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 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.     

Comments on the Grad procedure for the Fokker-Planck equation

If the kinetic equation of a macroscopic system is expanded with respect to the velocity in terms of orthogonal functions, e.g., in terms of Hermite functions, one obtains an infinite hierarchy of equations for the expansion coefficients. Grad's method consists in truncating this hierarchy and investigating the remaining finite system. In this paper we set up conditions under which this procedure is rigorously justified in case of the Fokker-Planck equation.     

On the description of dissipative collective motion

We use a recently developed time-dependent projection method to describe the dissipation of collective motion coupled to an intrinsic system. The underlying physical picture is similar to that of the linear response approach. Our approach is, however, different from the conceptual point of view. We do not resort to a quasistatic picture but use instead a time-dependent projector. Furthermore, we project on a model space which includes the intrinsic hamiltonian in addition to the collective subspace. In this way we obtain a Fokker-Planck equation for the collective variables which is coupled to a transport equation describing the evolution of the temperature of the intrinsic system.     

Consequences of coarse-grained Vlasov equations

The Vlasov equation is analyzed for coarse-grained distributions resembling a finite width of test particles as used in numerical implementations. It is shown that this coarse-grained distribution obeys a kinetic equation similar to the Vlasov equation, but with additional terms. These terms give rise to entropy production indicating dissipative features due to a nonlinear mode coupling. The interchange of coarse graining and dynamical evolution is discussed with the help of an exactly solvable model for the self-consistent Vlasov equation and practical consequences are worked out. By calculating analytically the stationary solution of a general Vlasov equation we can show that a sum of modified Boltzmann-like distributions is approached dependent on the initial distribution. This behavior is independent of degeneracy and only controlled by the width of test particles. The condition for approaching a stationary solution is derived and it is found that the coarse graining energy given by the momentum width of test particles should be smaller than a quarter of the kinetic energy. Observable consequences of this coarse graining are: (i) spatial correlations in observables, (ii) too large radii of clusters or nuclei in self-consistent Thomas–Fermi treatments, (iii) a structure term in the response function resembling vertex correction correlations or internal structure effects and (iv) a modified centroid energy and higher damping width of collective modes.     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

Coarse grained approach for volume conserving models

Volume conserving surface (VCS) models without deposition and evaporation, as well as ideal molecular-beam epitaxy models, are prototypes to study the symmetries of conserved dynamics. In this work we study two similar VCS models with conserved noise, which differ from each other by the axial symmetry of their dynamic hopping rules. We use a coarse-grained approach to analyze the models and show how to determine the coefficients of their corresponding continuous stochastic differential equation (SDE) within the same universality class. The employed method makes use of small translations in a test space which contains the stationary probability density function (SPDF). In case of the symmetric model we calculate all the coarse-grained coefficients of the related conserved Kardar–Parisi–Zhang (KPZ) equation. With respect to the symmetric model, the asymmetric model adds new terms which have to be analyzed, first of all the diffusion term, whose coarse-grained coefficient can be determined by the same method. In contrast to other methods, the used formalism allows to calculate all coefficients of the SDE theoretically and within limits numerically. Above all, the used approach connects the coefficients of the SDE with the SPDF and hence gives them a precise physical meaning.     

Dynamics of spinor condensates and stochastic approach

The dynamics of the collective spin for Bose-Einstein condensates with nonlinear interactions, is studied within the framework of the two-component spinor. We discuss the spin resonance when the system is submitted to a periodically-modulated magnetic field at the zero temperature. In this case, the nonlinearity parameter controls the critical change between a localized and a homogeneous spin state. When the temperature is finite – or a random magnetic field is considered – the movement of the collective spin is governed by the Landau-Lifshitz-Gilbert equation, from which the complete Fokker-Planck equation is derived. This equation is the essential tool to describe the time-evolution of the probability distribution function for the collective spin. The functional integral approach is used to solve analytically examples of BEC spin behavior in a static magnetic field at finite temperature. We show how such a method can lead effectively to the complete solution of the Fokker-Planck equation for this kind of problems.     

From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.     

Optical bistability due to a two-photon absorption resonance with fluctuations

Using the intensity-dependent complex dielectric function for a two-photon absorption resonance we derive the Langevin equation for the fluctuating light-field in the non-linear resonator. The corresponding Fokker-Planck equation is solved by expanding the distribution function in terms of products of trigonometric functions and generalized Laguerre polynomials. The expansion coefficients are calculated using the method of matrix continued fractions. Numerical results for the stationary case are given.     

