On the theory of fluctuations around non-equilibrium steady states: A generalized time-convolutionless projector formalism

A new method of finding nonlinear Langevin type equations of motion for relevant macrovariables and the corresponding master equation for systems far from thermal equilibrium is presented by generalizing the time-convolutionless formalism proposed previously for equilibrium hamiltoian systems by Tokuyama and Mori. The Langevin type equation consists of a fluctuating force, and the nonlinear drift coefficients which are always identical to those of the master equation. A simple formula which relates the drift coefficients to the time correlation of the fluctuating forces is derived. This is a generalization of the fluctuation-dissipation theorem of the second kind in equilibrium systems and is valid not only for transport phenomena due to internal fluctuations but also for transport phenomena due to externally-driven fluctuations. A new cumulant expansion of the master equation is also obtained. The conditions under which a Langevin and a Fokker-Planck equation of a generalized type for non-equilibrium open systems can be derived are clarified.The theory is illustrated by studying hydrodynamic fluctuations near the Rayleigh-Bénard instability. The effects of two kinds of fluctuations, internal fluctuations of irrelevant macrovariables and external (thermal) noises, on the convective instability are investigated. A stochastic Ginzburg-Landau type equation for the order parameter and the corresponding nonlinear Fokker-Planck equation are derived.     

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.     

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.     

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.     

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).     

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.     

Stochastic quantization method for spinning particles in a gravitational plane wave field

The exact and analytic Green functions for spinning relativistic particles in interaction with a gravitational plane wave field are obtained within the Stochastic Quantization Method of Parisi and Wu. We have separated the classical calculations from those related to the quantum fluctuations. The problem has been solved by using a perturbative treatment via the Langevin equation relying on phase and configuration spaces formulation.      

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.     

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.     

Numerical studies on the phase-space evolution of relative motion of two heavy ions

The two-dimensional Fokker-Planck equation describing the phase-space evolution of the reduced density matrixf(r, v; t) with respect to radial distancer and radial velocityv is numerically solved for various cases of inelastic heavy-ion collisions. A comparison of the exact solution with the global moment approach (GMA) used so far by several authors clearly shows that the GMA is not applicable for deeply inelastic collisions. A local moment approach (LMA) is proposed as an approximate solution to the Fokker-Planck equation.     

Exact solution to the cauchy problem for a generalized “linear” vectorial Fokker-Planck equation: Algebraic approach

The exact solution to the Cauchy problem for a generalized “linear” vectorial Fokker-Planck equation is found by using the disentangling techniques of Feynman and algebraic (operational) methods.     

Strategies for fluctuation renormalization in nonlinear transport theory

We are concerned here with the problems encountered in the derivation of nonlinear transport equations from a correspondingly nonlinear Langevin equation. A dynamical coupling between the time-dependent averages and the fluctuations must be accounted for by a procedure which leads to a renormalization of the nonlinear transport equation. Generalizing the familiar phenomenological approach to Brownian motion to nonlinear dynamics, we illustrate how the problem arises and show how the fluctuation renormalization can be obtained exactly by a formal procedure or approximately by more tractable methods.     

On the maximal noise for stochastic and QCD travelling waves

Using the relation of a set of nonlinear Langevin equations to reaction–diffusion processes, we note the existence of a maximal strength of the noise for the stochastic travelling wave solutions of these equations. Its determination is obtained using the field-theoretical analysis of branching-annihilation random walks near the directed percolation transition. We study its consequence for the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation. For the related Langevin equation modeling the quantum chromodynamic nonlinear evolution of gluon density with rapidity, the physical maximal-noise limit may appear before the directed percolation transition, due to a shift in the travelling-wave speed. In this regime, an exact solution is known from a coalescence process. Universality and other open problems and applications are discussed in the outlook.     

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.     

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.     

Nonlinear waves in media with fifth order dispersion

Nonlinear waves described of the fifth order dispersive nonlinear evolution equation are numerically investigated. The numerical method for boundary value problem for this equation is proposed. Exact solutions to nonlinear evolution equation of the fifth order are given. The numerical method was tested using some exact solutions. The influence of the fifth order dispersion on the propagation of nonlinear waves and formation of the periodic structures is studied.     

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.     

THE FOKKER-PLANCK EQUATiON AND THE SCHWI NGER-DYSON EQUATION IN LATTICE GAUGE THEORY

The stochastic quantization in Lattice Gauge Theory (LGT) is discussed by using Langevin equations and Fokker-Planck equations. It is shown that the evolution equation in stochastic process reduces to Schwinger-Dyson equation when lattice system reaches equilibrium.