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.     

The stochastic Boltzmann equation and hydrodynamic fluctuations

Based on the assumption of a kinetic equation in space, a stochastic differential equation of the one-particle distribution is derived without the use of the linear approximation. It is just the Boltzmann equation with a Langevin-fluctuating force term. The result is the general form of the linearized Boltzmann equation with fluctuations found by Bixon and Zwanzig and by Fox and Uhlenbeck. It reduces to the general Landau-Lifshitz equations of fluid dynamics in the presence of fluctuations in a similar hydrodynamic approximation to that used by Chapman and Enskog with respect to the Boltzmann equation.This work received financial support from the Alexander von Humboldt Foundation.     

Fluctuations in a fluid under a stationary heat flux. I. General theory

We present the basic formulas for a unified treatment of the correlation functions of the hydrodynamic variables in a fluid between two horizontal plates which is exposed to a stationary heat flux in the presence of a gravity field (Rayleigh-Bénard system). Our analysis is based on fluctuating hydrodynamics. In this paper (I) we show that in the nonequilibrium stationary state the hydrodynamic fluctuations evolve on slow and fast time scales that are widely separated. A time scale perturbation theory is used to diagonalize the hydrodynamic operator partially. This enables us to derive the eigenvalue equations for the nonequilibrium hydrodynamic modes. Therein we take into account the variation of the macroscopic quantities with position. The correlation functions are formally expressed in terms of the nonequilibrium modes. In paper II the slow hydrodynamic modes (viscous and viscoheat modes) will be determined explicitly for ideal heat-conducting plates with stick boundary conditions and used to compute the slow part of the correlation functions; in paper III the fast hydrodynamic modes (sound modes) will be explicitly determined for stick boundary conditions and used to compute the fast part of the correlation functions. In these papers we will also compute the shape and intensity of the lines measured in light scattering experiments.     

Brownian motion in a rotating flow

The Langevin equations for a particle of an arbitrary shape and the correlation functions for the fluctuating forces, torques, or force-torque acting on the particle in a rotating flow are derived from the semimicroscopic level of coarse graining by using fluctuating hydrodynamics. In order to obtain the solution of the Navier-Stokes Langevin equation valid over the entire flow region, use is made of the method of matched asymptotic expansions in ( _f a²/v)^1/2 1. The cases of slow and rapid rotation are analyzed. It is shown that the fluctuation-dissipation theorems hold up to the order of ( _f a²/v)^1/2 in both slow and rapid rotation, and that the diffusivity tensor depends on the angular velocity of the fluid and becomes anisotropic.     

Stochastic analyses of the dynamics of generalized Little-Hopfield-Hemmen type neural networks

Stochastic analyses are conducted of model neural networks of the generalized Little-Hopfield-Hemmen type, in which the synaptic connections with linearly embeddedp sets of patterns are free of symmetric ones, and a Glauber dynamics of a Markovian type is assumed. Two kinds of approaches are taken to study the stochastic dynamical behavior of the network system. First, by developing the method of the nonlinear master equation in the thermodynamic limitN, an exact self-consistent equation is derived for the time evolultion of the pattern overlaps which play the role of the order parameters of the system. The self-consistent equation is shown to describe almost completely the macroscopic dynamical behavior of the network system. Second, conducting the system-size expansion of the master equation for theN-body probability distribution of the Glauber dynamics makes it possible to analyze the fluctuations. In the course of the analysis, the self-consistent equation for the pattern overlaps is derived again. The main result of the rigorous fluctuation analysis is that as far as the fluctuations are concerned, the time course of the pattern overlap fluctuations behaves independently of the fluctuations in the remaining modes of the system's macrovariables, in accordance with the self-determining property of the macroscopic motion of the pattern overlaps for neural networks with linear synaptic couplings.     

Equations of fluctuating nonlinear hydrodynamics for normal fluids

The full set of fluctuating nonlinear hydrodynamic equations for normal fluids is derived from the conventional Langevin equations extended to include multiplicative noise. The equations describing the set of conserved variables (the mass density, the momentum densityg, the energy density) agree with those found by Morozov for a case of a driving free energy which is a local function of the hydrodynamic variables. We show here that if the standard form of the hydrodynamic equations is to hold in the absence of noise, then the driving free energy must be a local function ofg and, but it may have to be a nonlocal function of the mass density.     

