Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation

The Langevin equation (LE) for the one-dimensional relativistic Brownian motion is derived from a microscopic collision model. The model assumes that a heavy pointlike Brownian particle interacts with the lighter heat bath particles via elastic hard-core collisions. First, the commonly known, nonrelativistic LE is deduced from this model, by taking into account the nonrelativistic conservation laws for momentum and kinetic energy. Subsequently, this procedure is generalized to the relativistic case. There, it is found that the relativistic stochastic force is still $\delta$ correlated (white noise) but no longer corresponds to a Gaussian white noise process. Explicit results for the friction and momentum-space diffusion coefficients are presented and discussed.

Figure 1

The momentum-dependent, relativistic friction coefficient $\nu \left(P\right)$, divided by the total mean collision rate, $\nu \left(P\right)∕\left[n_{\mathrm{b}}C\left(P\right)\right]$, as calculated numerically for two different heat bath distributions $f_{\mathrm{b}}^1\left(p\right)$ and two different bath temperatures, is depicted versus the scaled momentum $P$. The solid lines refer to the standard Jüttner distribution with $\eta =0$, and the dotted lines to $\eta =1$ in Eq. (17b). (a) Weakly relativistic heat bath. In the limit $kT⪡m{c}^2$ the bath distributions (17b) approach a Maxwellian, and therefore the results for different $\eta$ practically coincide. In particular, for $P=0$ the nonrelativistic result is recovered. (b) Strongly relativistic heat bath. The friction coefficient increases with the temperature of the heat bath.

Figure 2

Relativistic one-particle collision coefficient $C\left(P\right)={⟨I_r(t,\tau )⟩}_{\mathrm{b}}L∕\tau$, numerically calculated for the same parameters as in Fig. 1. The solid lines refer to a standard Jüttner distribution with $\eta =0$, and the dotted lines to $\eta =1$ in Eq. (17b). (a) Weakly relativistic heat bath. At small temperatures, the zero-value $C\left(0\right)$ is approximately equal to the nonrelativistic Stokes value $\sqrt{kT∕\left(2\pi m\right)}$. (b) Strongly relativistic heat bath. For $\mid P\mid \to \infty$ the coefficient $C\left(P\right)$ converges to $1∕2$.

Figure 3

Mean value of the relativistic stochastic force $\mu \left(P\right)\equiv {⟨\chi \left(t\right)⟩}_{\mathrm{b}}$ calculated numerically for two different heat bath distributions $f_{\mathrm{b}}^1\left(p\right)$ and two different bath temperatures. Solid lines refer to a standard Jüttner distribution with $\eta =0$, and dotted lines to $\eta =1$ in Eq. (17b).

Figure 4

Relativistic diffusion coefficients $D\left(P\right)$ calculated numerically for different heat bath distributions $f_{\mathrm{b}}^1\left(p\right)$ for (a) weakly and (b) strongly relativistic heat. The solid lines refer to a standard Jüttner distribution with $\eta =0$, and the dotted lines to $\eta =1$ in Eq. (17b).