# Negative diffusion, or force that changes particle velocity distribution

Here is my problem: I am studying the dynamics of particles with a 1D "random" force.

This "random" force is expressed as a Fourier series (sum of cosines) with a random initial phase for each mode. This creates a force that looks random/turbulent but is not just some noise.

Theory tells us that particles will diffuse in the velocity direction, since this is a 1D-1V problem. By performing some calculations and averaging over a big set of particles, we can find an expression of this diffusion coefficient, additionally, this diffusion depends on the particles velocity. So for some velocities the coefficient of diffusion is larger or smaller than for other velocities.

A reference that explains the problem better is: F. Doveil and D. Gésillon, 1982, The Physics of Fluids, 25, "Statistics of charged particles in external random longitudinal electric fields".

In order to measure the diffusion I use two methods:

• First, i do the analytic calculation.
• Second, i wrote a particle solver using a Runge-Kutta of 4th order method. I solve newton equation for N particles and every time i do statistics to find the diffusion of particles.

I notice that depending on the amplitude of the force i get a curve, in which i can measure the diffusion, that is easier or harder to measure. Let me explain, diffusion is measured by the linear growth of that curve. For small amplitude the linear phase lasts for a very long time, hence is easy to measure. But for large amplitudes, the linear phase is really small. Well, since the diffusion depends on the velocity of the particles and the amplitude of the force, then particles diffuse faster to larger velocities, which have a different diffusion coefficient than at the start. Hence the diffusion i measure is a mix of all the diffusion coefficients, but what i want to measure is the diffusion around 1 particle velocity (This interval is such the diffusion is almost constant in that interval, but not outside it).

My objective is to find a Force that can keep particles from diffusing to larger (and smaller) velocities, so that particles velocities stay around the velocity i need to measure the diffusion. I tried just relocating the particles whose velocity goes outside the interval. But this causes problems because particles are no longer captured by a potential well, so they need time to be captured and start diffusing.

In resume, I need a force that generates a negative diffusion of particles so that they stay around the velocity I like.

Is it possible? If yes how can i achieve this? You don't need to explain I just need a reference that explains it.

-If $$\dot{x}(t)=\sqrt{2D}\xi(t)$$, then the particle is "fully difussive" and you could indeed measure $$\propto t$$.
-If $$\dot{x}(t)=\sqrt{2D}\xi(t)+F(x)$$, then the particles will only diffuse around the stable fixed points of the force F and you will not be able to measure the desired $$\propto t$$ behavior.
The fact that this velocity separation disables you to measure a consistent diffusion constant for the whole system is not telling that this is a bad fit? Anyway, any force of the form $$F(|\dot{x}(t)-v_i|)$$ will produce that all the particles will eventually have the same velocity $$v_i$$ (but I would say that adding this force would drive you to misleading conclusions regarding your problem).