Correlations in classical and quantum systems far from thermal equilibrium

We consider classical systems described by a Fokker-Planck equation or a generalized Fokker-Planck equation and quantum systems described by a density matrix equation or by a generalized Fokker-Planck equation using the principle of quantum classical correspondence. We split the corresponding operators of the equation of motion into a part which refers to the proper system and another one which describes the coupling of the proper system to the external world (reservoirs). We demonstrate that by use of conservation laws, referring to the proper systems, exact relations hold for certain moments, valid for all temperatures and coupling constants of the reservoirs. Using the concepts of a previous paper we describe then a perturbation theoretical approach which allows in a simple manner to determine a number of important correlation functions (moments of the total system). The time dependent case is briefly discussed. The applicability and usefulness of the present procedure is demonstrated by the example of the single-mode laser yielding e.g. expressions for the atom-field correlation.     

Generalized hydrodynamics of systems of Brownian particles

A generalized hydrodynamic theory is developed for systems of interacting Brownian particles on the basis of a Fokker-Planck equation. General results are derived for correlation functions, frequency- and wave-vector dependent transport coefficients. Explicit expressions for moments, cumulants and the hydrodynamic limits of the transport coefficients are given. For the special cases of overdamped systems with and without hydrodynamic interaction the general results are simplified. As an example for the application of this approach the system of charged spherical polystyrene spheres in aqueous solution is treated in detail. The generalized transport functions are evaluated in mode-mode coupling approximation and detailed numerical results are presented for various collective and single-particle properties. Finally, the relationship to a corresponding Smoluchowski approach is discussed.     

Quantum mechanical solutions of the laser masterequation

We treat a laser consisting of one mode described by a running wave and a set of atoms with two optically active levels which are homogeneously broadened. We start from the laser density matrix equations ofWeidlich andHaake and define a distribution functionf for lightfield and atomic variables, where we use for the lightfield the coherent state representation and for the atomic system a modified version of the distribution function used bySchmid andRisken in a previous paper. We derive a partial differential equation forf which is completely exact and is of the type of a generalized Fokker-Planck equation, i.e. it contains higher derivatives. Using a recently stated theorem ofHaken andWeidlich we show that this distribution function allows to calculate single-time as well as multitime quantum mechanical correlation functions. If the leading terms of the generalized Fokker-Planck equation are retained we find the semiclassical Fokker-Planck equation ofRisken,Schmid andWeidlich. Our treatment can be extended to several modes connected with standing waves and multilevel atoms.     

Kinetic Approach to Long time Behavior of Linearized Fast Diffusion Equations

We show that the rate of convergence towards the self-similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self-similar solution at a fixed time. As a consequence, we find an improved rate of convergence to self-similarity in terms of a Fourier based distance between two solutions. The results are based on the asymptotic equivalence of a collisional kinetic model of Boltzmann type with a linear Fokker-Planck equation with nonconstant coefficients, and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous Boltzmann equation for Maxwell molecules.     

Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling; the effect of a stochastic temporal variation in the frequencies is also included. The Fokker-Planck equation for the coupled Langevin system is reduced to a kinetic equation for the oscillator distribution function. Instabilities of the phase-incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes. Depending on the coupling, the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected. A detailed analysis of these singularities to all orders in the normal form expansion is presented. Physically, the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization. These predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions.     

The Fokker-Planck equation of a laser with many modes and multi-level atoms

Using the Glauber-SudarshanP-representation for the field modes and a quasi-distribution function recently presented for arbitrary quantum systems we derive an exact generalized Fokker-Planck equation for a multi-mode laser containing a set of multi-level atoms with homogeneous and inhomogeneous level broadening. By introduction of suitable collective atomic coordinates this generalized Fokker-Planck equation is reduced to an ordinary one which may serve as a basis for the adequate treatment of laser light statistics.     

Generalized Fokker-Planck equation approach to optical parametric processes II. Photocounting statistics

We study the photocounting statistics of a system of three interacting one-mode boson fields based on results of the recursion procedure of solving the generalized Fokker-Planck equation developed in the first part of this paper. If some simplifications are adopted or losses neglected, we are able to obtain explicit solutions for the photocounting generating function, the photocounting distribution and its factorial moments in terms of the multimode superposition of coherent and chaotic fields, the chaotic component being negative in the anticorrelation 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).