the success of the tracking method for chaotic systems. For signals from deterministic systems that are sufficiently complex in a sense that we make precise, typical drift observers provide poor tracking performance and require the drift to be particularly slow. In contrast, our theory shows that the optimality criterion seeks out a special drift observer that both provides better tracking performance and allows the drift to be appreciably faster. For periodic or quasiperiodic systems (no chaos), good tracking is easily achievable and the present method is irrelevant. For stochastic systems (no determinism), the optimal tracking method does not asymptotically improve tracking performance. Exhaustive numerical simulations of a simple drifting chaotic map, first without and then with stochastic forcing, show agreement with theoretical predictions of tracking performance and validate the theory.     

Conversion of a chaotic attractor into a strange nonchaotic attractor in an one dimensional map and BVP oscillator

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.     

涨落作用下周期驱动的分数阶过阻尼棘轮模型的混沌输运现象

考虑涨落作用下周期驱动的过阻尼分数阶棘轮模型, 通过模型的数值求解, 研究确定性棘轮的混沌特性与噪声的作用对输运行为的影响, 进而讨论过阻尼分数阶分子马达反向输运的机理. 分析表明: 随着势垒高度、 势不对称性与模型记忆性的变化, 随机棘轮的反向输运并不必然地要求确定性棘轮也反向输运; 随着模型阶数的减小, 亦即分数阻尼介质记忆性的增强, 确定性棘轮在反向输运之前会经历一个周期倍化导致的混沌状态, 但在噪声作用下, 反向流的发生会提前, 即混沌状态的确定性棘轮在噪声的作用下即可进行反向输运. 也就是说, 噪声能定性地改变棘轮的输运状态: 从无噪声时的混沌运动到有噪声时的定向输运. 这是过阻尼随机棘轮反向输运的一种机理, 也是噪声在定向输运过程中发挥积极作用的一个体现.     

Chaotic dynamics,fluctuations, nonequilibrium ensembles

The ideas and the conceptual steps leading from the ergodic hypothesis for equilibrium statistical mechanics to the chaotic hypothesis for equilibrium and nonequilibrium statistical mechanics are illustrated. The fluctuation theorem linear law and universal slope prediction for reversible systems is briefly derived. Applications to fluids are briefly alluded to. (c) 1998 American Institute of Physics.     

Statistics of power injection in a plate set into chaotic vibration

A vibrating plate is set into a chaotic state of wave turbulence by either a periodic or a random local forcing. Correlations between the forcing and the local velocity response of the plate at the forcing point are studied. Statistical models with fairly good agreement with the experiments are proposed for each forcing. Both distributions of injected power have a logarithmic cusp for zero power, while the tails are Gaussian for the periodic driving and exponential for the random one. The distributions of injected work over long time intervals are investigated in the framework of the fluctuation theorem, also known as the Gallavotti-Cohen theorem. It appears that the conclusions of the theorem are verified only for the periodic, deterministic forcing. Using independent estimates of the phase space contraction, this result is discussed in the light of available theoretical framework.     

用一种少参数非线性自适应滤波器自适应预测低维混沌时间序列

基于混沌动力系统的相空间延迟坐标重构,利用混沌序列固有的确定性和非线性,提出了用 于混沌时间序列预测的一种少参数非线性自适应滤波预测模型.该预测模型在Volterra自适 应滤波器的基础上引入sigmoid函数来减少待定参数.实验研究表明,这种少参数非线性自适 应滤波预测器仅需用50个样本经20次预训练后,就能有效地预测一些低维混沌序列,且这种 少参数非线性自适应滤波预测器更便于工程实现. 关键词： 混沌 非线性自适应预测 少参数非线性自适应滤波器 自适应算法