# Correlation of position and velocity in Brownian motion

There are two definitions of the term "Brownian motion": a physical science definition based on how things such as Brownian particles move, and a mathematical definition as a certain continuous limit of a random walk. In the mathematical definition I think (please correct if I am wrong) the path is a fractal curve and therefore the instantaneous velocity is hard to define; it may even be undefined (?). But a physical thing such as a Brownian particle always has a well-defined velocity $${\bf v}(t)$$ at each instant of time $$t$$ (in classical not quantum physics).

I think it is this instantaneous velocity which satisfies the equipartition theorem, so that for a particle in a fluid at temperature $$T$$ we have $$\frac{1}{2} m \langle v_x^2 \rangle = \frac{1}{2} k_{\rm B} T$$ and similarly for $$y$$- and $$z$$-components. The average here could be read either as the average over many similar particles at some given time, or as the average over time for a single particle. (It is not self-evident that these two types of average agree, but I think they do in conditions of thermal equilibrium when the average in question is not itself changing with time.)

Another well known property of the motion is $$\langle x^2 \rangle = 2 D t$$ where $$D$$ is a constant. Here I think the average has to be over many realisations of the experiment: this is not a time-average.

We also have $$\langle x \rangle = 0, \;\;\;\;\; \langle v_x \rangle = 0.$$

My question is: what is the value of $$\langle x v_x \rangle ?$$

This is related to another recent question from me: Understanding mean rate of change in Brownian motion

• @ChiralAnomaly see other question (linked in the above) for the background. Each particle wanders such that its mean $x^2$ grows with time, so in this sense the sample is expanding. Sep 23 '21 at 15:59

Almost all paths $$t\mapsto B_t(\omega)$$ of the Wiener process are in fact not differentiable. Therefore, if we take $$B_t$$ as position of the particle $$x(t)$$ we end up having an unphysical particle whose velocity is not well defined. As far as I know, this led Langevin, Ornstein and Uhlenbeck to develop a dynamical theory of Brownian motion by which the velocity $$v(t)=\dot x(t)$$ is 'driven' by a Wiener process $$v(t)=e^{-\beta t}v_0+e^{-\beta t}\int_0^te^{\beta s}\,dB_s$$ which is described in Nelson [1]. The parameter $$\beta$$ describes friction. The larger that is the faster the velocity tends to zero. If $$\beta=0$$ we have $$v(t)=v_0+B_t\,,\quad\quad x(t)=x_0+\int_0^tv(s)\,ds=x_0+v_0t+\int_0^tB_s\,ds\,.$$ In this theory the mean (over many particles) of $$x(t)v(t)$$ is \begin{align} \mathbb E[x(t)v(t)]&=\mathbb E\left[B_t\int_0^tB_s\,ds\right]=\mathbb E\left[B_t\int_0^t(t-s)\,dB_s\right]=\int_0^t(t-s)\,ds=t^2/2. \end{align} This is a covariance. The correlation turns out to be $$\frac{\mathbb E[x(t)v(t)]}{\sqrt{\mathbb E[x^2(t)]}\sqrt{\mathbb E[v^2(t)]}}=\frac{1}{2\sqrt{3}}\,.$$ Note that the Langevin/OU theory is not compatible with $$\langle x^2\rangle=2Dt$$. That equation is due to Einstein and Smoluchovski which Nelson calls a highly idealized treatment in his Chapter 9. In fact in Nelson's notation, Einstein Smoluchovski corresponds to $$x(t)=\sqrt{2D}B_t$$ which Langevin/OU discarded as unphysical. Nelson also writes that the predictions of the OU theory are numerically indistinguishable of the Einstein Smoluchovsik theory.
I also want to mention the approach using white noise by which the derivative of the Wiener process exists in the sense of distributions (delta functions): $$w(t)=\dot B_t=\frac{dB_t}{dt}\,,\quad\quad E[w(t)w(s)]=\delta(t-s)\,.$$ Informally we can write this as $$\mathbb E[dB_t\,dB_s]=\delta(t-s)\,dt\,ds$$ which (informally) yields $$\mathbb E[dB_t\,B_t]=\int_0^t\delta(t-s)\,dt\,ds=dt\,.$$ Therefore, the covariance is $$\mathbb E[\dot B_tB_t]=1\,.$$ In this theory we could set the position $$x(t)$$ to be $$B_t$$ but, of course, have the unphysical velocity $$v(t)=\dot B_t$$ (delta function). Note that the variance $$\mathbb E[(\dot B_t)^]$$ is infinite by the definition of white noise. I could not figure out quickly if the mathematicians therefore define the correlation of $$\dot B_t$$ and $$B_t$$ to be zero or if the let it be undefined.
• Thanks v. much for this. It seems I should read Nelson's book, but I would appreciate clarification of your comment on $\langle x^2 \rangle = 2Dt$ (where the average is over all possible motions for a given time interval). That is an experimentally observed datum so how can Langevin/OU theory be not compatible with it? A theory which does not reproduce this observation must be wrong. Sep 23 '21 at 15:57