Theory of relativistic Brownian motion: The $(1+1)$-dimensional case

We construct a theory for the $(1+1)$-dimensional Brownian motion in a viscous medium, which is (i) consistent with Einstein’s theory of special relativity and (ii) reduces to the standard Brownian motion in the Newtonian limit case. In the first part of this work the classical Langevin equations of motion, governing the nonrelativistic dynamics of a free Brownian particle in the presence of a heat bath (white noise), are generalized in the framework of special relativity. Subsequently, the corresponding relativistic Langevin equations are discussed in the context of the generalized Ito (prepoint discretization rule) versus the Stratonovich (midpoint discretization rule) dilemma: It is found that the relativistic Langevin equation in the Hänggi-Klimontovich interpretation (with the postpoint discretization rule) is the only one that yields agreement with the relativistic Maxwell distribution. Numerical results for the relativistic Langevin equation of a free Brownian particle are presented.

Figure 1

Velocity curves $v\left(t\right)$, corresponding to Eq. (24), for the purely damped motion of a relativistic particle in the rest frame of the heat bath (laboratory frame). Especially at high velocities $\mid v\mid \lesssim c$, the relativistic velocity curves deviate from the exponential decay, predicted by the Newtonian theory.

Figure 2

Stationary solutions ${\varphi }_{\mathrm{I}∕\mathrm{S}∕\mathrm{HK}}\left(v\right)$ of the relativistic Fokker-Planck equations, according to Ito (I:solid line), Stratonovich (S:dotted), and Hänggi-Klimontovich (HK:dashed-dotted). For low temperatures, i.e., for $\beta ⪢1$, a Gaussian shape is approached [see (a)]. On the other hand, for very high-temperature values, corresponding to $\beta ⩽1$, the distributions exhibit a bistable shape, and the quantitative deviations between ${\varphi }_{\mathrm{I}∕\mathrm{S}∕\mathrm{HK}}\left(v\right)$ increase significantly as $\beta \to 0$.

Figure 3

These diagrams show a comparison between numerical and analytical results for the stationary cumulative distribution function $F\left(v\right)$ in the laboratory frame ${\Sigma }_0$. (a) In the nonrelativistic limit $\beta ⪢1$ the stationary solutions of the three different FPE are nearly indistinguishable. (b)–(c) In the relativistic limit case $\beta ⩽1$, however, the stationary solutions exhibit deviations from each other. Because our simulations are based on an Ito-discretization scheme, the numerical values (squares) are best fitted by the Ito solution (solid line).

Figure 4

(a) Mean-square displacement divided by time $t$ as numerically calculated for different $\beta$-values in the laboratory frame ${\Sigma }_0$ (rest frame of the heat bath). As evident from this figure, for the relativistic Brownian motion the related asymptotic mean-square displacement grows linearly with $t$. (b) The coordinate space diffusion constant $D_{100}^x\left(\beta \right)$ was numerically determined at time $t=100{\nu }^{-1}$. The dashed line corresponds to the empirical fitting formula $D^x\left(\beta \right)=c^2{\nu }^{-1}{(\beta +2)}^{-1}$, which reduces to the classical nonrelativistic result $D^x\simeq c^2∕\left(\nu \beta \right)=kT∕\left(m\nu \right)$ for $\beta ⪢2$.