Free Energy Calculation
=================================

:Date: 29 July 2019





Derivation
-------------

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

.. math::
	\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 (:math:`e^{-\beta \Delta E}`) is too sensitive, :math:`\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 :math:`\Delta E` is small

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