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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: :$U\left(\lambda\right) = U_A + \lambda\left(U_B - U_A\right)$ 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: :