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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 $-PV$?
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