In Case Study 1 in Frenkel and Smit (2008, 2nd ed, p.51), we calculate the pressure of a Lennard-Jones fluid by a NVT Monte Carlo simulation. We find the pressure from the virial theorem by calculating

$$ \frac{P}{k_BT} = \rho + \frac{\text{vir}}{Vk_BT} $$


$$ \text{vir} = \frac{\sum\limits_{i} \sum\limits_{j>i} \textbf{f}(\textbf{r}_{ij}) \cdot \textbf{r}_{ij} }{3} $$

But why are we not calculating the pressure from the virial expansion at second order by

$$ \frac{P}{k_BT} = \rho + B_2 \rho^2 $$

with $B_2$ being, assuming a spherically symmetric potential and using spherical polar coordinates

$$ B_2 = 2\pi \int\limits_{0}\limits^{\infty} \left[ 1 - e^{-u(r)/k_BT} \right] r^2 dr $$

This integral seems like something we can sample by a Monte Carlo simulation, right?


1 Answer 1


The virial theorem provides an exact expression for the pressure in a classical gas controlled by di-atomic forces. This expression can be sampled in MD simulations, resulting in an exact (within numerical errors) equation of state for the interacting (classical) gas. Note that the viral in your formula should be replaced by its ensemble average.

The virial expansion (as the name suggests) is an approximate result, valid in a low density (classical or quantum) gas. The expression you quote is valid to second order in the fugacity of a classical gas. There is no need to sample this integral, it is just a one-dimensional integral which is trivial to compute. The partition function of the fully interacting gas, which is what the MD simulation samples, is a very high dimensional (6N) integral.


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