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.     

Nonequilibrium fluctuations from a nematic under a thermal gradient and a gravity field*

We use a fluctuating hydrodynamics (FH) approach to study the fluctuations of the hydrodynamic variables of a thermotropic nematic liquid crystal (NLC)in a nonequilibrium steady state (NESS). This NESS is produced by an externally imposed temperature gradient and a uniform gravity field. We calculate analytically the equilibrium and nonequilibrium seven modes of the NLC in this NESS. These modes consist of a pair of sound modes, one orientation mode of the director and two visco-heat modes formed by the coupling of the shear and thermal modes. We find that the nonequilibrium effects produced by the external gradients only affect the longitudinal modes. The analytic expressions for the visco-heat modes show explicitly how the heat and shear modes of the NLC are coupled. We show that they may become propagative, a feature that also occurs in the simple fluid and suggests the realization of new experiments. We show that in equilibrium and in the isotropic limit of the NLC, our modes reduce to well-known results in the literature. For the NESS considered we point out the differences between our our modes and those reported by other authors. We close the paper by proposing the calculation of other physical quantities that lend themselves to a more direct comparison with possible experiments for this system.     

The elimination of fast variables in complex chemical reactions. II. Mesoscopic level (reducible case)

The master equation for chemical reactions that proceed through a number of steps (complex reactions) is considered. Examples are studied in which the reaction constant of one of the steps is much larger than the others, and a reduced master equation is derived by means of a projection operator formalism. This reduction amounts to an elimination of intermediates. The consistency of the scheme is shown by means of the-expansion.     

One-dimensionalXY model: Ergodic properties and hydrodynamic limit

We prove theorems on convergence to a stationary state in the course of time for the one-dimensionalXY model and its generalizations. The key point is the well-known Jordan-Wigner transformation, which maps theXY dynamics onto a group of Bogoliubov transformations on the CARC ^*-algebra overZ ¹. The role of stationary states for Bogoliubov transformations is played by quasifree states and for theXY model by their inverse images with respect to the Jordan-Wigner transformation. The hydrodynamic limit for the one-dimensionalXY model is also considered. By using the Jordan-Wigner transformation one reduces the problem to that of constructing the hydrodynamic limit for the group of Bogoliubov transformations. As a result, we obtain an independent motion of normal modes, which is described by a hyperbolic linear differential equation of second order. For theXX model this equation reduces to a first-order transfer equation.     

Numerical integration of the fluctuating hydrodynamic equations

An approach to numerically integrate the Landau-Lifshitz fluctuating hydrodynamic equations is outlined. The method is applied to one-dimensional systems obeying the nonlinear Fourier equation and the full hydrodynamic equations for a dilute gas. Static spatial correlation functions are obtained from computer-generated sample trajectories (time series). They are found to show the emergence of long-range behavior whenever a temperature gradient is applied. The results are in very good agreement with those obtained from solving the correlation equations directly.     

Source function extracted from single-event HBT correlation functions of hydrodynamical sources

We investigate the single-event two-pion correlation functions for the hydrodynamic particle-emitting sources with the fluctuating initial conditions generated by the Heavy Ion Jet Interaction Generator (HIJING). Using a three-dimension fast Fourier transform (FFT), we further extract the source functions from the single-event correlation functions. It is found that the inhomogeneity of the hydrodynamic sources with the fluctuating initial conditions lead to event-by-event fluctuations of the correlation functions and source functions.     

