Analytical and numerical study on the nonequilibrium relaxation by the simplified Fokker–Planck equation

Abstract

Nonequilibrium relaxation by the simplified Fokker–Planck equation is studied by calculating the relaxation of Grad’s moments. From calculated relaxation of moments, we propose two simplified Fokker–Planck-type equations. For the quantitative validation of nonequilibrium relaxation by using the simplified Fokker–Planck-type equations, we solve a rarefied shock layer problem. To study the stronger diffusion of the distribution function than the simplified Fokker–Planck-type equation, we consider diffusion of the distribution function due to the gain term of the Boltzmann equation, because the Boltzmann equation can be described by a partial differential equation with higher-order terms than the simplified Fokker–Planck equation. The effect of the gain term of the Boltzmann equation is discussed based on comparisons with our proposed Bhatnagar–Gross–Krook equation with velocity-dependent collision frequency.

Fast and accurate calculation of dilute quantum gas using Uehling–Uhlenbeck model equation

“…which corresponds to the forward regime of the shock wave. As observed in the profile off (ṽ x ) for the classical gas [24],f (ṽ x ) ≫f eq (ṽ x ) is obtained at the negative velocity in the forward regime of the shock wave. Additionally,f (ṽ x ) ≃f eq (ṽ x ) is obtained at point (B), which corresponds to the backward regime of the shock wave and location of the peak value ofZ.…”

Fokker-Planck model of hydrodynamics

“…Such a framework is quite useful for numerical implementation and may provide an alternate to methods such as DSMC or the recent set of works where a nonlinear extension to Ref. [14] was proposed for dilute gases [7][8][9]. In particular, our model needs to be contrasted with Ref.…”

“…In this regard, it was pointed out that a computationally attractive option is to solve the Fokker-Planck collision model via the associated Langevin equation [7][8][9]. The computational motivation behind the FP approach is that unlike the direct simulation Monte Carlo (DSMC) method [10,11], collision dynamics is simpler and the associated Langevin dynamics is highly computationally efficient [12,13].…”

Collective dynamics of Stokesian swimmers with impulsive acceleration

“…where E n (x) = ∞ 1 e −tx t −n dt and A 3 is the factor for normalization. Figure 8 shows the DSMC results of P ∞ (v) versus v fitted by equation (27), when S m = 10, 100 and 500 are used in the DBI. P ∞ (v) is approximated by equation (27) with appropriate parameters d 1 , d 2 , m and n, which are shown in table 2.…”