Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit

A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Grüner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.     

Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems

A new definition for the approximate symmetries of Itô dynamical system is given. Determining systems of approximate symmetries for Itô and Stratonovich dynamical systems have been obtained. It has been shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a Hamiltonian. The results have been applied to an example.     

Resonant response of a thermalized ensemble of nonlinear oscillators

The analysis is carried out of the response of the center of gravity (dipole moment) of the distribution of noninteracting thermalized nonlinear oscillators to a sinusoidal driving force. Heat bath coupling is modeled by damping and noise. The driving is weak, but the frequency is resonant, so that there is a nonlinear resonance in the phase space. The response has a linear part that can be obtained from the perturbation analysis and a small nonlinear correction that is specific for the resonant structure.     

Looking at Fokker-Planck dynamics with a noisy instrument

We consider the class of experiments which can be characterized by a Fokker-PIanck dynamics corresponding to the overdamped motion of a state point in a suitable stochastic potential. We assume that the general form of the potential is known (or can be guessed with reasonable accuracy), but that its parameters are to be determined experimentally by measurements made with a noisy instrument. This possible method for determining the potential parameters, which exploits the system's own internal stochastic motion in order to explore rapidly its available parameter space, is substantially more efficient than traditional methods involving time averages of single point measurements, and yet does not appear to have been previously considered. The method could be important when, for example, the experiment must be completed in a limited time owing either to the expense of the experimental materials or to the temporary stationarity of the preparation, situations which are commonly encountered in experimental biochemistry and biology.     

Projector formalism of generalized Brownian motion theory applied to dissipative and noisy systems

The projector formalism of Zwanzig-Mori type is extended to obtain generalized Fokker-Planck and generalized nonlinear Langevin equations for coarse-grained variables when the underlying microscopic dynamics is dissipative and noisy (stochastic).     

Melnikov's criterion for nondifferentiable weak-noise potentials

The stationary probability density of Fokker-Planck models with weak noise is asymptotically of the form exp[–1 /(q)]. If is smooth, it satisfies a Hamilton-Jacobi equation at zero energy and can be interpreted as the action of an associated Hamiltonian system. Under this assumption, has the properties of a Liapounov function, and can be used, e.g., as a thermodynamic potential in nonequilibrium steady states. We consider systems having several attractors and show, by applying Melnikov's method to the associated Hamiltonian, that in general is not differentiable. A small perturbation of a model with differentiable leads to a nondifferentiable . The method is illustrated on a model used in the treatment of the unstable mode in a laser.     

ISSDE: A Monte Carlo implicit simulation code based on Stratonovich SDE approach of Coulomb collision

A Monte Carlo implicit simulation program, Implicit Stratonovich Stochastic Differential Equations(ISSDE), is developed for solving stochastic differential equations(SDEs) that describe plasmas with Coulomb collision. The basic idea of the program is the stochastic equivalence between the Fokker–Planck equation and the Stratonovich SDEs. The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision. The cases of Lorentzian plasma, Maxwellian plasma and arbitrary distribution function of background plasma have been considered. The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge–Kutta methods. ISSDE is built with C++ and has standard interfaces and extensible modules. The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE, which shows its correctness and reliability.     

非平衡定态化学反应体系对细致平衡偏离的不可逆性之随机热力学量度

在随机描述的层次上,成功地构筑了非平衡定态化学反应体系对细致平衡偏离的不可逆性之随机热力学判据.基于Fokker-Planck方程建立了连续Markov过程系统的随机熵平衡方程,发现在随机态空间中随机熵的时间变率亦可分为源项和流项两部分贡献.对于化学反应体系,作为态空间源项的随机熵产生可作为偏离细致平衡不可逆性的合适度量,此泛函量的零值标志着细致平衡.进一步借助按系统广度量的Ω展开法,通过对随机力及共轭的随机流的分析揭示态空间中的随机熵产生仅源于状态对细致平衡的偏离,并主要来自非Poisson涨落的贡献,因而可作为随机热力学量去判别和量度化学反应体系对细致平衡的偏离.     

A stochastic theory for clustering of quenched-in vacancies—2. A solvable model

The model introduced for clustering of quenched-in vacancies in the first part of this series of papers is considered. Using a generating function, the rate equations are converted into a first order partial differential equation for the generating function coupled to a differential equation for the rate of change of the concentration of single vacancy units. A decoupling scheme is effected which gives an exponentially decaying solution with a very short time constant for the concentration of single vacancy units. The differential equation for the generating function is solved for times larger than the time required for the concentration of single vacancy units to reach its asymptotic value. The distribution for the size of the clusters is obtained by inverting the solution thus obtained. Several results that follow are shown to be in reasonably good agreement with the experimental results.     

The first,the biggest,and other such considerations

We investigate the relation between the underlying dynamics of randomly evolv ing systems and the extrema statistics for such systems. Independent processes, Fokker-Planck processes and Lévy processes are considered.