Approximate Relations for Potentials of Mean Force

In some cases, one may not be able to collect enough trajectories for a histogram analysis. Instead, one can use moments [15]. In particular, the derivative of the free energy is approximately given by the average force (weighted by the Boltzmann factor of the accumulated work) 

This approximation is valid if r(t) is approximately Gaussian distributed. 

we obtained the potential of mean force by collecting work and r values. Formally, we can also use the free energy AA(t) measured at all times t to extract the underlying potential of mean force. If we multiply (5.56) by exp[—/ fc(r(z) — r(t))2/2] and integrate over r, we obtain 

Effectively, this constitutes a Fredholm integral equation of the first kind for exp[—f3G r) where we know the left-hand side, exp(—/M.4(7,)) = 

For very stiff pulling springs (where r is almost a coupling parameter under external control), we can instead pursue the so-called stiff-spring approximation of Park et al. [45]. A Fourier representation of the spring Boltzmann factor on the right-hand side of (5.60) results in 

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