# Why is potential energy used and not total energy when calculating free energy?

I have run a replica-exchange molecular dynamics simulation, using a number, $$n$$, of replicas initiated at the same number, $$n$$, of unique temperatures. I have calculated the free energy profile over this range of temperatures using the potential energy, giving me $$n-1$$ free energy points as this is calculated w.r.t. to the lowest temperature.

My question is, and I suspect I've misunderstood something crucial, why is it only the potential energy which is considered in the calculation?

In the free energy difference:

$$\tag{1}\Delta F_{ij} = -k_{\text{B}} T_j \ln \left( \frac{Z_j}{Z_i} \right),$$

in which $$i$$ is the reference. I get that the kinetic part of $$Z$$ cancels if they're at the same temperature but that's not the case here is it?

My explanation as of now is that since I calculated the free energy using the configurations from each replica, not from temperature data, in each calculation using the configurations from the replicas they all have the same "effective" temperature and so the kinetic cancels. This is because I'm not really calculating free energy difference between different temperatures but more so of different states, but I feel like this is a weak explanation since these states differ only by temperature?

• If you read section 1.2 of this paper, they show how the detailed balance condition is enforced in REMD. When you enforce detailed balance, the kinetic energy ends up cancelling out so that the Metropolis condition only depends on the potential energy. This is good because it simplifies things but also because one should only ever need potential energies to compute ensemble averages: see here Jan 20 at 0:46
• Yes, for the Metropolis condition I agree in that with appropriate scaling of the momenta, the kinetic terms cancel in a swap but how does that relate to the free energy calculation? For ensemble averages yes, but the free energy isn't an ensemble average, no? Jan 20 at 0:51
• When you say, "I have calculated the free energy profile over this range of temperatures using the potential energy", what do you mean? Cause it sounds like you have maybe just computed the average potential energy at different temperatures. Sorry, I might just be adding to the confusion cause I may have misunderstood your initial question. The equation you have written will depend on the difference in kinetic energy at the two different temperatures. Jan 20 at 1:19
• Maybe the answer is that REMD is just a way of sampling from the canonical ensemble at a single temperature. The extra temperatures are just there as a way of proposing Monte Carlo moves. So, in the end, you get a set of configurations from which you can compute ensemble averages at a fixed temperature. Computing a free energy difference between states at different temperatures would require using some kind of thermodynamic integration or some other technique. It is rarely possible to actually compute an absolute free energy. Maybe that's part of the confusion? Jan 20 at 1:25
• @jheindel Sorry for being unclear, I used histogram reweighting technique, WHAM specifically, in which the sampled potential energies were taken per replica trajectory. I followed this paper, eqns 54 & 66 in particular, in which they explicitly only refer to the potential energy. Jan 20 at 1:38