A variant of Einstein’s partial differential equation for Brownian motion

In a series of papers from 1905, his `annus mirabilis’, Albert Einstein analysed Brownian motion and derived the following partial differential equation (known as the diffusion equation) from physical principles to describe the process:

$\displaystyle D \frac{\partial^2 f}{\partial x^2} = \frac{\partial f}{\partial t}$

This equation has as a solution the probability density function

$\displaystyle f(x, t) = \frac{1}{\sqrt{4 \pi Dt}} \exp\bigg(-\frac{x^2}{4Dt}\bigg)$

(see Einstein, A., 1956, Investigations on the Theory of Brownian Movement, New York: Dover, ISBN 978-0-486-60304-9, pp. 15-16).

Writing $\displaystyle D = \frac{\sigma^2}{2}$ this formula can be rewritten as

$\displaystyle f(x, t) = \frac{1}{\sqrt{2 \pi \sigma^2 t}} \exp\bigg(-\frac{x^2}{2 \sigma^2 t}\bigg)$

This is the density of the $\displaystyle N(0, \sigma^2 t)$ distribution of the increments of a Wiener process $\displaystyle \{X(t): t \geq 0, X(0) = 0\}$ , also commonly referred to as Brownian motion. This continuous-time stochastic process is symmetric about zero, continuous, and has stationary independent increments, i.e., the change from time $\displaystyle u$ to time $\displaystyle u + t$ , given by the random variable $\displaystyle X(u, u + t) = X(u + t) - X(u)$ , has the same $\displaystyle N(0, \sigma^2 t)$ probability distribution as the change from time $\displaystyle 0$ to time $\displaystyle t$ , given by the random variable $\displaystyle X(0, t) = X(t)$ , and the change is also independent of the history of the process before time $\displaystyle u$ .

A continuous-time stochastic process which is also symmetric about zero and continuous like the Wiener process, but which has non-stationary increments, has a probability density function of the form

$\displaystyle f(x, t) = \frac{1}{\sqrt{2 \pi \sigma^2 t^2}} \exp\bigg(-\frac{x^2}{2 \sigma^2 t^2}\bigg)$

The fact that time appears as a squared term in this formula rather than linearly is enough to destroy the increment-stationarity property of the Wiener process. This can be demonstrated by observing that increment-stationarity requires

$\displaystyle X(u, u + t) \,{\buildrel d \over =}\, X(t)$

and therefore by definition of $\displaystyle X(u, u + t)$ we must also have

$\displaystyle X(u + t) \,{\buildrel d \over =}\, X(u, u + t) + X(u) \,{\buildrel d \over =}\, X(t) + X(u) \sim N(0, \sigma^2 (t^2 + u^2))$

but this not true here since

$\displaystyle X(u + t) \sim N(0, \sigma^2(u + t)^2)$

We have a contradiction and must therefore conclude that

$\displaystyle X(u, u + t) \,{\buildrel d \over \neq}\, X(t)$

which means that this alternative continuous-time process does not have the increment-stationarity property of the Wiener process. (Note that this kind of contradiction does not arise with the Wiener process: we have $\displaystyle X(u, u + t) + X(u) \sim N(0, \sigma^2(t + u))$ which is the same as the distribution of $\displaystyle X(u + t)$ ).

Is there a partial differential equation describing this alternative continuous-time process analogous to the partial differential equation derived by Einstein for the standard Brownian motion process? I did indeed find such a partial differential equation for the alternative process, as follows. Taking partial derivatives of the probability density function

$\displaystyle f(x, t) = \frac{1}{\sqrt{2 \pi \sigma^2 t^2}} \exp\bigg(-\frac{x^2}{2 \sigma^2 t^2}\bigg)$

we obtain

$\displaystyle \frac{\partial f}{\partial t} = \frac{1}{\sqrt{2 \pi \sigma^2 t^2}} \exp\bigg(-\frac{x^2}{2 \sigma^2 t^2}\bigg)\bigg\{\frac{x^2}{\sigma^2 t^3} - \frac{1}{t}\bigg\}$

$\displaystyle \frac{\partial f}{\partial x} = \frac{1}{\sqrt{2 \pi \sigma^2 t^2}} \exp\bigg(-\frac{x^2}{2 \sigma^2 t^2}\bigg)\bigg\{-\frac{x}{\sigma^2 t^2}\bigg\}$

$\displaystyle \frac{\partial^2 f}{\partial x^2} = \frac{1}{\sqrt{2 \pi \sigma^2 t^2}} \exp\bigg(-\frac{x^2}{2 \sigma^2 t^2}\bigg)\bigg\{\frac{x^2}{(\sigma^2 t^2)^2} - \frac{1}{\sigma^2 t^2}\bigg\}$

Comparing the expressions for $\displaystyle \frac{\partial f}{\partial t}$ and $\displaystyle \frac{\partial^2 f}{\partial x^2}$ we see that

$\displaystyle \sigma^2 \frac{\partial^2 f}{\partial x^2} = \frac{1}{t}\frac{\partial f}{\partial t}$

and this is the required variant of Einstein’s partial differential equation for the alternative continuous-time process.

