Maxwellian relaxation of elastic particles in one dimension
Because of complete energy and momentum exchange during an elastic collision between two equal mass particles in one dimension, they can be viewed as being transparent to each other.
Plot of ξ = (1 − 6m + m 2)/(1 + 2m + m 2) as a function of the mass ratio m. Note that the roots of ξ are at , which are the reciprocals of each other.
Temporal evolution of the normalized velocity distribution of mass 1 particles in a binary mixture with a mass ratio of m = 2. The results were obtained by Monte Carlo simulations. Note the switching behavior of the distribution function between unimodal and bimodal forms at odd and even times, respectively.
Normalized velocity distribution of mass 1 particles after 20 time steps in a binary mixture with m = 2. The jagged line is from Monte Carlo simulations and the smooth solid line is from equipartition of energy.
Temporal evolution of the normalized velocity distribution function of mass 1 particles in Fourier space, in a binary mixture with m = 2. The continuous curves are theoretical results from the nonlinear Boltzmann equations, and the markers are from Monte Carlo simulations. Note the difference between the functional forms of F(k, t) at even and odd times.
Standard deviations of the velocity distributions of (a) mass 1 and (b) mass m particles as a function of time. The circles are the computer simulation data. The continuous curves are from Eq. (33a), with a value of τ calculated by a least-squares fit of Eq. (34) to the mass 1 data. Note that the same relaxation time generates curves that fit well to the other three branches. In this case, m = 2 and τ = 3.982. The theoretical value of τ from Eq. (33a) for m = 2 is 3.979.
Dependence of the relaxation time of the standard deviations of the velocity distribution functions on m. The continuous curve is the theoretical result and the circles are the simulation data.
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