Relativistic diffusion processes and random walk models

The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As is well known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (noncontinuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the nonrelativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

Figure 1

Spreading of the Gaussian PDF $\rho (t,x)=p(t,x|0)$ from Eq. (2) at different times $t$, where $t$ is measured in units of $D/c^2$. At initial time $t=0$, the PDF corresponds to a $\delta$-function centered at the origin.

Figure 2

Transition PDFs ${\rho }_N(x)=p_N(N\tau ,x|0,0)$ for the relativistic time-discrete RW model from Sec. 2c1. The PDFs were numerically calculated at five different times $t=N\tau$ for two different parameter values $\chi =m{c}^2/(k_{\mathrm{B}}T)=\tau /(2D)$. (a) Weakly relativistic velocity distribution with $\chi >1$. The numerical solutions look qualitatively similar to a (nonrelativistic) Gaussian, but have a finite support $[-ct,ct]$. (b) Strongly relativistic velocity distribution with $\chi <1$. The peaks reflect the discrete time steps of the model, and are associated with the backward/forward-scattering of the fastest particles; cf. explanation in the text.

Figure 3

Transition PDF $\rho (t,x)=p(t,x|0)$ for the one-dimensional ($d=1$) relativistic diffusion process (23a) at different times $t$ (measured in units of $D/c^2$).

Figure 4

Comparison of the mean square displacements $⟨X^2(t)⟩$, divided by $2Dt$, for the one-dimensional ($d=1$) nonrelativistic Wiener process (2) and its relativistic generalization from Eq. (23a) with initial condition $(t_0,x_0)=(0,0)$.