Thermodynamic integration

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In molecular dynamics simulation, it is necessary, sometimes, to compare the thermal quantity difference of two given phases. For example, free energy differences are not simply functions of the phase space coordinates of the system, but instead, they are related to the canonical partition function Q(N,V,T) and cannot be directly measured in a simulation. Thermodynamic cycles are designed and integration along the path are done to calculate these thermal quantities. Such paths can either be real chemical process, or alchemical processes. A good example of the alchemical process the Kirkwood's coupling parameter method^[1].

As free energy can be expressed by

F = - k B T l n Q (N, V, T)

where in the partition function take the derivate of F, we will get that it equals the derivative of potential energy. thus the free energy difference of different state can be used to calculate the difference of potential energy.

$F(\lambda=1)-F(\lambda=0) = \int_{\lambda=0}^{\lambda=1}d <\frac{\partial U(\lambda)}{\partial\lambda}>_{\lambda}$