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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?

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    $\begingroup$ I am leaving this as a comment as it might help you (or not) A long time ago I did LDA measurements, and they did a velocity bias correction. I.e., it corrects for higher probability of measuring faster seeding particles. Of course this was discrete data, but maybe that is good enough. So in your case, is there a way to divide the MB distribution by the particle velocity and draw from that? $\endgroup$
    – Bernhard
    Dec 29 '20 at 7:07
  • $\begingroup$ Can you please explain why simulating Brownian motion as a random walk is not adequate? I thought the whole point of Brownian motion was that the details of the forces producing it are essentially irrelevant. $\endgroup$
    – G. Smith
    Dec 29 '20 at 7:08
  • $\begingroup$ @G.Smith Sure! Basically, there are other forces acting on my particles in addition to the randomly impacting water molecules, and I'm interested in force-dependent effects in addition to just the positions. $\endgroup$
    – Alex I
    Dec 29 '20 at 7:21
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    $\begingroup$ @Bernhard Thank you - reading now. The fact that "velocity bias" even exists means I'm apparently on the right track :) $\endgroup$
    – Alex I
    Dec 29 '20 at 7:23
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    $\begingroup$ Definitely. When reading your first paragraph I thought, sound a bit like a Maxwell distribution. I think you are definitely getting somewhere. To considerations that you can ignore for now. 1. I think the Maxwell distribution is only valid for gasses? 2. You will have to consider the direction and velocity of your particle as well in the velocity bias. Because it is small I suppose it may get significant? You can call this drag ;) Good luck, sound like a cool project you're working on! $\endgroup$
    – Bernhard
    Dec 29 '20 at 8:03
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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:
enter image description here
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:
enter image description here
(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 $\Delta t$.
enter image description here
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.

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  • $\begingroup$ Thomas - thank you! "you don't need to know anything else about ξ(t)" - unfortunately, I think I do. If I was trying to look at the path of the particle, where forces (and acceleration/velocity) can be averaged on fairly long timescales, then perhaps that would be true. In my case, the particle is just part of another simulation, which deals with fairly fast events - let me try to explain ... $\endgroup$
    – Alex I
    Dec 31 '20 at 1:27
  • $\begingroup$ I'm looking at molecular dynamics simulations of stuff that happens at the surface of the particle. The characteristic timescales are ~100ps; the timestep is a few fs. The particle is anchored, and the way it's anchored is not trivially describable by a model like a linear spring (think: atomic friction/adhesion model); the interaction between the brownian forces and the anchor is essentially what I'm trying to look at. I'd like to avoid fully simulating the particle itself; but I need to replace it with a realistic series of bumps (and not just positions/velocities) $\endgroup$
    – Alex I
    Dec 31 '20 at 1:34

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