In order to elucidate the relationship between the drift and diffusion terms, this study commences with an examination of random walks. Among various biased random walk scenarios, we specifically focus on a situation where the probability of a particle moving in one of the six possible directions in a 3-dimensional space is denoted as p, while the probability in the remaining five directions is given by ${\displaystyle \frac{1-p}{5}}$, subject to the condition ${\displaystyle \frac{1}{6}}<p<1$. Although the case when $0<p<{\displaystyle \frac{1}{6}}$ presents an intriguing scenario, it falls outside the purview of this study. Given a particle’s jump speed as c, the collective or group velocity of all particles can be described by the equation: This can be expressed equivalently in the form $\left\{X_1\left(t\right),X_2\left(t\right),X_3\left(t\right),t\in T\right\}$, where $X_1\left(t\right),X_2\left(t\right)$ and $X_3\left(t\right)$ represent 1-dimensional random walks with a jump speed of ${\displaystyle \frac{c}{\sqrt{3}}}$. We introduce a 3-dimensional rectangular coordinate system, aligning its (1, 1, 1) direction parallel to the group velocity $\mathit{u}$. Consequently, the 1-dimensional random walk model along the $x_i$-axis, which is one of the three equivalent coordinate axes, is given by: where $n\in \mathbb{N}$. The corresponding unbiased 1-dimensional random walk model on the $x_i$-axis among the three equivalent coordinate axes is where $n\in \mathbb{N}$. The relative standard deviation of the two 1-dimensional random walk processes along the two equivalent $x_i$-axes is Consider the 3-dimensional random walk vectors, representing velocities, formed by $X_1\left(t\right),X_2\left(t\right)$ and $X_3\left(t\right)$ over a unit time interval. Their norms follow the discrete Maxwell distribution, and the average norm is proportional to the standard deviation of any component vector along the equivalent axes[2]. Consequently, the average velocity magnitude of the biased 3-dimensional random walk is ${\displaystyle \frac{\sqrt{c^2-u^2}}{c}}$ times that of its unbiased counterpart, with u defined in Eq. 4 (see Part 1 of the Supplementary Information for the detailed Mathematica code). Evidently, the biased case represents a decelerative process. This also illuminates the relationship between the drift term and the attenuation rate of diffusion in a continuous process with constant energy from an alternative perspective. For a more rigorous depiction of this relationship in continuous time, further proofs are warranted. Yet, directly proving this for a continuous-time stochastic process poses challenges. We can transpose to the following strategy.

When particles in swarm $\mathcal{A}$ possess speeds governed by a Maxwell distribution with a mean of c, the associated scale parameter for this distribution is ${\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{2}}}c$. Particles in $\mathcal{A}$ can be categorized into subgroups ${\mathcal{A}}_i$, based on their speeds, where each subgroup’s velocity vectors terminate on a series of spheres of radius $r_i$ ($i=1,2,3,\cdots$). The endpoints of these vectors for particles in ${\mathcal{A}}_i$ are uniformly distributed over the sphere with radius $r_i$. The number of particles in $\mathcal{A}$ distributed across these spheres follows a Maxwell distribution with respect to r, with the scale parameter ${\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{2}}}c$ and mean radius $\overline{r}=c$. Hence, the scenario where particle speeds are uniformly c can be viewed as a specialized case ($r_i=c$). The collective motion characteristics of particles whose velocity vectors terminate on spheres of radius $r_i$ embody the motion traits of particles with speeds dictated by the Maxwell distribution. The particle counts in both $\mathcal{A}$ and ${\mathcal{A}}_i$ are substantial, with a uniform distribution. When a subgroup $\mathcal{A}$ of the parent particle swarm—where the magnitudes of the particle velocities follow a Maxwell distribution—moves along the z-axis (with a reference frame based on the parent particle swarm) at an average speed of u, it is equivalent to each sub-particle group ${\mathcal{A}}_i$ ($i=1,2,3,\cdots$) moving along the z-axis with an average speed of ${\displaystyle \frac{r_{i}u}{c}}$.

Let R follow a Maxwell distribution with a scale parameter $\lambda$. Then $aR$ will follow a Maxwell distribution with a scale parameter $a\lambda$. Specifically, along the z-axis, all the component speeds $u_i$ in particle swarm ${\mathcal{A}}_i$, given by $u_i={\displaystyle \frac{r_{i}u}{c}}$ (where $i=1,2,3,\cdots$), follow a Maxwell distribution with a mean value of u. Moreover, the standard deviations ${\sigma }_i={\displaystyle \frac{r_i\sqrt{1-u^2/c^2}}{\sqrt{3}}}$ (where $i=1,2,3,\cdots$) of the mixed distributions (we can regard the particles in ${\mathcal{A}}_i$ as a particle swarm possessing a mixed distribution with weight $w={\displaystyle \frac{c+u}{2c}}$) across all spherical layers also conform to a Maxwell distribution. For the case along the x- or y-axis, the particles in ${\mathcal{A}}_i$ can also be viewed as a particle swarm possessing a mixed distribution with weight $w={\displaystyle \frac{c+u}{2c}}$. Adopting a method analogous to that in my previous work[2], it can be demonstrated that the ratio of the standard deviation of the mixed distribution along the x- or y-axis to that of the unbiased moving particle on the same axis satisfies the relationship ${\displaystyle \frac{\sqrt{c^2-u^2}}{c}}$, and the ratio of the standard deviation along the z-axis also satisfies this relationship (see Part 2 of the Supplementary Information for the detailed Mathematica code).

The above results indicate that for particles possessing a group velocity of u and with speeds following the Maxwell distribution, their speeds within ${\mathcal{R}}_u$ exhibit a deceleration characterized by a Lorentz-like factor. Despite this change in speed, their distributions still follow the Maxwell distribution within ${\mathcal{R}}_u$. This behavior permits their motion to be described by the following Itô equation when viewed from ${\mathcal{R}}_0$. It should be noted that the definitions of ${\mathcal{R}}_0$ and ${\mathcal{R}}_u$ align with those presented in my previous study[2]. where $u_1$, $u_2$ and $u_3$ are the drift speeds and $\sqrt{{u_1}^2+{u_2}^2+{u_3}^2}=u$; c is the average speed of particles; $w_1\left(t\right)$, $w_2\left(t\right)$ and $w_3\left(t\right)$ are Brownian motion processes; and $\sigma$ is their standard deviation. Then, the slice distribution of Eq. 8 is where i takes the values 1, 2 and 3, in accordance with the Einstein summation convention. Moreover, the forward equation of Eq. 8 is where $p(\mathit{x},t)$ is the probability of a particle at position $\mathit{x}(x_1,x_2,x_3)$ and time t. This is also a drift-diffusion (or advection-diffusion) equation. Only the diffusion coefficient of this equation is limited by the advection term.

This result (3-dimensional biased Brownian motion) can be simulated by three 1-dimensional biased Brownian motions $\left\{X_1\left(t\right),X_2\left(t\right),X_3\left(t\right),t\in T\right\}$. Here, the trajectories of 300 particles randomly moving for 10 s with a time step of 0.01 s are presented in Figure 1. There is no significant difference from the situation that is unbiased and not affected by the special relativistic-like effect.