Editing Inverse Monte Carlo
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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 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). | 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 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: | 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]]; | |||
: <math> \beta \Phi_{12 | : <math> \beta \Phi_{12} (r) \equiv \frac{ \Phi_{12}(r) }{ k_B T} </math> | ||
where <math> \lambda_s </math> is greater than zero and depends on the stage <math> s </math> | === Procedure === | ||
The simulation | The simulation procedure is divided in several stages. First the effective interaction is modified through the simulation in each stage, <math> s </math>, to bias the current result of | ||
the radial distribution function, <math> g_{inst}(r) </math> to the target <math> g_0(r) </math> by using: | |||
: <math> \beta \Phi_{12}^{new}(r) = \beta \Phi_{12}^{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> | |||
The simulation in 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 | 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)) | radial distribution function over the stage, is achieved (See Ref. 1)) | ||
When the simulation | When the simulation on one stage is finished a new stage starts with a smaller value of <math> \lambda </math>: | ||
: <math> | : <math> \lambda_{s+1} = \alpha \lambda_s </math> with: <math> 0 < \alpha < 1 </math> | ||
At the final | At the final stages, with <math> \lambda </math> being small enough, one can obtain an effective pair potential compatible with the input <math> g_0(r) </math> | ||
== References == | == References == |