# Question about the evolution of an ensemble with non hamiltonian equation of motion

1. Mar 27, 2009

### 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.