Exponential Averaging

Free Energy Fundamentals
Theory Definitions Exponential Averaging / Zwanzig Thermodynamic Integration Bennett Acceptance Ratio WHAM MBAR
Methods of Free Energy Simulations Thermodynamic Cycle Intermediate States (Constructing) Soft Core Potentials Running Simulations Extracting Simulation Information Analysis / Bootstrap Sampling
Improving Sampling and Accelerating Simulations Umbrella Sampling Hamiltonian Replica Exchange [math]\displaystyle{ \lambda }[/math]-dynamics / Expanded Ensemble

Free Energy How-to's
Example Calculations Absolute Solvation Free Energy Relative Binding Affinity Absolute Binding Affinity
Tutorials Ethanol Solvation - GROMACS Expanded Ensemble - GROMACS Absolute Binding Free Energy - GROMACS

Exponential Averaging (EXP) is one of the earliest free energy methods available to researchers. Mostly used to evaluate between only two states of interest, it has an exact solution where as many others require approximations. Other names for this technique are the "Zwanzig Relationship" (named after the person who derived it) and "free energy perturbation (FEP)." To avoid confusion, the later term should be avoided as it can easily be confused with other definitions in the scientific field.

Derivation

Starting from the statistical relation for free energy of

[math]\displaystyle{ \displaystyle A = -\beta^{-1} \mathrm{ln}\, Q }[/math]

the free energy difference between two states with potentials [math]\displaystyle{ U_0(\vec{q}) }[/math] and [math]\displaystyle{ U_1(\vec{q}) }[/math] is simply

[math]\displaystyle{ \displaystyle A_1 - A_0 = -\beta^{-1} \left[\mathrm{ln}(Q_1) - \mathrm{ln}(Q_0)\right] = -\beta^{-1} \mathrm{ln}\left[\frac{Q_1}{Q_0}\right] }[/math].

By then adding and subtracting [math]\displaystyle{ e^{-beta U_0(\vec{q})} }[/math] from the numerator, we get

[math]\displaystyle{ \Delta A_{10} = -\beta^{-1} \mathrm{ln} \left[\frac{\int e^{-\beta \left(U_1(\vec{q})-U_0(\vec{q})+U_0(\vec{q})\right)} d\vec{q}}{Q_0}\right] = -\beta^{-1} \mathrm{ln} \left[\frac{\int e^{-\beta \left(U_1(\vec{q})-U_0(\vec{q})\right)} e^{-\beta U_0(\vec{q})} d\vec{q}}{Q_0}\right] }[/math]

which gives the final relationship of:

<math> \displaystyle \Delta A_{10} = \beta^{-1} \mathrm{ln} \left\langle e^{-\beta \left(U_1(\vec{q})-U_0(\vec{q})\right)} \right\rangle_0 = \beta^{-1} \mathrm{ln} \left\langle e^{-\beta \Delta U(\vec{q})} \right\rangle_0