4. Pressure

Internal pressure is calculated from internal energy according to

\[\begin{split}P_a = \frac{1}{\mathcal{V}} \left( 2T_a - \text{Vir}_a \right) \\ \text{Vir}_a = L_a \frac{\partial \mathcal{U}}{\partial L_a} \\ \mathcal{U} = \sum_{i=1}^M \mathcal{U}_0( \{ \mathbf{r}\}_i ) + W[\{ \tilde\phi \} ]\end{split}\]

where \(\mathcal{V}\) is the simulation volume, \({T_a}\) is the kinetic energy and \(L_a\) the length of the box in the Cartesian direction \(a\), Vir is the virial of the total interaction energy \(\mathcal{U}\).

\(\mathcal{U}\) comprises intramolecular bonded terms \(\mathcal{U}_0\) (see Intramolecular bonds for details), and field terms \(W[\{ \tilde\phi \} ]\) (see Theory for details).

Using the above expressions, the following form for internal pressure is obtained:

\[\begin{split}P_a = \frac{2 T_a}{\mathcal{V}} -\frac{L_a}{\mathcal{V}} \sum_{i=1}^N \frac{\partial \mathcal{U}_{0i}}{\partial L_a} + P^{(3)}_a \\\end{split}\]

\[P^{(3)}_a = \frac{1}{\mathcal{V}}\left ( -W[\{ \tilde\phi(\mathbf{r}) \}] + \int \sum_t \bar{V}_t(\mathbf{r})\tilde\phi_t(\mathbf{r})d\mathbf{r} + \int \sum_t \sigma^2\bar{V}_t(\mathbf{r})\nabla_a^2\tilde\phi_t(\mathbf{r}) d\mathbf{r} \right)\]

where \(\bar{V}_t(\mathbf{r}) = \frac{\partial w(\{\tilde\phi\})}{\partial\tilde\phi_t}\) and \(σ\) is a coarse-graining parameter (see Filtering for details). Note that the above expression is obtained for a Gaussian filter which is the most natural choice in HhPF theory.