Fokker-Planck description of multi-degree-of-freedom systems with correlated fluctuations

A Fokker-Planck equation is derived for a many-degree-of-freedom nonlinear Langevin equation driven by parametric gaussian fluctuations with finite correlation times. An oscillator with a fluctuating frequency is presented as an example.     

Augmented Langevin approach to fluctuations in nonlinear irreversible processes

A Fokker-Planck equation derived from statistical mechanics by M. S. Green [J. Chem. Phys. 20:1281 (1952)] has been used by Grabertet al. [Phys. Rev. A 21:2136 (1980)] to study fluctuations in nonlinear irreversible processes. These authors remarked that a phenomenological Langevin approach would not have given the correct reversible part of the Fokker-Planck drift flux, from which they concluded that the Langevin approach is untrustworthy for systems with partly reversible fluxes. Here it is shown that a simple modification of the Langevin approach leads to precisely the same covariant Fokker-Planck equation as that of Grabertet al., including the reversible drift terms. The modification consists of augmenting the usual nonlinear Langevin equation by adding to the deterministic flow a correction term which vanishes in the limit of zero fluctuations, and which is self-consistently determined from the assumed form of the equilibrium distribution by imposing the usual potential conditions. This development provides a simple phenomenological route to the Fokker-Planck equation of Green, which has previously appeared to require a more microscopic treatment. It also extends the applicability of the Langevin approach to fluctuations in a wider class of nonlinear systems.     

Microdynamics and equations of motion for macrovariables

By means of a new time-dependent projection operator an exact generalized Langevin equation for the macrovariables of a system is derived. This equation is in general nonlinear and also valid far from equilibrium. The projection operator picks up the macroscopic part of an observable which is defined in such a way that it's mean value depends only on the macroscopic state given by the mean values of the considered macrovariables. The exact equation can be separated into an evolution equation for the mean values and an equation for the fluctuations. The second equation contains a nonlinear random force and a term which shows up to be the linearization of the mean value equations around the mean path. The connection with previous works is discussed.     

A master equation description of local fluctuations

A theory of fluctuations of macrovariables in nonequilibrium systems based on a nonlinear master equation is outlined. This equation takes into account, via a “mean field” type of approximation, the effect of the spatial extension of fluctuations. A comparison with the birth and death formalism reveals several unsatisfactory features of the latter.     

Basic concepts of synergetics

The paper presents a brief outline of microscopic as well as of macroscopic synergetics. In microscopic synergetics we start from evolution equations for microscopic variables or densities in which fluctuating forces and control parameters are included. When control parameters are changed, the systems are studied close to instability points. The concepts of order parameters, enslaving, critical fluctuations, and critical slowing down are presented. In macroscopic synergetics unbiased estimates on distribution functions and underlying processes are made based on observed moments or correlation functions. In such a case, a Fokker-Planck equation or a corresponding Langevin equation may be derived.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

A stochastic approach to the kinetic theory of gases

A theory of fluctuations in non-equilibrium diluted gases is presented. The velocity distribution function is treated as a stochastic variable and a master equation for its probability is derived. This evolution equation is based on two processes: binary hard sphere collisions and free flow. A mean-field approximation leads to a non-linear master equation containing explicitly a parameter which represents the spatial correlation length of the fluctuations. An infinite hierarchy of equations for the successive moments is found. If the correlation length is sufficiently short a truncation after the first equation is possible and this leads to the Boltzmann kinetic equation. The associated probability distribution is Poissonian. As to the fluctuation of the macroscopic quantities, an approximation scheme permits to recover the Langevin approach of fluctuating hydrodynamics near equilibrium and its fluctuation-dissipation relations.     

