# Pressure of a system of particles

I'll derive the pressure of a system of particles, which would be used in a MD simulation for example, according to Allen's Computer Simulation of Liquids and show where my issue is.

Consider a Hamiltonian system with $$N$$ interacting particles and generalized coordinates $$q_k$$. The virial theorem is:

$$\langle q_k \frac{\partial H}{\partial q_k} \rangle = kT$$

This implies that

$$\frac{1}{3} \langle \sum_i \mathbf{r}_i \cdot \mathbf{f}_i^{tot} \rangle = NkT$$

where $$\mathbf{f}_i^{tot}$$ is the total force including interparticle and external forces. So we can split this force into internal and external contributions, $$\mathbf{f}_i^{tot} = \mathbf{f}_i^{ext} + \mathbf{f}_i^{int}$$. The external part of the force defines the pressure on the system via

$$\frac{1}{3} \langle \sum_i \mathbf{r}_i \cdot \mathbf{f}_i^{ext} \rangle = -PV$$

• What is the reasoning behind this equation?
• Why does the sum of dot products between particle position vectors and external forces equal $$-PV$$?