# Thermodynamic integration

Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) whose potential energies $U_{A}$ and $U_{B}$ have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the Boltzmann-weighted integral over phase space (i.e. partition function), the free energy difference between two states cannot be calculated directly from the potential energy of just two coordinate sets (for state A and B respectively). In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Kirkwood's coupling parameter method.

## Derivation

Consider two systems, A and B, with potential energies $U_{A}$ and $U_{B}$. The potential energy in either system can be calculated as an ensemble average over configurations sampled from a molecular dynamics or Monte Carlo simulation with proper Boltzmann weighting. Now consider a new potential energy function defined as:
    $U(\lambda )=U_{A}+\lambda (U_{B}-U_{A})$
Here, $\lambda$ is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of $\lambda$ varies from the energy of system A for $\lambda =0$ and system B for $\lambda =1$. In the canonical ensemble, the partition function of the system can be written as:
    $Q(N,V,T,\lambda )=\sum _{s}\exp[-U_{s}(\lambda )/k_{B}T]$
In this notation, $U_{s}(\lambda )$ is the potential energy of state $s$ in the ensemble with potential energy function $U(\lambda )$ as defined above. The free energy of this system is defined as:
    $F(N,V,T,\lambda )=-k_{B}T\ln Q(N,V,T,\lambda )$,
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.
    ${\begin{aligned}\Delta F(A\rightarrow B)&=\int _{0}^{1}{\frac {\partial F(\lambda )}{\partial \lambda }}d\lambda \\&=-\int _{0}^{1}{\frac {k_{B}T}{Q}}{\frac {\partial Q}{\partial \lambda }}d\lambda \\&=\int _{0}^{1}{\frac {k_{B}T}{Q}}\sum _{s}{\frac {1}{k_{B}T}}\exp[-U_{s}(\lambda )/k_{B}T]{\frac {\partial U_{s}(\lambda )}{\partial \lambda }}d\lambda \\&=\int _{0}^{1}\left\langle {\frac {\partial U(\lambda )}{\partial \lambda }}\right\rangle _{\lambda }d\lambda \\&=\int _{0}^{1}\left\langle U_{B}(\lambda )-U_{A}(\lambda )\right\rangle _{\lambda }d\lambda \end{aligned}}$
The change in free energy between states A and B can thus be computed from the integral of the ensemble averaged derivatives of potential energy over the coupling parameter $\lambda$. In practice, this is performed by defining a potential energy function $U(\lambda )$, sampling the ensemble of equilibrium configurations at a series of $\lambda$ values, calculating the ensemble-averaged derivative of $U(\lambda )$ with respect to $\lambda$ at each $\lambda$ value, and finally computing the integral over the ensemble-averaged derivatives.
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.