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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 |
| Given the structural information (distribution functions) the inverse [[Monte Carlo |Monte Carlo technique]] tries to compute the corresponding [[Intermolecular pair potential |interaction potential]].
| | so-called inverse problem in fluids. |
| More information can be found in the review by Gergely Tóth (see reference 4).
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| == An inverse Monte Carlo algorithm using a [[Wang-Landau method|Wang-Landau]]-like algorithm ==
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| A detailed explanation of the procedure can be found in reference 1. Here an outline description for a simple fluid system is given:
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| ==== Input information ====
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| #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>
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| #An initial guess for the effective interaction [[Intermolecular pair potential |(pair) potential]], i.e.
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| ::<math> \beta \Phi_{12} (r) \equiv \frac{ \Phi_{12}(r) }{ k_B T} </math>
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| ==== Procedure ====
| | * Given the structural information (distribution functions) the inverse Monte |
| 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
| | Carlo technique tries to compute the corresponding interaction potential |
| the radial distribution function, <math> g_{inst}(r) </math> towards the target <math> g_0(r) </math> by using:
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| : <math> \beta \Phi_{12}^{new}(r) = \beta \Phi_{12}^{old}(r) + \left[ g_{\mathrm{inst}}(r) - g_0(r) \right] \lambda_s </math>,
| | * Some example(s) of these techniques can be found in the following reference(s) |
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| where <math> \lambda_s </math> is greater than zero and depends on the stage <math> s </math> at which one is at.
| | == References == |
| The simulation for each stage proceeds until some convergence criteria (that takes into account
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| the precision of the values of <math> g_0(r) </math>) for the global result of the
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| radial distribution function over the stage, is achieved (See Ref. 1))
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| When the simulation for a particular stage have finished a new stage is initiated, with a smaller value of <math> \lambda </math>:
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| : <math> \left. \lambda_{s+1} \right.= \alpha \lambda_s </math> with: <math> 0 < \alpha < 1 </math> | | #[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", Phys. Rev. E 68, 011202 (2003) (6 pages)] |
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| 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).
| | #[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", Phys. Rev. E 70, 021203 (2004) (5 pages) ] |
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| == References ==
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| #[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)]
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| #[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)]
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| #[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)]
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| [[category: Monte Carlo]]
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