reweighting

This is part of the analysis module

Calculate free energies from a biassed/higher temperature trajectory. Calculate free energies from a biassed/higher temperature trajectory.

We can use our knowledge of the Boltzmann distribution in the cannonical ensemble to reweight the data contained in trajectories. Using this procedure we can take trajectory at temperature \(T_1\) and use it to extract probabilities at a different temperature, \(T_2\), using:

\[ P(s',t) = \frac{ \sum_{t'}^t \delta( s(x) - s' ) \exp\left( +( \left[\frac{1}{T_1} - \frac{1}{T_2}\right] \frac{U(x,t')}{k_B} \right) }{ \sum_{t'}^t \exp\left( +\left[\frac{1}{T_1} - \frac{1}{T_2}\right] \frac{U(x,t')}{k_B} \right) } \]

where \(U(x,t')\) is the potential energy of the system. Alternatively, if a static or pseudo-static bias \(V(x,t')\) is acting on the system we can remove this bias and get the unbiased probability distribution using:

\[ P(s',t) = \frac{ \sum_{t'}^t \delta( s(x) - s' ) \exp\left( +\frac{V(x,t')}{k_B T} \right) }{ \sum_t'^t \exp\left( +\frac{V(x,t')}{k_B T} \right) } \]