Free Energy Calculation

Date:

29 July 2019

Derivation

Step 1 A straight forward result of Canonical Ensemble And Boltzmann Factor: Zwanzig equation

\[\begin{align*} &\Delta F = -k_B T \ln{z_2\over z_1} =-k_B T \ln \langle e^{-\beta \Delta E} \rangle \end{align*}\]

Problem of using Zwanzig equation: Exponential (\(e^{-\beta \Delta E}\)) is too sensitive, \(\Delta E\) must be small for it to work

Step 2 Solution to this problem: Perturbation Theory

idea: Take many steps instead of one, at each step \(\Delta E\) is small

\[\begin{align*} &\Delta F = \int_0^1 \langle {\partial H \over \partial \lambda} \rangle \; d\lambda \end{align*}\]