Corrections to Einstein's relation for Brownian motion in a tilted periodic potential

In this paper we revisit the problem of Brownian motion in a tilted periodic potential. We use homogenization theory to derive general formulas for the effective velocity and the effective diffusion tensor that are valid for arbitrary tilts. Furthermore, we obtain power series expansions for the velocity and the diffusion coefficient as functions of the external forcing. Thus, we provide systematic corrections to Einstein's formula and to linear response theory. Our theoretical results are supported by extensive numerical simulations. For our numerical experiments we use a novel spectral numerical method that leads to a very efficient and accurate calculation of the effective velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic

Stochastic Gravity

Gravity is treated as a stochastic phenomenon based on fluctuations of the metric tensor of general relativity. By using a (3+1) slicing of spacetime, a Langevin equation for the dynamical conjugate momentum and a Fokker-Planck equation for its probability distribution are derived. The Raychaudhuri equation for a congruence of timelike or null geodesics leads to a stochastic differential equation for the expansion parameter $\theta$ in terms of the proper time $s$. For sufficiently strong metric fluctuations, it is shown that caustic singularities in spacetime can be avoided for converging geodesics. The formalism is applied to the gravitational collapse of a star and the Friedmann-Robertson-Walker cosmological model. It is found that owing to the stochastic behavior of the geometry, the singularity in gravitational collapse and the big-bang have a zero probability of occurring. Moreover, as a star collapses the probability of a distant observer seeing an infinite red shift at the Schwarzschild radius of the star is zero. Therefore, there is a vanishing probability of a Schwarzschild black hole event horizon forming during gravitational collapse.Comment: Revised version. Eq. (108) has been modified. Additional comments have been added to text. Revtex 39 page

Some relations between Lagrangian models and synthetic random velocity fields

We propose an alternative interpretation of Markovian transport models based on the well-mixedness condition, in terms of the properties of a random velocity field with second order structure functions scaling linearly in the space time increments. This interpretation allows direct association of the drift and noise terms entering the model, with the geometry of the turbulent fluctuations. In particular, the well known non-uniqueness problem in the well-mixedness approach is solved in terms of the antisymmetric part of the velocity correlations; its relation with the presence of non-zero mean helicity and other geometrical properties of the flow is elucidated. The well-mixedness condition appears to be a special case of the relation between conditional velocity increments of the random field and the one-point Eulerian velocity distribution, allowing generalization of the approach to the transport of non-tracer quantities. Application to solid particle transport leads to a model satisfying, in the homogeneous isotropic turbulence case, all the conditions on the behaviour of the correlation times for the fluid velocity sampled by the particles. In particular, correlation times in the gravity and in the inertia dominated case, respectively, longer and shorter than in the passive tracer case; in the gravity dominated case, correlation times longer for velocity components along gravity, than for the perpendicular ones. The model produces, in channel flow geometry, particle deposition rates in agreement with experiments.Comment: 54 pages, 8 eps figures included; contains additional material on SO(3) and on turbulent channel flows. Few typos correcte

Numerical self-consistent stellar models of thin disks

We find a numerical self-consistent stellar model by finding the distribution function of a thin disk that satisfies simultaneously the Fokker-Planck and Poisson equations. The solution of the Fokker-Planck equation is found by a direct numerical solver using finite differences and a variation of Stone's method. The collision term in the Fokker-Planck equation is found using the local approximation and the Rosenbluth potentials. The resulting diffusion coefficients are explicitly evaluated using a Maxwellian distribution for the field stars. As a paradigmatic example, we apply the numerical formalism to find the distribution function of a Kuzmin-Toomre thin disk. This example is studied in some detail showing that the method applies to a large family of actual galaxies.Comment: 12 pages, 9 figures, version accepted in Astronomy & Astrophysic

The linear Fokker-Planck equation for the Ornstein-Uhlenbeck process as an (almost) nonlinear kinetic equation for an isolated N-particle system

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the infinite particle limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation.Comment: Minor revisions and corrections (including the title

The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence

We present an overview of recent works on the statistical description of turbulent flows in terms of probability density functions (PDFs) in the framework of the Lundgren-Monin-Novikov (LMN) hierarchy. Within this framework, evolution equations for the PDFs are derived from the basic equations of fluid motion. The closure problem arises either in terms of a coupling to multi-point PDFs or in terms of conditional averages entering the evolution equations as unknown functions. We mainly focus on the latter case and use data from direct numerical simulations (DNS) to specify the unclosed terms. Apart from giving an introduction into the basic analytical techniques, applications to two-dimensional vorticity statistics, to the single-point velocity and vorticity statistics of three-dimensional turbulence, to the temperature statistics of Rayleigh-B\'enard convection and to Burgers turbulence are discussed.Comment: Accepted for publication in C. R. Acad. Sc