Tagged particle fluctuations in uniform shear flow

The nonlinear Boltzmann and Boltzmann-Lorentz equations are used to describe the dynamics of a tagged particle in a nonequilibrium gas. For the special case of Maxwell molecules with uniform shear flow, an exact set of equations for the average position and velocity, and their fluctuations, is obtained. The results apply for arbitrary magnitude of the shear rate and include the effects of viscous heating. A generalization of Onsager's assumption of the regression of fluctuations is found to apply for the relationship between the equations for the average dynamics and those for the time correlation functions. The connection between fluctuations and dissipation is described by the equations for the equal-time correlation function. The source term in these equations indicates that the “noise” in this nonequilibrium state is qualitatively different from that in equilibrium, or even local equilibrium. These equations are solved to determine the velocity autocorrelation function as a function of the shear rate.     

Canonically invariant formulation of Langevin and Fokker-Planck equations

We present a canonically invariant form for the generalized Langevin and Fokker-Planck equations. We discuss the role of constants of motion and the construction of conservative stochastic processes. Received : 24 July 1997 / Revised : 30 October 1997 / Accepted : 26 January 1998     

Noise-intensity fluctuation in Langevin model and its higher-order Fokker-Planck equation

In this paper, we investigate a Langevin model subjected to stochastic intensity noise (SIN), which incorporates temporal fluctuations in noise-intensity. We derive a higher-order Fokker-Planck equation (HFPE) of the system, taking into account the effect of SIN by the adiabatic elimination technique. Stationary distributions of the HFPE are calculated by using the perturbation expansion. We investigate the effect of SIN in three cases: (a) parabolic and quartic bistable potentials with additive noise, (b) a quartic potential with multiplicative noise, and (c) a stochastic gene expression model. We find that the existence of noise-intensity fluctuations induces an intriguing phenomenon of a bimodal-to-trimodal transition in probability distributions. These results are validated with Monte Carlo simulations.     

Conformational fluctuations of a tethered polymer in uniform flow

The decay of correlations in the conformational fluctuations of a tethered polymer subjected to a uniform flow is analyzed in terms of relaxation times and associated normal modes. These quantities are calculated numerically from Brownian dynamics simulations of several bead spring polymer models. In this way, the influence of different effects like a finite extensibility of the springs and excluded-volume as well as hydrodynamic interactions between the beads on the decay of fluctuations is identified. Moreover, by comparison of the simulation results to analytically tractable blob models with corresponding assumptions, the capability of the tensile-blob picture to predict relaxation times and modes is assessed. For excluded-volume and hydrodynamic interactions a crossover to Rouse-like behavior occurs when the flow velocity and hence the polymer deformation exceeds a certain value. For finitely extensible springs, in contrast, the relaxation times decrease monotonically with increasing polymer deformation. This latter behavior differs from assumptions often used in rheological modeling by dumbbells and is not captured by the blob model.Received: 4 April 2003, Published online: 12 August 2003PACS: 83.80.Rs Polymer solutions - 83.10.Mj Molecular dynamics, Brownian dynamics - 36.20.Ey Conformation (statistics and dynamics) - 47.50. + d Non-Newtonian fluid flows     

Power fluctuations in a closed turbulent shear flow

We study experimentally the power consumption P of a confined turbulent flow at constant Reynolds number Re. We analyze in details its temporal dynamics and statistical properties, in a setup that covers two decades in Reynolds numbers. We show that nontrivial power fluctuations occur over a wide range of amplitudes and that they involve coherent fluid motions over the entire system size. As a result, the power fluctuations do not result from averaging of independent subsystems and its probability density function Pi(P) is strongly non-Gaussian. The shape of Pi(P) is Reynolds number independant and we show that the relative intensity of fluctuations decreases very slowly as Re increases. These results are discussed in terms of an analogy with critical phenomena.     

Incompatibility of the Schrödinger equation with Langevin and Fokker-Planck equations

Quantum mechanics posits that the wave function of a one-particle system evolves with time according to the Schrödinger equation, and furthermore has a square modulus that serves as a probability density function for the position of the particle. It is natural to wonder if this stochastic characterization of the particle's position can be framed as a univariate continuous Markov process, sometimes also called a classical diffusion process, whose temporal evolution is governed by the classically transparent equations of Langevin and Fokker-Planck. It is shown here that this cannot generally be done in a consistent way, despite recent suggestions to the contrary.     

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.     

Note on Boltzmann Langevin equation and hydrodynamic fluctuations

A microscopic derivation presented of the generalization of the linearized Boltzmann equation with the Langevin fluctuation force, which has earlier been postulated by Bixon and Zwanzig and by Fox and Uhlenbeck in their kinematical discussions on the hydrodynamic fluctuations.     

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.     

Comparison between the Boltzmann and BGK equations for uniform shear flow

The exact fourth-degree moments derived in the preceding paper from the Boltzmann equation for Maxwell molecules under uniform shear flow are compared with those obtained from the BGK and the Gaussian approximations. It is shown that the BGK results are closer to the exact ones than the Gaussian results. However, the deviations become significant as the shear rate increases.     

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.     

Exact moment solution of the Boltzmann equation for uniform shear flow

The Ikenberry-Truesdell exact solution to the Boltzmann equation for Maxwell molecules is revisited. This solution refers to a state characterized by a linear profile of the velocity flow and spatially uniform density and temperature. The solution is extended to include explicit expressions for the fourth-degree moments. It is shown that if the shear rate is larger than a certain critical value, the fourth-degree moments do not reach stationary values, even when the temperature is kept constant. The explicit shear-rate dependence of the moments below this critical value are obtained.     

Orientational correlation and velocity distributions in uniform shear flow of a dilute granular gas

Using particle simulations of the uniform shear flow of a rough dilute granular gas, we show that the translational and rotational velocities are strongly correlated in direction, but there is no orientational correlation-induced singularity at perfectly smooth (beta=-1) and rough (beta=1) limits for elastic collisions (e=1); both the translational and rotational velocity distribution functions remain close to a Gaussian for these two limiting cases. Away from these two limits, the orientational as well as spatial velocity correlations are responsible for the emergence of non-Gaussian high-velocity tails. The tails of both distribution functions follow stretched exponentials, with the exponents depending on normal (e) and tangential (beta) restitution coefficients.     

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.     

Light scattering by critical fluids in the presence of a uniform shear flow

Some predictions are made on dynamic light scattering by critical fluids in the presence of a shear flow. A Doppler shift broadening is predicted to occur whenever the scattering vector has a component along the direction of flow.