How much energy should I give into each particle in this equation:

Question

I am trying to recreate the results of an article (membrane simulation) and I have the following line:

Both particles have the same soft radius,

$U_{rep} (r)/\epsilon = \text{exp}\left\{ -20 (r/\sigma -1 )\right \}$

With $\epsilon$ being the energy scale, $\sigma$ being the length scale, and $r$ is the distance between two particles. This is a two body interactions. If I calculate the interactions between two particles, should I put $U_{rep}$ or half of it into each particle?

Answer 1

Whatever is the nature of the interaction one should take it into account in the Hamiltonian only once for each distinct pair of particles: $$ U_\text{int} = \sum_{i>j} U_1 (\mathbf{r}_i, \mathbf{r}_j) = \sum_{i<j} U_1 (\mathbf{r}_i, \mathbf{r}_j). $$ Usually these sums are merged to make the Hamiltonian of the system more symmetric. In that case the total sum should be divided by two: $$ U_\text{int} = \frac{1}{2}\sum_{i\neq j} U_1 (\mathbf{r}_i, \mathbf{r}_j). $$ Now the total hamiltonian $$ H = \sum_i \frac{p_i^2}{2m} + \frac{1}{2}\sum_{i\neq j} U_1 (\mathbf{r}_i, \mathbf{r}_j) $$ can be rewritten as a sum of "one-particle" Hamiltonians: $$ H = \sum_i\left( \frac{p_i^2}{2m} + \frac{1}{2}\sum_{j \neq i} U_1 (\mathbf{r}_i, \mathbf{r}_j) \right). $$ If you mean this when say "put energy into each particle" then, yes, you should put a half of the interaction energy.