Virial expansion: Potential that depends on velocities

[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 R_i 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?

 

[gtoguy97]

Yes, it is possible to find the second virial coefficient numerically for a certain temperature. The easiest way to do this is to use a numerical integration technique such as Monte Carlo integration. This involves randomly sampling points from the potential energy function and calculating the integral over a range of temperatures. The second virial coefficient can then be calculated from the integration results.