Most random phenomena in nature exhibit characteristics of bias${}^1$. This includes random motion, as such phenomena tend not to occur in a strictly neutral or uniform manner. In a group composed of randomly moving particles, the motions of particles in its sub-groups tend to be biased. Among these biased random motions, several distinct motion processes warrant particular attention. These special motions include particles with a higher probability of movement in a specific direction, while maintaining an equivalent yet lower probability of movement in other directions${}^2$; particles with greater probabilities of movement toward a point, while maintaining an equivalent yet lower probability of movement in other directions${}^3$; and biased random processes of particles capable of generating rotation${}^4$. The study of such phenomena typically begins with the simplest biased random processes.

Biased random processes, akin to the first scenario previously described, have been the subject of extensive research[5]. The nature of the bias can shape the form of the expression for a random process. For example: where $w\left(t\right)$ denotes a Brownian motion process. In Eq. 1, when the drift (u) and diffusion ($\sigma$) terms grow synchronously according to $S_t$, geometric Brownian motion will be produced. There will be fractional Brownian motion when particles follow the motion law in Eq. 2. where $\theta$ is the regression speed. Additionally, there is a simpler scenario where particles with the same jump speed have a higher probability of moving in a specific direction and equal probabilities of moving in other directions. This is a common biased phenomenon in physics. In a system with constant kinetic energy, it is generally challenging to alter the distribution of particle speeds, while the directions of motion can change more readily. When describing such systems using Eq. 3 the system’s kinetic energy will increase as the drift term u increases. It is obvious that such an equation does not reflect real scenarios. In contrast, when the drift term u increases, it may be a better solution to reduce the diffusion term $\sigma$ accordingly. Unfortunately, there is currently no clear quantitative relationship available. This article will delve into this issue in detail and derive a quantitative relationship of Lorentz-like factor between the drift (u) and diffusion ($\sigma$) terms. This kind of problem is similar to the problem that the particle speed or time slows down in a moving frame in special relativity. Some researchers${}^{6--9}$ demonstrated the diffusion process (random walk) with a special relativistic effect. However, it is the diffusion or random walk of particles when their speed is affected by the special relativity effect. The case involved in this study is that the particle velocity is determined, and the diffusion in the biased random particle swarm formed by them is restricted by the Lorentz-like factor. They are not the same kind of problem.

In my previous research${}^2$, I demonstrated the special relativistic-like effect of biased random motion when particles move at a uniform speed of c. This article extends that work by considering particles whose speeds follow a Maxwell distribution with equal average speed c. I establish that such biased random motion retains its special relativistic-like effect. Additionally, I present the corresponding Itô equations and simulate the motion of related examples. My findings offer insights into the intricate relationship between biased random processes and the special relativity effect.

For a particle swarm composed of particles with a fixed energy, assumed to be in reference frame ${\mathcal{R}}_0$, the magnitudes of the particle velocities follow a Maxwell distribution with parameter ${\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{2}}}c$, yielding an average speed of c. When a sub-swarm of particles (assumed to be in reference frame ${\mathcal{R}}_u$), exhibits a higher probability of motion in a specific direction and equal yet lower probabilities in other directions (with the group velocity of this sub-swarm denoted as u), as observed from ${\mathcal{R}}_0$, the velocities of particles within this sub-swarm decelerate, adhering to a pattern determined by the Lorentz-like factor ${\displaystyle \frac{\sqrt{c^2-u^2}}{c}}$. Consequently, the biased motion of such particles can be described by a set of Itô equations (Eq. 8). Both the time slice distribution and the forward equation for these biased particles differ from those in the reference frame ${\mathcal{R}}_0$.