Thermodynamic integration

Thermodynamic integration is a method used to compare the difference in the thermodynamic quantity of two given states (e.g., A and B) in molecular dynamics simulation. Free energy difference is one quantity commonly computed in this way; since they are not simply functions of the phase space coordinates of the system, but are related to the canonical partition function Q(N,V,T), they cannot be directly measured in a simulation. These differences are usually calculated by designing a thermodynamic cycle and integrating along the relevant paths. Such paths can either be real chemical processes or alchemical processes. A good example of the alchemical process is the Kirkwood's coupling parameter method.[1]

Free energy can be expressed by

F(N,V,T,\lambda)=-k_{B}T \ln Q(N,V,T,\lambda),

where λ=0 represents state A and λ=1 state B. If we take the derivative of F with respect to λ, we will get that it equals the ensemble average of the derivative of potential energy with respect to λ.

\Delta F = \int_0^1 d\lambda \frac{\partial F(\lambda)}{\partial\lambda} = -\int_0^1 d\lambda \frac{k_{B}T}{Q} \frac{\partial Q}{\partial\lambda} = \int_0^1 d\lambda \left\langle\frac{\partial U(\lambda)}{\partial\lambda}\right\rangle_{\lambda}

Thus the free energy difference of different states can be computed from the difference of potential energy. Umbrella sampling is a related free energy method. It adds a bias to the potential energy. In the limit of an infinite strong bias it is equivalent to thermodynamic integration.[2]

See also

References

  1. ^ J. G. Kirkwood. Statistical mechanics of fluid mixtures, J. Chem. Phys., 3:300-313,1935
  2. ^ J Kästner et al. (2006). "QM/MM Free-Energy Perturbation Compared to Thermodynamic Integration and Umbrella Sampling: Application to an Enzymatic Reaction". JCTC 2 (2): 452–461. doi:10.1021/ct050252w.  edit