Empirical Calculation of Mean Free Paths

Introduction

Mean free path is an important parameter for the transport properties of a gas. The transport properties include thermal conductivity, gas mixing, and viscosity. These properties involve diffusion which is basically a random walk process. In this random walk process, the mean step size is the mean free path. To compute the rate of diffusion, one has to know the number of collisions per second as well. See Appendix for information on mixing rate versus time.

How the mean free path is computed in this program

Basically the mean free path is the average distance traveled by an atom between hard sphere collisions. For finite size containers this distance must include the distance from the last atom-atom collision to a collision with a wall as well as the distance between the wall collision and the next atom-atom collision. For this reason the program saves the times of atom-atom collisions and the speed after that collision and the new mean free distance becomes the result:

$l = s (t' - t)$

(1.1)

where l is the distance, s is the speed, t' is the time of the new collision and t is the time of the old collision.

Of course, an atom-atom collision always involves 2 atoms so there are 2 such distances, l₁ and l₂, for each collision. To compute the mean free path we add both l's to a distance sum:

$l_{s u m}^{'} = l_{s u m} + l_{1} + l_{2}$

(1.2)

We also increment a collision counter,

$n_{c}^{'} = n_{c} + 1$

(1.3)

The mean free path, λ, is then computed, after a very large number of collisions, as:

$λ = \frac{l_{s u m}^{'}}{n_{_{c}}^{'}}$

(1.4)

Theory for mean free path

The usual expression in 3 dimensions is that the mean free path is

$λ = \frac{1}{n σ}$

(1.5)

where n is the density of atoms per unit volume and σ is their collisional cross section area. For 3 dimensions the expression for σ for hard spheres is

$σ = 4 π r^{2}$

(1.6)

where r is the radius of the hard sphere. For a mixture of two different atoms of different radii, the expression of σ is:

$σ = π {(r_{a} + r_{b})}^{2}$

(1.7)

When the atomic species have different densities then n becomes:

$n = \frac{2 n_{a} n_{b}}{n_{a} + n_{b}}$

(1.8)

In 2 dimensions the density is the number of atoms per unit area, n₂, and the collision area becomes a simple length. Then the expression for mean free path becomes:

$λ_{2 D} = \frac{1}{4 n_{2} (r_{a} + r_{b})}$

(1.9)

Appendix

If we have an initial setup where atoms of a given type are all concentrated at x=0 while atoms of the other type are spread out over the entire x domain, then the rate of the first type's spreading, mixing or heat conduction will be proportional to the square root of the rate of collisions that are made. For a single atom, the rate of collisions will be

$\dot{n} = \frac{< v_{x} >}{λ}$

(1.10)

where <v_x> is the average x speed of the atom. The number of collisions in a given time interval, t, will then be:

$N (t) = \dot{n} t$

(1.11)

The rate of spreading is proportional to the square root of N(t) times the step size which here is the mean free path:

$δ x (t) = N (t) λ = \sqrt{\frac{< v_{x} >}{λ} t} λ = \sqrt{< v_{x} > t λ}$

(1.12)