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Slow-growth approach

The free-energy profile along a geometric parameter $ \xi$ can be scanned by an approximate slow-growth approach [84]. In this method, the value of $ \xi$ is linearly changed from the value characteristic for the initial state (1) to that for the final state (2) with a velocity of transformation $ \dot{\xi}$. The resulting work needed to perform a transformation $ 1 \rightarrow 2$ can be computed as:

$\displaystyle w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt.$ (6.31)

In the limit of infinitesimally small $ \dot{\xi}$, the work $ w^{irrev}_{1 \rightarrow 2}$ corresponds to the free-energy difference between the the final and initial state. In the general case, $ w^{irrev}_{1 \rightarrow 2}$ is the irreversible work related to the free energy via Jarzynski's identity [74]:

$\displaystyle {\rm exp}\left\{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T} \right...
...xp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} \bigg\rangle.$ (6.32)

Note that calculation of the free-energy via eq.(6.32) requires averaging of the term $ {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \}$ over many realizations of the $ 1 \rightarrow 2$ transformation. Detailed description of the simulation protocol that employs Jarzynski's identity can be found in Ref. [78].

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