Overlapping distribution method

For two N particle systems 0 and 1 with partition function $Q\_\{0\}$ and $Q\_\{1\}$ ,

from $F(N,V,T)\; =\; -\; k\_\{B\}T\; \backslash ln\; Q$

get the thermodynamic free energy difference is $\backslash Delta\; F\; =\; -k\_\{B\}T\; \backslash ln\; (Q\_\{1\}/Q\_\{0\})\; =\; -\; k\_\{B\}\; T\; \backslash ln\; (\backslash frac\{\backslash int\; ds^\{N\}\backslash exp[-\backslash beta\; U\_\{1\}(s^\{N\})]\}\{\backslash int\; ds^\{N\}\backslash exp[-\backslash beta\; U\_\{0\}(s^\{N\})]\})$

For every configuration visited during this sampling of system 1 we can compute the potential energy U as a function of the configuration space, and the potential energy difference is

$\backslash Delta\; U\; =\; U\_\{1\}(s^\{N\})\; -\; U\_\{0\}(s^\{N\})$

Now construct a probability density of the potential energy from the above equation:

$p\_\{1\}(\backslash Delta\; U)\; =\; \backslash frac\{\backslash int\; ds^\{N\}\backslash exp(-\backslash beta\; U\_\{1\})\backslash delta(U\_\{1\}-U\_\{0\}-\backslash Delta\; U)\}\{Q\_\{1\}\}$

where in $p\_\{1\}$ is a configurational part of a partition function

$$

p_{1}(\Delta U) = \frac{\int ds^{N}\exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U)}{Q_{1}} = \frac{\int ds^{N}\exp[-\beta(U_{0}+\Delta U)]\delta(U_{1}-U_{0}-\Delta U)}{Q_{1}}

$=\; \backslash frac\{Q\_\{0\}\}\{Q\_\{1\}\}\; \backslash exp\; (-\backslash beta\; \backslash Delta\; U)\; \backslash frac\{\backslash int\; ds^\{N\}\backslash exp(-\backslash beta\; U\_\{0\})\backslash delta(U\_\{1\}-U\_\{0\}-\backslash Delta\; U)\}\{Q\_\{0\}\}\; =\; \backslash frac\{Q\_\{0\}\}\{Q\_\{1\}\}\; \backslash exp\; (-\; \backslash beta\; \backslash Delta\; U)\; p\_\{0\}(\backslash Delta\; U)$

since

$\backslash Delta\; F\; =\; -k\_\{B\}T\; \backslash ln\; (Q\_\{1\}/Q\_\{0\})$

$\backslash ln\; p\_\{1\}(\backslash Delta\; U)\; =\; \backslash beta(\backslash Delta\; F\; -\backslash Delta\; U)\; +\; \backslash ln\; p\_\{0\}(\backslash Delta\; U)$

now define two functions:

$f\_\{0\}(\backslash Delta\; U)\; =\; \backslash ln\; p\_\{0\}(\backslash Delta\; U)\; -\; \backslash frac\{\backslash beta\backslash Delta\; U\}\{2\}$

f_{1}(\Delta U) = \ln p_{1}(\Delta U) + \frac{\beta\Delta U}{2}

thus that

$f\_\{1\}(\backslash Delta\; U)\; =\; f\_\{0\}(\backslash Delta\; U)\; +\; \backslash beta\backslash Delta\; F$

and$\backslash Delta\; F$ can be obtained by fitting $f\_\{1\}$ and $f\_\{0\}$

{{Reflist}}

Category:Potentials

Category:Chemical thermodynamics