Theory of thermostatted inhomogeneous granular fluids: A self-consistent density functional description

Abstract

The authors present a study of the non equilibrium statistical properties of a one dimensional hard-rod fluid dissipating energy via inelastic collisions and subject to the action of a Gaussian heat bath, simulating an external driving mechanism. They show that the description of the fluid based on the one-particle phase-space reduced distribution function, in principle necessary because of the presence of velocity dependent collisional dissipation, can be contracted to a simpler description in configurational space. Indeed, by means of a multiple-time scale method the authors derive a self-consistent governing equation for the particle density distribution function. This equation is similar to the dynamic density functional equation employed in the study of colloids, but contains additional terms taking into account the inelastic nature of the fluid. Such terms cannot be derived from a Liapunov generating functional and contribute not only to the relaxational properties, but also to the non equilibrium steady state properties. A validation of the theory against molecular dynamics simulations is presented in a series of cases, and good agreement is found.

Beyond dynamic density functional theory: the role of inertia

“…Among the advantages of the present approach over the standard time-dependent density functional method are: (a) the possibility to treat assemblies of inelastic particles [4], i.e. systems with interactions which do not conserve the energy, (b) the ability to deal with non-isothermal systems and (c) a better description of systems in the presence of time-dependent potentials.…”

Hydrodynamics of a Granular Gas in a Heterogeneous Environment

“…We can model this interaction by means of a fluctuating volume force [40], that we denote as F st (r) and fulfills the conditions [32,35,40,41] …”

“…being ξ 2 (r) the fluctuating force intensity [32,41] (that has dimensions of velocity squared times time), 1 the unit matrix in d dimensional space, δ ij is the Kronecker delta function and A indicates average of magnitude A over a sufficiently long time interval (long compared with the characteristic microscopic time scale). It has been shown in the case of the large mass limit and homogeous energy injection, that Equation (1) for a particle can be written as a Langevin equation corresponding to a granular brownian particle [42].…”

Kinetic Density Functional Theory: A Microscopic Approach to Fluid Mechanics

“…These corrections can be computed by an inverse friction expansion or by multiple time scale methods and give rise to a more complex DDFT equation as shown in refs. [22,23] , which takes into account momentum and energy fluctuations.…”

“…The rigorous mathematical procedure to derive this result employs the multiple time scale analysis [22,23], which exploits the time scale separation between the zeroth mode associated with the density fluctuations and the remaining modes, which also include the momentum and energy fluctuations. Due to the action of the heat bath these are fast relaxing modes because the velocities of the particles rapidly relax towards the equilibrium distribution, in a time of order of the inverse friction time τ = 1/γ.…”