I was intrigued to find that a slight generalisation of this framework makes it applicable to quantum wave-packet dispersion. To see this, let $\displaystyle \sigma(t^2) = \sqrt{a + bt^2}> 0$ where $a$ and $b$ are some parameters. Then the partial differential equation

$\displaystyle b \frac{\partial^2 f}{\partial x^2} = \frac{1}{t}\frac{\partial f}{\partial t}$

has as a solution the probability density function

$\displaystyle f(x, t) = \frac{1}{\sqrt{2 \pi \sigma^2(t^2)}} \exp\bigg(-\frac{x^2}{2 \sigma^2(t^2)}\bigg)$

as can be verified by comparing the partial derivatives

$\displaystyle \frac{\partial f}{\partial t} = \frac{1}{\sqrt{2 \pi \sigma^2 (t^2)}} \exp\bigg(-\frac{x^2}{2 \sigma^2 (t^2)}\bigg)\bigg\{\frac{bx^2 t}{\sigma^4(t^2)} - \frac{bt}{\sigma^2(t^2)}\bigg\}$

and

$\displaystyle \frac{\partial^2 f}{\partial x^2} = \frac{1}{\sqrt{2 \pi \sigma^2(t^2)}} \exp\bigg(-\frac{x^2}{2 \sigma^2(t^2)}\bigg)\bigg\{\frac{x^2}{\sigma^4(t^2)} - \frac{1}{\sigma^2(t^2)}\bigg\}$

(As a check, note that all of this reduces to the previously obtained differential equation and probability density function when $\displaystyle a = 0$ and $\displaystyle b = \sigma^2$ ). Now, in quantum mechanics a wave representation of a moving body is obtained as a wave-packet consisting of a superposition of individual plane waves of different wavelengths (or equivalently, different wave numbers $k = \frac{2 \pi}{\lambda}$ ) in the form

$\displaystyle \psi(x, t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \Phi(k) e^{i(kx - i\hbar k^2 t/2m)}dk$

where $\displaystyle \Phi(k)$ is the Fourier transform of the $x$ -space wavefunction $\displaystyle \psi(x, t)$ at $\displaystyle t = 0$ , i.e.,

$\displaystyle \Phi(k) = \int_{-\infty}^{\infty} \psi(x', 0)e^{-ikx'}dx'$

The wave-packet $\displaystyle \psi(x, t)$ disperses over time and it has been shown that the probability density as a function of time of such a moving body as the wave-packet disperses, given by $\displaystyle |\psi(x, t)|^2$ , always becomes Gaussian (irrespective of the original shape of the wave-packet) and has the form of the probability density function above, i.e.,

$\displaystyle |\psi(x, t)|^2 = \frac{1}{\sqrt{2 \pi \sigma^2(t^2)}} \exp\bigg(-\frac{x^2}{2 \sigma^2(t^2)}\bigg)$

See, for example, Mita, K, 2007, Dispersion of non-Gaussian free particle wave packets, Am. J. Phys. 75 (10), who derives an expression for $\displaystyle |\psi(x, t)|^2$ like the one above with $\displaystyle x^2$ replaced by $\displaystyle (x - \gamma_1)^2$ and

$\displaystyle \sigma^2(t^2) = \frac{\gamma_2^2 + (\hbar t/m)^2}{2\gamma_2}$

(see equations (15) to (18) on page 952 of the paper). Therefore the partial differential equation

$\displaystyle b \frac{\partial^2 f}{\partial x^2} = \frac{1}{t}\frac{\partial f}{\partial t}$

has as a solution the probability density of a moving body undergoing quantum wave-packet dispersion as time progresses (with $b = \frac{(\hbar/m)^2}{2\gamma_2}$ in the above set-up).

A variant of Einstein’s partial differential equation for Brownian motion

Published by Dr Christian P. H. Salas

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