Thermodynamic integration

A detailed description of thermodynamic integration is given in reference .

The free energy of a fully interacting system can be written as the sum of the free energy a non-interacting reference system and the difference in the free energy of the fully interacting system and the non-interacting system

[math]\displaystyle{ F_{1} = F_{0} + \Delta F_{0\rightarrow 1} }[/math].

Using thermodynamic integration the free energy difference between the two systems is written as

[math]\displaystyle{ \Delta F_{0\rightarrow 1} = \int\limits_{0}^{1} d\lambda \langle U_{1}(\lambda) - U_{0}(\lambda) \rangle_{\lambda} }[/math].

Here [math]\displaystyle{ U_{1}(\lambda) }[/math] and [math]\displaystyle{ U_{0}(\lambda) }[/math] describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The coupling strength of the systems is controlled via the coupling parameter [math]\displaystyle{ \lambda }[/math]. It is neccessary that the connection of the two systems via the coupling constant is reversible. The notation [math]\displaystyle{ \langle \ldots \rangle_{\lambda} }[/math] denotes an ensemble average of a system driven by the following classical Hamiltonian

[math]\displaystyle{ H_{\lambda}= \lambda H_{1} + (1-\lambda) H_{0} }[/math].

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