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'''Inverse Monte Carlo''' refers to the numerical techniques to solve the so-called inverse problem in fluids.
'''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]].
Given the structural information (distribution functions) the inverse Monte
More information can be found in the review by Gergely Tóth (see reference 4).
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 method|Wang-Landau]]-like algorithm ==
== 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 ====
A detailed explanation of the procedure can be found in reference 1. A sketchy description for a simple
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  
fluid system is given below:
the radial distribution function, <math> g_{inst}(r) </math> towards the target <math> g_0(r) </math>  by using:
 
=== Input information ===
 
*''Experimental'' Radial distribution function <math> g_0(r) </math> at given conditions of temperature, <math> T </math>  and density <math> \rho </math>
 
*Initial guess for the effective interaction (pair) potential;
 
: <math> \beta \Phi (r) \equiv \frac{ \Phi(r) }{ k_B T} </math>
 
=== Procedure ===  
 
* The simulation procedure is divided in several stages
 
* The effective interaction is modified through the simulation in each stage, <math> s </math>,  to bias the current result of
the radial distrbution function, <math> g_{inst}(r) </math> to the target <math> g_0(r) </math>  by using:
 
: <math> \beta \Phi^{new}(r) = \beta \Phi^{old}(r) +  \left[  g_{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>


: <math> \beta \Phi_{12}^{new}(r) = \beta \Phi_{12}^{old}(r) +  \left[  g_{\mathrm{inst}}(r) - g_0(r) \right] \lambda_s </math>,  
* The simulation in each stage proceeds until some convergence criterium (that takes into account
the precission 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)


where <math> \lambda_s </math> is greater than zero and depends on the stage <math> s </math> at which one is at.
* When the simulation on one stage is finished a new stage starts with a smaller value of <math> \lambda </math>:
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>
: <math> \lambda_{s+1} = \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).
* At the final stages, with <math> \lambda </math> being small enough, we can get an effective pair potential compatible with the input <math> g_0(r) </math>


== References ==
== References ==
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