Steady-state representation of the homogeneous cooling state of a granular gas

Abstract

The properties of a dilute granular gas in the homogeneous cooling state are mapped to those of a stationary state by means of a change in the time scale that does not involve any internal property of the system. The new representation is closely related with a general property of the granular temperature in the long time limit. The physical and practical implications of the mapping are discussed. In particular, simulation results obtained by the direct simulation Monte Carlo method applied to the scaled dynamics are reported. This includes ensemble averages and also the velocity autocorrelation function, as well as the self-diffusion coefficient obtained from the latter by means of the Green-Kubo representation. In all cases, the obtained results are compared with theoretical predictions.

Role of Molecular Chaos in Granular Fluctuating Hydrodynamics

“…The main hypothesis used here is that the system, starting in a uniform equilibrium state with granular temperature T g (0) = T 0 , evolves to the Homogeneous Cooling State (HCS), which is characterized by a single time-scale measured by the temperature T g (t): any other quantity depends on time only through T g (t). Apparently, as observed in many previous studies [8], in a homogeneous setting, Molecular Chaos is sufficient to guarantee this hypothesis. The HCS is unstable against spatial fluctuations: this instability appears at scales larger than a critical length L c which depends on α and on the mean free path, therefore it can be avoided by taking the linear size of the system L < L c [19].…”

“…with ω 0 ant t 0 arbitrary constants, implying the definition of rescaled velocitiesṽ(τ ) = v(t)ω 0 t. It is easy to see [8] that observing the system on this new time scale is equivalent to apply a positive continuous drag to all particles ∂ τṽ (τ ) = ω 0ṽ (τ ). This naturally leads to define also the rescaled…”

“…Since the system is cooling, the typical velocity of the particles becomes smaller and smaller, leading to increasing rounding errors and other numerical problems. These problems can be avoided with the use of a procedure which, in the HCS, maps the dynamics to a steady state by means of a time-rescaling [8,23]. In the HCS, the granular temperature obeys the following equation:…”

“…(2.9), it is necessary to know the value of ω 0 = ζ H (0)/2. This is computed, in the homogeneous case, giving the theoretical expressions [8] ζ…”

“…Such a choice, however, at the chosen packing fraction nσ 2 = 10 −2 gives N c ∼ 2.5, which is so low that the Molecular Chaos could be invalidated. A first check is to study the sensitivity of the rescaling procedure (2.7) (tested in [8] in the homogeneous DSMC simulations and in [7]) to the parameter m c (or equivalently N c ). Having this aim, we have characterized the entire system analyzing its granular temperature T g as function of the rescaled time τ for different choices of the cell number m c at α and N fixed (see Fig.…”