The basic concepts of diffusion and self-diffusion are discussed. We derive relations between the mean square displacement of a diffusing particle, the diffusion coefficient and the velocity-time correlation function. These relations are generalized to the case of non-equilibrium granular gases.

12.1 Transport in granular gases

Transport processes in their conventional meaning imply macroscopic currents of mass, momentum, energy, etc. which occur correspondingly due to the presence of (even small) gradients of density, average velocity, temperature, etc. According to their definition, there are no such transport phenomena in the homogeneous cooling state.

In spite of the lack of macroscopic currents, individual-particle transport still exists in the homogeneous cooling state. The gas may consist of uniformly distributed particles (host particles) and of sparsely scattered guest particles, which differ in size, mass, material properties, etc. The stochastic motion of the guest particles, caused by collisions with the host particles is called Brownian motion. In isotropic systems, the average displacement of the guest particles is zero (since there is no preferred direction for the particles to move). The mean square displacement, however, increases with time, so that over a period of time the Brownian particle shifts more and more from its initial location. This kind of motion, which can occur even without macroscopic gradients, is called diffusion.

The concepts of diffusion and Brownian motion have been introduced for molecular systems in thermodynamic equilibrium, where particles suffer elastic collisions. For the description of granular gases, which are non-equilibrium systems, these concepts have to be generalized.

Consider a gas which consists of N_B uniformly distributed host particles B and of N_A ≪ N_B sparsely distributed guest particles A. The total number density n = N/V = (N_A + N_B)/V ≃ N_B /V is supposed to be homogeneous, while the local concentration $n_A(\overrightarrow{r},\hspace{0.17em}t)\hspace{0.17em}\equiv \hspace{0.17em}\Delta N_A(\overrightarrow{r},\hspace{0.17em}t)/\Delta V$ is not necessarily uniform (here $\Delta N_A(\overrightarrow{r},\hspace{0.17em}t)$ denotes the number of particles A at time t in a small volume ΔV located at point $\overrightarrow{r}$). If the concentration of the guest particles is not homogeneous, a diffusion flux of particles A arises, which is directed opposite to the concentration gradient. This process is described by (12.1)

(p. 124 ) where ${\overrightarrow{J}}_A\hspace{0.17em}\text{(}\overrightarrow{r})$ is the current of the guest particles A at $\overrightarrow{r}$ against the density gradient $\overrightarrow{\nabla }{n}_A\hspace{0.17em}\text{(}\overrightarrow{r})$ and D_A is the diffusion coefficient. With the continuity equation (12.2) we obtain the diffusion equation (12.3)

If the particles A and B are mechanically identical, however, somehow distinguishable (e.g. by colour), the process is called self-diffusion. In this case, the system is uniform and the particles A are called tracers.

12.2 Diffusion coefficient and mean square displacement

The coefficient of diffusion (and self-diffusion)¹² is closely related to the mean square displacement of tracers with time. Assume at time t = 0 the tracer particles are located at $\overrightarrow{r}$ = 0. The mean square displacement of particles at time t is then (12.4) where N_A = $\int d\overrightarrow{r}{n}_A\hspace{0.17em}\text{(}\overrightarrow{r},\hspace{0.17em}t\text{)}$ = const. is the total number of tracers. Now we multiply both sides of (12.3) by ${\overrightarrow{r}}^2/N_A$ and integrate over $d\overrightarrow{r}$. With (12.4) the LHS reads (12.5)

For the RHS we apply Green's theorem (Abramowitz and Stegun, 1965) to the functions n_A and r²: (12.6) where S denotes the surface which confines the system of interest and $\overrightarrow{l}$ is the normal of this surface. If we put the surface S to infinity, where the concentration of the host particles n_A vanishes, the surface integral in (12.6) becomes zero. Hence, we obtain for the RHS of (12.3) (12.7) where we use the relation ${\overrightarrow{\nabla }}^2{r}^2\hspace{0.17em}\text{=}\hspace{0.17em}\overrightarrow{\nabla }\cdot \hspace{0.17em}\left(\overrightarrow{\nabla }{r}^2\right)\hspace{0.17em}\text{=}\hspace{0.17em}\overrightarrow{\nabla }\cdot (2\overrightarrow{r})\hspace{0.17em}\text{=}\hspace{0.17em}\text{6}$ and the normalization condition for n_A. Combining (12.5) and (12.7) we obtain a relation between

