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 the references)

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. A sketchy description for a simple fluid system is given below:

Input information

• Experimental Radial distribution function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_0(r) } at given conditions of temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } and density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho }

• Initial guess for the effective interaction (pair) potential;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \Phi (r) \equiv \frac{ \Phi(r) }{ k_B T} }

Procedure

• The simulation procedure is divided in several stages

• The effective interaction is modified through the simulation in each stage, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s } , to bias the current result of

the radial distrbution function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{inst}(r) } to the target $g_{0}(r)$ by using:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \Phi^{new}(r) = \beta \Phi^{old}(r) + \left[ g_{inst}(r) - g_0(r) \right] \lambda_s } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_s } is greater than zero and depends on the stage Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s }

• The simulation in each stage proceeds until some convergence criterium, 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 samller value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } :

<math> \lambda_{s+1} = \alpha \lambda_s; \textmath{with:} 0 < \alpha < 1 </mat>

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)