Equation for Brownian Motion Trajectory

[H Quizzagan]

I am trying to understand how one can simulate the trajectory of a Brownian particle as a function of time. I am only able to do it with the assumption that I can simply generate random values of x and then take the cumulative sums of these values to get the trajectory of the Brownian particle.

But, are there different sets of equations that guides or regarding the function x(t) that I can easily use to simulate the trajectory of a Brownian particle? Thank you so much!

 

[vanhees71]

The equation of motion for a Brownian particle is called Langevin equation, which is a stochastic differential equation. In the most simple one-dimensional form it reads
$$\dot{p}+\gamma p + \sqrt{2B} \xi(t)=0,$$
where $B=m \gamma T$ (where $m$ is the mass of the Brownian particle, $T$ the temperature of the fluid the particle is moving in), $\gamma$ is the friction coeffcient, and $\xi$ is normalized Gaussian white noise,
$$\langle \xi(t) \xi(t') \rangle=\delta(t-t').$$

 

[H Quizzagan]

Indeed, the Langevin equation is useful in describing Brownian motion. Correct me if I am wrong, but is it that solving this differential equation yields for an equation of the mean-square displacement. This mean-square displacement which is a function of time is the analytical way to solve for the trajectory/position of the Brownian particle over time?

 

[vanhees71]

Yes, you can easily solve the equation analytically in the sense that you can derive the phase-space distribution functions. The point is that adding many Gaussian distributed independent random numbers you get again a Gaussian distribution. Thus both the momentum and the position are Gaussian distributions. Thus you have to evaluate the mean (for given initial conditions) $(\langle x(t) \rangle, \langle p(t) \rangle)$ and the covariance matrices $\langle x_i(t) x_j(t)$, $\langle p_i(t) p_j(t)$, and $\langle x_i(t) p_j(t)$.

The calculation is a bit lengthy for the forum, but the idea is as follows: You calculate the Green's function of the deterministic part of the equation, i.e.,
$$\dot{G}+\gamma G=\delta(t) \; \Rightarrow \; G(t)=\Theta(t) \exp(-\gamma t)$$
Then the solution for the stochastic equation reads
$$p(t)=-\sqrt{2B} \int_0^t \mathrm{d} t' G(t-t') \xi(t')+p_0 \exp(-\gamma t).$$
Because $\dot{x}=p/m$ you need to integrate this only once more for $x(t)$, and then you can evaluate all the needed averages using
$$\langle \xi(t) \rangle=0, \quad \langle \xi(t) \xi(t') \rangle=\delta(t-t').$$

 

[H Quizzagan]

@vanhees71 oh okay. thanks a lot!