# Virial expansion: Potential that depends on velocities

1. Feb 12, 2016

### Korbid

I'm studying this potential that depends on positions and velocities. http://motion.cs.umn.edu/PowerLaw/
$$E(\tau)=\frac{k}{\tau^2}e^{-\tau/\tau_0}$$
where τ is the time to collision.
By extrapolation $$\tau=\frac{b-\sqrt{d}}{a}$$
$$a=||\vec{v}_{ij}||^2$$
$$b=-\vec{r}_{ij}\cdot\vec{v}_{ij}$$
$$c=||\vec{r}_{ij}||^2 - (R_i+R_j)^2$$
$$d=b-ac$$
Where Ri is the radius of the particles and κ,τ0 characteristic parameters.

I need to find the second virial coefficient for a certain temperature, even numerically. Is it possible?

2. Feb 17, 2016