Fluctuating Nonlinear Hydrodynamics Research Articles

Stochastic differential equations for non-linear hydrodynamics

We formulate the stochastic differential equations for non-linear hydrodynamic fluctuations. The equations incorporate the random forces through a random stres tensor and random heat flux as in the Landau and Lifshitz theory. However, the equations are non-linear and the random forces are non-Gaussian. We provide explicit expressions for these random quantities in terms of the well-defined increments of the Wienner process.

Nonlinear fluctuating hydrodynamics and sequence of time scales of relaxation in supercooled liquids

Nonlinear fluctuating hydrodynamic (NFH) models for relaxation in the supercooled liquid are considered. Recent results on self consistent mode coupling theory for the slow relaxation of density fluctuations are analyzed to explain the glassy dynamics. The relaxation mechanisms for different types of models with and without wave vector dependences are discussed. For the schematic models where all wave vector dependences are dropped a sequence of time scales enters the relaxation process. For the non-ergodicity parameter very close to the ideal transition point is scaled by an exponent equal to 1/2. This is demonstrated here through an analysis of the mode-coupling equations for the wave vector dependent models that follow from equations of NFH.

Statistical theory of nonlinear hydrodynamic fluctuations in ionic systems

A system of equations for time correlation functions in ionic systems is obtained and solved on the basis of the Fokker-Planck equation for the distribution function of coarse-grained dynamic variables such as mass density, charge density, momentum density, current density, and total energy. Statistical weight, thermodynamic forces and hydrodynamic velocities are calculated both in Gaussian and higher approximations. Expressions for generalized transfer coefficients depending onk andw, with separate fluctuation contributions, are found.

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.

Non-linear hydrodynamic fluctuation theory for a charged two component fluid in equilibrium

A scheme introduced recently for the description of non-linear hydrodynamic fluctuations of a one-component fluid is extended to the case of a two-component charged fluid. The random fluxes which are introduced are assumed to be Gaussian processes with white noise. It is found that the usual expressions for the systematic parts of the dissipative fluxes are only consistent with this assumption, if the Onsager coefficients are constant.

Non-linear hydrodynamic fluctuations around equilibrium

Non-linear hydrodynamic fluctuations around equilibrium

Non-linear hydrodynamic fluctuations around equilibrium

We formulate a scheme describing the fluctuations in a system obeying the non-linear hydrodynamic equations. The random fluxes are assumed to be Gaussian processes with white noise. It is shown that the usual expressions for the systematic parts of the dissipative fluxes are consistent with this assumption, provided that the Onsager coefficients are constants. The linear response of the system to a small external force field is studied and the relevant fluctuation- dissipation theorems are derived.