# Molecular Simulation/Potential of mean force

The potential of mean force is one of the 3 major forces considered in stochastic dynamics models. The potential of mean force provides a free energy profile along a preferred coordinate, be it a geometric or energetic coordinate, such as the distance between two atoms or the torsional angle of a molecule. This free energy profile describes the average force of all possible configurations of a given system (the ensemble average of the force) on particles of interest.. The potential of mean force can be determined in both Monte Carlo Simulations as well as Molecular Dynamics.

## Relation to Radial Distribution Function

For a liquid, the potential of mean force is related to the radial distribution function,

$w(r)=-k_{B}T\ln \left(g(r)\right)$

The potential of mean force can be calculated directly by using histogram analysis of a trajectory from an MD or MC simulation. This analysis calculates the probability of each possible configuration, and determines the free energy change associated with that state. In systems where parts of the coordinate of interest will not be sufficiently sampled from a naive simulation, umbrella sampling, free energy perturbation, or other enhanced sampling techniques can be used.

### Derivation

$\langle F\rangle =-\langle {{\frac {d}{dr}}U(r^{N})}\rangle _{r_{1},r_{2}}={\frac {-\int dr_{3}\ldots dr_{N}{\frac {dU}{dr_{1}}}e^{-\beta U}}{\int dr_{3}\ldots dr_{N}e^{-\beta U}}}$
$=+k_{B}T{\frac {{\frac {d}{dr_{1}}}\int dr_{3}\ldots dr_{N}e^{-\beta U}}{\int dr_{3}\ldots dr_{N}e^{-\beta U}}}$
$=+k_{B}T{\frac {d}{dr_{1}}}\ln \int dr_{3}\ldots dr_{N}e^{-\beta U}$
$=+k_{B}T{\frac {d}{dr_{1}}}\ln(N(N-1){\frac {\int dr_{3}\ldots dr_{N}e^{-\beta U}}{\int dr^{N}e^{-\beta U}}}$
$=+k_{B}T{\frac {d}{dr_{1}}}\ln \left(g(r)\right)$

By integrating this function over 2 distance values, the reversible work associated with moving the two particles can be found. This work is equal to the change in Helmholtz energy associated with the change in the two states.

$W_{1\rightarrow 2}=\int _{r_{1}}^{r_{2}}\langle F_{1}\rangle dr_{1}=\Delta A_{1\rightarrow 2}$

## Sources

1. Chandler, D. Introduction To Modern Statistical Mechanics, Oxford University Press, 1987, QC174.8.C47

2. Tuckerman, M. Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press, 2010