(p. 125 )

Fig. 12.1. Transformation of the integration variables t₁, t₂ into the variables τ, τ₁. The integration is performed with the condition t₂ ≥ t₁; the correspondence between the lines confining the integration domains in both sets of variables, t₁, t₂ and τ, τ₁ is indicated.

the mean square displacement and the diffusion coefficient for three-dimensional systems (12.8)

12.3 Diffusion coefficient and velocity-time correlation function

Using the kinematic relation $\overrightarrow{r}(t)\hspace{0.17em}\text{=}\hspace{0.17em}\underset{0}{\overset{t}{\int }}\overrightarrow{\upsilon }\hspace{0.17em}(t_1)\hspace{0.17em}d{t}_1$ (again for $\overrightarrow{r}$ = 0 at t = 0) we write (12.9) and encounter with the velocity autocorrelation function (12.10)

The obvious relation K_υ (t₁, t₂) = K_υ (t₂, t₁) allows us to rewrite (12.9) with the condition t₂ ≥ t₁ imposed: (12.11)

For gases in equilibrium, the velocity autocorrelation function depends on the time lag τ = t₂ − t₁ only and decays with a characteristic time τ_υ. First, we assume that the gas is in equilibrium, K_υ (t₂, t₁) = K_υ(τ), and transform the variables, t₁, t₂ → τ, τ₁ with τ₁ = t₁. The Jacobian of this transformation equals unity and the change of the integration area is illustrated in Fig. 12.1. In the new variables we obtain (12.12)

(p. 126 ) and, correspondingly, the mean square displacement reads (12.13)

For longer time periods as compared with the characteristic time, that is, for t ≫ τ_υ, we may neglect τ/t as compared with 1. With (12.8): (12.14) which implies that at t ≫ τ_υ, the diffusion coefficient has the general expression (12.15)

Equation (12.15) expresses the transport coefficient D_A as the time-integral of the velocity correlation function. This is the simplest fluctuation-dissipation relation, which relates a fluctuating quantity (particle velocity) to a dissipative quantity (diffusion coefficient). Similar expressions exist for other transport coefficients (viscosity, thermal conductivity, etc.) and the corresponding fluctuating quantities (Resibois and de Leener, 1977).

We recall that all the above discussion refers to the case of equilibrium gases. Granular gases are a priori non-equilibrium systems, nevertheless, the concept of the diffusion coefficient may be generalized for such systems. Obviously, this refers only to dilute systems (Esipov and Pöschel, 1997) where the particles are highly mobile. Whereas the diffusion coefficient D for equilibrium systems is just a constant, the time dependence of temperature of a cooling granular gas causes the diffusion coefficient to be time-dependent as well. Therefore, the natural generalization of the diffusion coefficient for non-equilibrium systems is the diffusivity (12.16)

The brackets <···> denote averaging over the non-equilibrium ensemble, whose evolution is described by the time-dependent N-particle distribution function $\rho (\overrightarrow{r},\hspace{0.17em}\dots \hspace{0.17em}{\overrightarrow{r}}_N,\hspace{0.17em}{\overrightarrow{\upsilon }}_1,\hspace{0.17em}\dots \hspace{0.17em}{\overrightarrow{\upsilon }}_N,\hspace{0.17em}t)$ (for simplicity, we have retained the same notation as for the equilibrium average). We will show that for non-equilibrium systems, such as granular gases, the diffusivity can also be expressed in terms of the velocity-time correlation function. To this end we need an appropriate tool to describe the detailed particle dynamics for the case of dissipative systems. The technique of the pseudo-Liouville and binary-collision operators occurs to be very convenient for this purpose. Let us consider this in more detail.