Slow-growth approach
The free-energy profile along a geometric parameter [math]\displaystyle{ \xi }[/math] can be scanned by an approximate slow-growth approach[1]. In this method, the value of [math]\displaystyle{ \xi }[/math] is linearly changed from the value characteristic for the initial state (1) to that for the final state (2) with a velocity of transformation [math]\displaystyle{ \dot{\xi} }[/math]. The resulting work needed to perform a transformation [math]\displaystyle{ 1 \rightarrow 2 }[/math] can be computed as:
- [math]\displaystyle{ w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. }[/math]
In the limit of infinitesimally small [math]\displaystyle{ \dot{\xi} }[/math], the work [math]\displaystyle{ w^{irrev}_{1 \rightarrow 2} }[/math] corresponds to the free-energy difference between the the final and initial state. In the general case, [math]\displaystyle{ w^{irrev}_{1 \rightarrow 2} }[/math] is the irreversible work related to the free energy via Jarzynski's identity[2]:
- [math]\displaystyle{ exp^{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T}}= \bigg \langle exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \bigg\rangle. }[/math]
Note that calculation of the free-energy via this equation requires averaging of the term [math]\displaystyle{ {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} }[/math] over many realizations of the [math]\displaystyle{ 1 \rightarrow 2 }[/math] transformation. Detailed description of the simulation protocol that employs Jarzynski's identity can be found in reference [3].