Slow-growth approach: Difference between revisions
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can be computed as: | can be computed as: | ||
<math> | ::<math> | ||
w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. | w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. | ||
</math> | </math> | ||
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In the limit of infinitesimally small <math>\dot{\xi}</math>, the work <math>w^{irrev}_{1 \rightarrow 2}</math> | In the limit of infinitesimally small <math>\dot{\xi}</math>, the work <math>w^{irrev}_{1 \rightarrow 2}</math> | ||
corresponds to the free-energy difference between the the final and initial state. | corresponds to the free-energy difference between the the final and initial state. | ||
In the general case, <math>w^{irrev}_{1 \rightarrow 2} | In the general case, <math>w^{irrev}_{1 \rightarrow 2}</math> is the irreversible work related | ||
to the free energy via Jarzynski's identity<ref name="jarzynski1997"/>: | to the free energy via Jarzynski's identity<ref name="jarzynski1997"/>: | ||
<math> | ::<math> | ||
exp^{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T}}= | |||
\bigg \langle exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \bigg\rangle. | \bigg \langle exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \bigg\rangle. | ||
</math> | </math> | ||
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can be found in reference <ref name="oberhofer2005"/>. | can be found in reference <ref name="oberhofer2005"/>. | ||
== References == | == References == | ||
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</references> | </references> | ||
---- | ---- | ||
[[Category: | [[Category:Advanced molecular-dynamics sampling]][[Category:Theory]] | ||
Latest revision as of 13:54, 16 October 2024
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].