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.     

A new formula for the saddle-to-scission time

We propose a new formula for the saddle-to-scission time that is more general that the one based on Kramers' approach. Its validity and applicability is then studied in detail. Such a formula is useful for the evaluation of the fission time of very heavy nuclei.     

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).     

The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

Two models for the Freedericksz transition in a fluctuating magnetic field are considered: one is based on a dichotomic and the other on a nonlinear Gaussian noise. Both noises are characterized by a finite correlation time. It is shown that the linear response assumption leading to the best Fokker-Planck approximation in the dichotomic and nonlinear Gaussian cases can be trusted only up to the order ¹ and ⁰, respectively. The role of the corrections to the linear response approximation is discussed and it is shown how to replace the non-Fokker-Planck terms stemming from these corrections with equivalent terms of standard type. This technique is shown to produce perfect agreement with the exact analytical results (dichotomic noise) and to satisfactorily fit the results of analog simulation (nonlinear Gaussian noise).     

The projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator into a perturbation ₁ and an unperturbed part ₀. The standard Fokker-Planck structure is recovered at the second order in ₁, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in ₁ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time, a resummation up to infinite order in must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in ₁, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order in ₁ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady-state distributions. Therefore, over the whole range 0 the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in ₁ vanish. In the short- region the LL leads to results virtually coincident with those of the BFPA. In the large- region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.     

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.     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.