Forces acting on particle in brownian motion

Question

I need to simulate the motion of a small particle (100nm rigid sphere) in water. For the purposes of this I'm only interested in the forces acting on the particle, not its position. I need to generate random forces drawing from a physically realistic distribution.

I've read a few chapters on classical Brownian motion (eg: http://physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/kap6.pdf and https://www.stat.berkeley.edu/~aldous/205B/bmbook.pdf) and I'm if anything more confused about it than when I started. There's plenty of material on the distribution of positions (random walk), but not so much of forces. It seems that each collision with a water molecule lasts on the order of picoseconds during which momentum is transferred (no idea what the exact force over time profile for an individual collision is, but hopefully there are enough collisions in overlapping at any point in time that it would smoothe out the sum; and I assume the collision is fully elastic); and the overall force is the sum of a fairly high number of collisions like this happening at random (assumed to be independent) times.

The tricky parts: if each water molecule moved at the same speed, then the number of collisions per unit time would be simply given by the Poisson distribution; but of course the molecules would have a Maxwell–Boltzmann distribution of velocities, and it seems faster molecules are more likely to collide per unit time (essentially: since they travel further in that amount of time), so the distribution of collisions per unit time is not Poisson, and the distribution of velocities of colliding molecules is not Maxwell–Boltzmann. The total force averaged over any time interval would be the (vector) sum of the momentum of all colliding particles divided by the time, but neither the distribution of colliding particle velocities not the distribution of number of particles colliding per unit time is obvious (and the two distributions are not independent).

How do I produce a random time series that correctly represents the forces acting on a particle in Brownian motion?

Answer 1

I think your confusion probably begins when chapter 6 of Lennart Sjögren introduces the random force:

$$\begin{align} \frac{dx(t)}{dt}&=v(t) \\ \frac{dv(t)}{dt}&=-\frac{\gamma}{m}+\frac{1}{m}\xi(t) \end{align} \tag{6.3}$$ This is the Langevin equations of motion for the Brownian particle.
The random force $\xi(t)$ is a stochastic variable giving the effect of background noise due to the fluid on the Brownian particle.

So how does this random force $\xi(t)$ actually look like?

The force is caused by all the little bumps from the water molecules hitting the Brownian particle under observation. For a small Brownian particle (i.e. hit only by a few water molecules) it would look like this:

Each of the force bumps is very short (a few pico-seconds) and thus can be well approximated by Dirac delta functions: $\xi(t)=\sum_i A_i\delta(t-t_i).$

For a bigger Brownian particle (i.e. hit by many water molecules) it is essentially the same, but there are many more bumps and the force will look more like this:

_{(image from White noise)}

The actual bump strengths ($A_i$) and their time points ($t_i$) are not known. Only statistical statements can be made (also given in equations (6.8) of chapter 6 of Lennart Sjögren):

1. The time average of $\xi(t)$ is zero, because there are equally many bumps from the left as there are from the right. Hence $$\langle\xi(t)\rangle=0.$$
2. The force at one time $\xi(t_1)$ is statistically uncorrelated with the force at another time $\xi(t_2)$, at least for $|t_1-t_2| >$ few pico-seconds. Hence $$\langle\xi(t_1)\xi(t_2)\rangle=g\delta(t_1-t_2).$$ Here again Dirac's delta function pops up.

In signal theory such a $\xi(t)$ is called white noise.

The good thing is: In addition to the two requirements above (including the constant $g$ there) you don't need to know anything else about $\xi(t)$.

All other details (e.g. the Maxwell/Boltzmann distribution for the velocities of the water molecules) are not needed to determine the statistical effect on velocity $v(t)$ of the Brownian particle.

So for simulations, instead of a realistic force $\xi(t)$ (like in the images above), you may choose any white noise function which is computational easier to handle. For example: You can use a function $\xi(t)$ only having the value $+B$ or $-B$, randomly chosen anew after every small time step $\Delta t$.

Even then you will still get the correct statistical behavior for the velocity $v(t)$, and hence for the position $x(t)$ of the Brownian particle.