Inverse Monte Carlo

Inverse Monte Carlo refers to the numerical techniques to solve the so-called inverse problem in fluids. Given the structural information (distribution functions) the inverse Monte Carlo technique tries to compute the corresponding interaction potential. More information can be found in the review by Gergely Tóth (see reference 4).

Uniqueness theorem

The uniqueness theorem is due to Henderson (Ref. 3).

An inverse Monte Carlo algorithm using a Wang-Landau-like algorithm

A detailed explanation of the procedure can be found in reference 1. Here an outline description for a simple fluid system is given:

Input information

• The experimental radial distribution function $g_{0}(r)$ at given conditions of temperature, $T$ and density $\rho$

• An initial guess for the effective interaction (pair) potential;

$\beta \Phi _{12}(r)\equiv {\frac {\Phi _{12}(r)}{k_{B}T}}$

Procedure

The simulation procedure is divided in several stages. First the effective interaction is modified through the simulation in each stage, $s$, to bias the current result of the radial distribution function, $g_{inst}(r)$ to the target $g_{0}(r)$ by using:

$\beta \Phi _{12}^{new}(r)=\beta \Phi _{12}^{old}(r)+\left[g_{inst}(r)-g_{0}(r)\right]\lambda _{s}$,

where $\lambda _{s}$ is greater than zero and depends on the stage $s$ The simulation in each stage proceeds until some convergence criteria (that takes into account the precision of the values of $g_{0}(r)$) for the global result of the radial distribution function over the stage, is achieved (See Ref. 1)) When the simulation on one stage is finished a new stage starts with a smaller value of $\lambda$:

$\lambda _{s+1}=\alpha \lambda _{s}$ with: $0<\alpha <1$

At the final stages, with $\lambda$ being small enough, one can obtain an effective pair potential compatible with the input $g_{0}(r)$

References

1. N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E 68 011202 (6 pages) (2003)
2. N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E 70 021203 (5 pages) (2004)
3. R. L. Henderson "A uniqueness theorem for fluid pair correlation functions", Physics Letters A 49 pp. 197-198 (1974)
4. Gergely Tóth, "Interactions from diffraction data: historical and comprehensive overview of simulation assisted methods", Journal of Physics: Condensed Matter 19 335220 (2007)