Editing 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 |Monte Carlo technique]] tries to compute the corresponding [[Intermolecular pair potential |interaction potential]].
More information can be found in the review by Gergely Tóth (see reference 4).
== An inverse Monte Carlo algorithm using a [[Wang-Landau method|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 |radial distribution function]] <math> g_0(r) </math> at given conditions of [[temperature]], <math> T </math>  and [[density]] <math> \rho </math>

* An initial guess for the effective interaction [[Intermolecular pair potential |(pair) potential]], i.e.

: <math> \beta \Phi_{12} (r) \equiv \frac{ \Phi_{12}(r) }{ k_B T} </math>

=== Procedure === 
The simulation procedure is divided into several stages. First, simulations are performed to modify the  effective interaction at each stage, <math> s </math>, in order  to bias the 
the radial distribution function, <math> g_{inst}(r) </math> towards the target <math> g_0(r) </math>  by using:

: <math> \beta \Phi_{12}^{new}(r) = \beta \Phi_{12}^{old}(r) +  \left[  g_{\mathrm{inst}}(r) - g_0(r) \right] \lambda_s </math>, 

where <math> \lambda_s </math> is greater than zero and depends on the stage <math> s </math> at which one is at.
The simulation for each stage proceeds until some convergence criteria (that takes into account
the precision  of the values of <math> g_0(r) </math>) for the global result of the
radial distribution function over the stage,  is achieved (See Ref. 1))
When the simulation for a particular stage have finished a new stage is initiated, with a smaller value of <math> \lambda </math>:

: <math> \left. \lambda_{s+1} \right.= \alpha \lambda_s </math> with: <math>   0 < \alpha < 1 </math>

At the final stage, with a sufficiently small <math> \lambda </math>, one can obtain an effective pair potential compatible with the input radial distribution function <math> g_0(r) </math>. One knows that this effective pair potential is valid due to the uniqueness theorem of Henderson (Ref. 3).

== References ==
#[http://dx.doi.org/10.1103/PhysRevE.68.011202 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)]
#[http://dx.doi.org/10.1103/PhysRevE.70.021203  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)]
#[http://dx.doi.org/10.1016/0375-9601(74)90847-0 R. L. Henderson  "A uniqueness theorem for fluid pair correlation functions", Physics Letters A  '''49''' pp. 197-198 (1974)]
#[http://dx.doi.org/10.1088/0953-8984/19/33/335220  Gergely Tóth, "Interactions from diffraction data: historical and comprehensive overview of simulation assisted methods", Journal of Physics: Condensed Matter '''19'''  335220 (2007)]
[[category: Monte Carlo]]