Non-equilibrium Thermodynamics
=================================

:Date: 28 July 2019

Dependency
-------------

- `Boltzmann Distribution`_

.. _Boltzmann Distribution: 


What is the problem ?
-------------------------



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

**Step 1** **Crooks Fluctuation Theorem** relates **work** :math:`W` of a non-equilibrium process to **free energy** :math:`\Delta G` (an equilibrium property)

.. math::
	\begin{align*}
	&\boxed{
		{P_{A\to B}(W) \over P_{B\to A}(-W)} = e^{\beta (W-\Delta G)}
	}\\
	& where\\
	& P = \text{Probability}\\
	& A,B = \text{States}
	\end{align*}

**Step 2** Proof of **Step 1**

*TO BE ADDED*

**Step 3** **Jarzynski Equality**, a consequence of Crooks Fluctuation Theorem

.. math::
	\begin{align*}
	&{P_{A\to B}(W) e^{-\beta W} = P_{B\to A}(-W)} e^{-\beta \Delta G}\\
	& \text{Integrate over } P\\
	& \Rightarrow \boxed{
		\langle e^{-\beta W} \rangle = e^{-\beta \Delta G}
	}
	\end{align*}

What is the significance ?
-----------------------------
In order to calculate free energy:

Previously we need to simulate molecules in equilibrium process which is time consuming.
Now we can just use Non-equilibrium process, saves a lot of time!

Perform multiple Non-equilibrium simulations, then:

- Using Crooks Fluctuation Theorem, find :math:`W` such that :math:`P_{A\to B}(W) = P_{B\to A}(-W)`, then :math:`\Delta G=W`

- Using Jarzynski Equality, :math:`\Delta G= {1\over -\beta}\ln \langle e^{-\beta W} \rangle`