On the Langevin formalism for nonlinear and nonequilibrium hydrodynamic fluctuations

The Landau-Lifshitz method of fluctuating hydrodynamics is generalized to the cases of nonlinear and nonequilibrium fluctuations. For a simple one-component fluid, the multiplicative random fluxes are constructed by using universal Gaussian variables with variances independent of the specific parameters of a fluid. It is shown that the nonlinear Langevin formalism proposed is equivalent to the approach based on the hydrodynamic Fokker-Planck equation derived earlier by statistical-mechanical methods. Then, the scheme is extended to the case of two-component fluids, where cross effects must be taken into account. In conclusion, the connection of the present formalism with the Keizer approach to nonequilibrium fluctuations is discussed.     

Microdynamics and nonlinear stochastic processes of gross variables

Starting from classical Hamiltonian mechanics, we derive for the dynamics of gross variables in nonequilibrium systems exact nonlinear generalized Fokker-Planck and Langevin equations in which the effect of the initial preparation is taken into account explicitly. This latter concept allows for the construction of a uniquely determined projection operator. The memory functions occurring in the Langevin equations are related to the random forces by a fluctuation-dissipation theorem of the second kind. We discuss the connection with the generalized Fokker-Planck equation. The known results for equilibrium fluctuations are recovered as a special case.Supported in part by the National Science Foundation, Grant CHE78-21460.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.     

The quantum-fluctuations of the optical parametric oscillator. II

We set up an effective Hamiltonian for an optical parametric oscillator. It contains the Bose operators of the three modes, signal, idler, and pump and their coupling to heat baths. This Hamiltonian is shown to be equivalent to a set of equations of motion, derived in a previous paper (I) from a microscopically exact Hamiltonian, provided that the heat baths are chosen in an adequate way. The comparison with the laser Hamiltonian makes clear the close analogy of the underlying elementary processes of spontaneous emission from atoms and spontaneous parametric emission from light modes in nonlinear media. The Hamiltonian is used to derive a master equation for the statistical operator of the three-mode system. In the coherent state representation this master equation transforms into an equivalentc-number Fokker-Planck equation without any approximation. The solution is obtained below threshold by linearization and above threshold by quasilinearization of the nonlinear dissipation coefficients. The results agree with those which were obtained by quantum mechanical Langevin methods in a previous paper (I).     

Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

Quantum fluctuations,master equation and Fokker-Planck equation

In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.     

A systematic derivation of exact generalized Brownian motion theory

We present here a simple unified derivation of the exact Fokker-Planck equation obtained earlier by Zwanzig and the exact Langevin and transport equations derived by Mori. The derivation, based on the use of a Hilbert space formulation of the dynamics, leads to substantial generalizations of these results in a straightforward manner. We obtain nonlinear Langevin equations for classical systems and discuss the extension of the theory to driven transport and to quantum dynamics based either on the use of density matrices or Γ-space densities as suggested by Wigner. Remaining limitations of the theory are pointed out.     

Nonlinear effects of colored nonstationary noise: Exact results

The master equation is derived for random systems under nonlinear time-dependent conditions. The (non-Markov) process is of such a type that with a time-dependent state transformation the dynamics can be modelled by a nonlinear but drift-free Langevin equation. The focus is on the statistical content of resulting master equation. The existence of stationary solutions and the quality of approximative results is discussed.     

Microscopic theory of brownian motion: III. The nonlinear Fokker-Planck equation

The nonlinear Fokker-Planck equation for the momentum distribution of a brownian particle of mass M in a bath of particles of mass m is derived. The contribution to this equation arising from initial deviation from bath equilibrium is analysed. This contribution is free of slow M-dependent decays and with certain restrictions leads to an effective shift in the initial value of the B particle momentum. The nonlinear Fokker-Planck equation for an initial bath equilibrium state is analyzed in terms of its predictions for momentum relaxation and mode coupling effects. It is found that in addition to nonlinear renormalization of the type previously found for the momentum correlation function, mode coupling leads to long-lived memory of the initial momentum state.     

Kinetic and hydrodynamic models of chemotactic aggregation

We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean-field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean-field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.