The continuum limit

Let us consider the continuum limit of the 1d random walk. If there is an equal probability of taking a step to the right or left, the random walk can be described by a simple ``master equation''

$$P_n(i)=\frac{1}{2}P_{n-1}(i+1)+\frac{1}{2}P_{n-1}(i-1)$$ (105)

where $P_n(i)$ is the probability that the walker is at site $i$ after $n$ steps. In order to obtain a differential equation for the probability density $P(x,t)$, we set $t=n \tau$, $x=ia$, and $P_n(i)=aP(x,t)$, where $\tau$ is the time between steps, and $a$ is the lattice spacing. Now we can rewrite (106) as

$$P(x,t)=\frac{1}{2}P(x+a,t-\tau)+\frac{1}{2}P(x-a,t-\tau).$$ (106)

We substract $P(x,t-\tau)$ from both sides, we divide by $\tau$, and rewrite (107) as:

$$\frac{P(x,t)-P(x,t-\tau)]}{\tau}=\frac{a^2}{2\tau}\frac{[P(x+a,t-\tau) % - 2P(x,t-\tau)+P(x-a,t-\tau)]}{a^2}.$$ (107)

Talking the limit $a \rightarrow 0$ and $\tau \rightarrow 0$ with the ratio $D=a^2/2\tau$ finite, we obtain the diffusion equation

$$\frac{\partial P(x,t)}{\partial t}=D\frac{\partial ^2 P(x,t)}{\partial x^2}$$ (108)

The generalization to three dimensions is

$$\frac{\partial P(\mathbf{r},t)}{\partial t}=D\nabla ^2 P(\mathbf{r},t).$$ (109)

This is the ``diffusion equation'' or ``Fokker-Planck equation'' and is frequently used to describe the dynamics of fluid molecules. The solution in free space can be shown to be a Gaussian

$$P(x,t)=\sqrt{\frac{1}{2 \pi D t}} e^{-x^2/4Dt}$$

where $P(x,t)$ is the probability of finding a particle at position $x$ at time $t$ if is started from $x=0$ at $t=0$. From this equation we can obtain:

$$\langle x(t) \rangle = \int_{-\infty}^{\infty}{dx xP(x,t)} = 0,$$

and

$$\langle x^2(t) \rangle = \int_{-\infty}^{\infty}{dx x^2P(x,t)}=2Dt$$

The generalization to $d$ dimensions is $\langle R^2(t) \rangle = 2dDt$, where $R^2$ is the square of the displacement of the particle.