Exploring the Evolution of an Ensemble of Non-Hamiltonian Particles

[evantop]

Consider an ensemble of one particle systems each evolving in one spatial dimension according to the non Hamiltonian equation of motion:
dx/dt=-ax

where x(t) is the position of the particle at time t and a is a constant. The compressibility of this system is nonzero so the ensemble’s distribution function f(x,t) satisfies a Liouville equation of the form:
df/dt-ax(df/fx)=af

Where it was found that the distribution function in the form of
f(x,t)=exp(at)*exp(-c((x^2)*exp(2at)))
I need to describe the evolution of the ensemble distribution qualitatively and explain why it should evolve that way!
We can see that at t=0 it's acts like a regular gaussian.

Please help!

 

[BigJDubs]

The evolution of the ensemble distribution is such that the peak of the distribution will move in the negative x-direction at a rate proportional to a. The width of the distribution will also expand over time as the exponential term (exp(2at)) increases. This is due to the fact that the equation of motion for the particles is non-Hamiltonian, meaning that the particles do not follow a closed curve in phase space. Instead, they are constantly losing energy and dissipating, leading to an increase in the spread of the ensemble distribution.