Hydrodynamic Fluctuations in Laminar Fluid Flow. I. Fluctuating Orr-Sommerfeld Equation

In recent years it has become evident that fluctuating hydrodynamics predicts that fluctuations in nonequilibrium states are always spatially long ranged. In this paper we consider the application of fluctuating hydrodynamics to laminar fluid flow, using plane Couette flow as a representative example. Specifically, fluctuating hydrodynamics yields a stochastic Orr-Sommerfeld equation for the wall-normal velocity fluctuations, where spontaneous thermal noise acts as a random source.This stochastic equation needs to be solved subject to appropriate boundary conditions. We show how an exact solution can be obtained from an expansion in terms of the eigenfunctions of the Orr-Sommerfeld hydrodynamic operator. We demonstrate the presence of a flow-induced enhancement of the wall-normal velocity fluctuations and a resulting flow-induced energy amplification and provide a quantitative analysis how these quantities depend on wave number and Reynolds number.     

The elimination of fast variables in complex chemical reactions. III. Mesoscopic level (irreducible case)

The master equation for a complex chemical reaction cannot always be reduced to a simpler master equation, even if there are fast and slow individual reaction steps. Nevertheless the elimination of intermediates can be carried out with the help of the-expansion. This is illustrated with a well-known complex reaction: the dissociation of N₂O₅. It is shown that the intrinsic fluctuations in the N₂O₅ decay are larger than those implied by the master equation suggested by the macroscopic rate law.     

A scaling method for deriving kinetic equations from the BBGKY hierarchy

On the basis of the scale covariance of correlation functions under a coarsegraining in space and time, the Boltzmann equation for neutral gases, the Balescu-Lenard-Boltzmann-Landau equation for dilute plasmas, and linear equations for the variances of fluctuations are derived from the BBGKY hierarchy equations with no short-range correlations at the initial time. This is done by using Mori's scaling method in an extended form. Thus it is shown that the scale invariance of macroscopic features affords a useful principle in nonequilibrium statistical mechanics. It is also shown that there existtwo kinds of correlation functions, one describing the interlevel correlations of the kinetic level with its sublevels and the other representing the fluctuations in the kinetic level.Partially financed by the Scientific Research Fund of the Ministry of Education.     

External non-white noise and nonequilibrium phase transitions

Langevin equations with external non-white noise are considered. A Fokker Planck equation valid in general in first order of the correlation time of the noise is derived. In some cases its validity can be extended to any value of. The effect of a finite in the nonequilibrium phase transitions induced by the noise is analyzed, by means of such Fokker Planck equation, in general, for the Verhulst equation under two different kind of fluctuations, and for a genetic model. It is shown that new transitions can appear and that the threshold value of the parameter can be changed.     

On fluctuations in μ-space

A master equation is derived microscopically to describe the fluctuating motion of the particle density in . space. This equation accounts for the drift motion of particles and is valid for any inhomogeneous gas. The Boltzmann equation is obtained from the first moment of this equation by neglecting the second cumulant (the pair correlation function). The successive moments form coarse-grained BBGKY-like hierarchy equations, in which small spatial regions with r_ij < the force range are smeared out. These hierarchy equations are convenient for investigating the nonequilibrium long-range pair correlation function, which arises mainly from sequences of isolated binary collisions and gives rise to the much-discussed long-time tail and the logarithmic term in the density expansion of transport coefficients. It is shown to have a spatial long tail, like the Coulombic potential, in a steady laminar flow. The stochastic nature of the nonlinear Boltzmann-Langevin equation is also investigated; the random source term is found to be expressed as a linear superposition of Poisson random variables and to become Gaussian in special cases.     

Phenomenological equations for multicomponent fluids

Ahydrodynamic equation of motion for each component of a multicomponent fluid is derived on the basis of nonequilibrium thermodynamics. Special care has been directed to the choice of state variables. In some limiting cases, this equation leads to customary phenomenological equations, such as the equation for diffusion and the Navier-Stokes equation. The viscosity is a consequence of nonlocal coupling of forces and fluxes. The reciprocity between the linear coefficients is examined closely.     

Regression law and the renormalization of the transport coefficients

Using Mazur's lemma we show that the coarse-grained variables used in nonequilibrium statistical mechanics are the Onsager's regression variables. With this result we find a regression law for the fluctuations which is both non-Markovian and nonlinear. Considering the Markovian approximation and generalizing Onsager's ideas leading to the symmetry of the transport matrix, we formulate Mori and Fujisaka's method for the renormalization of transport coefficients due to nonlinear interactions.