Non-equilibrium Thermodynamics

Date:: 28 July 2019

Dependency

Boltzmann Distribution

What is the problem ?

Derivation

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

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

Step 2 Proof of Step 1

TO BE ADDED

Step 3 Jarzynski Equality, a consequence of Crooks Fluctuation Theorem

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

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 \(W\) such that \(P_{A\to B}(W) = P_{B\to A}(-W)\), then \(\Delta G=W\)
Using Jarzynski Equality, \(\Delta G= {1\over -\beta}\ln \langle e^{-\beta W} \rangle\)