Inverse Monte Carlo - Revision history

Nice and Tidy at 10:51, 6 September 2007

2007-09-06T10:51:18Z

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	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).
	== 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		A detailed explanation of the procedure can be found in reference 1. Here an outline description for a simple fluid system is given:
	fluid system is given:		==== Input information ====
	=== 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>
	* 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>

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

	~~: <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 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:		the radial distribution function, <math> g_{inst}(r) </math> towards the target <math> g_0(r) </math> by using:

Nice and Tidy at 10:22, 6 September 2007

2007-09-06T10:22:30Z

Nice and Tidy at 10:03, 6 September 2007

2007-09-06T10:03:14Z

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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		Given the structural information (distribution functions) the inverse [[Monte Carlo \|Monte Carlo technique]] tries to compute the corresponding [[Intermolecular pair potential \|interaction potential]].
	Carlo technique tries to compute the corresponding 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 the references~~)

	==Uniqueness theorem==		==Uniqueness theorem==
	The uniqueness theorem is due to Henderson (Ref. 3).		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
	A detailed explanation of the procedure can be found in reference 1. ~~A sketchy~~ description for a simple		fluid system is given:
	fluid system is given ~~below~~:

	=== Input information ===		=== 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>

	*~~''Experimental'' 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]];

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

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

	=== Procedure ===		=== Procedure ===
			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:

	* The simulation procedure is divided in several stages		: <math> \beta \Phi_{12}^{new}(r) = \beta \Phi_{12}^{old}(r) + \left[ g_{inst}(r) - g_0(r) \right] \lambda_s </math>,

	* 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>		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 simulation in each stage proceeds until some convergence ~~criterium~~ (that takes into account		the precision of the values of <math> g_0(r) </math>) for the global result of the
	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))
	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 smaller value of <math> \lambda </math>:

	* When the simulation on one stage is finished a new stage starts with a smaller value of <math> \lambda </math>:

	: <math> \lambda_{s+1} = \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 stages, with <math> \lambda </math> being small enough, we can ~~get~~ an effective pair potential compatible with the input <math> g_0(r) </math>		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 ==

Noe: /* Procedure */

2007-09-06T08:24:01Z

Procedure

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	* The simulation in each stage proceeds until some convergence criterium (that takes into account		* 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		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)		radial distribution function over the stage, is achieved (See Ref. 1)

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	: <math> \lambda_{s+1} = \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 stages, with <math> \lambda </math> being small enough, we can get an effective pair potential compatible		* 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>
	with the input <math> g_0(r) </math>

	== References ==		== References ==

Noe: /* Procedure */

2007-09-05T17:44:14Z

Procedure

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	where <math> \lambda_s </math> is greater than zero and depends on the stage <math> 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 criterium, for the global result of the		* 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)		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 <math> \lambda </math>:		* When the simulation on one stage is finished a new stage starts with a smaller value of <math> \lambda </math>:

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

			* 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 ==

Noe: /* Procedure */

2007-09-05T17:36:42Z

Procedure

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	* The simulation in each stage proceeds until some convergence criterium, for the global result of the		* 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)		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 <math> \lambda </math>:

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

	== References ==		== References ==

Noe: /* Input information */

2007-09-05T17:33:18Z

Input information

@@ Line 15: / Line 15: @@
 === Input information ===
-*Radial distribution function <math> g(r) </math> at given conditions of temperature, <math> T </math>  and density <math> \rho </math>
+*''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 interaction (pair) potential;
+*Initial guess for the effective interaction (pair) potential;
 == References ==

Noe: /* An inverse Monte Carlo algorithm using a Wang-Landau-like algorithm */

2007-09-05T17:15:49Z

An inverse Monte Carlo algorithm using a Wang-Landau-like algorithm

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	== 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. A sketchy description is given below:		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 ===

			*Radial distribution function <math> g(r) </math> at given conditions of temperature, <math> T </math> and density <math> \rho </math>

			*Initial guess for the interaction (pair) potential;

	== References ==		== References ==

Noe: /* An inverse Monte Carlo algorithm using a Wang-Landau-like algorithm */

2007-09-05T17:11:03Z

An inverse Monte Carlo algorithm using a Wang-Landau-like algorithm

← Older revision		Revision as of 19:11, 5 September 2007
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	== 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. A sketchy description is given below:

	== References ==		== References ==

Noe: /* Uniqueness theorem */

2007-09-05T17:09:33Z

Uniqueness theorem

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	The uniqueness theorem is due to Henderson (Ref. 3).		The uniqueness theorem is due to Henderson (Ref. 3).

	== An inverse Monte Carlo algorithm using a [[Wang-Landau]]-like algorithm		== An inverse Monte Carlo algorithm using a [[Wang-Landau method\|Wang-Landau]]-like algorithm ==

	== References ==		== References ==

← Older revision		Revision as of 12:22, 6 September 2007
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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		A detailed explanation of the procedure can be found in reference 1. Here an outline description for a simple
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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>		* 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]];		* 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>		: <math> \beta \Phi_{12} (r) \equiv \frac{ \Phi_{12}(r) }{ k_B T} </math>

	=== Procedure ===		=== Procedure ===
	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 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> to the target <math> g_0(r) </math> by using:		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_{inst}(r) - g_0(r) \right] \lambda_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>,

	where <math> \lambda_s </math> is greater than zero and depends on the stage <math> 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 in each stage proceeds until some convergence criteria (that takes into account		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		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 ~~on one~~ stage is finished a new stage ~~starts~~ with a smaller value of <math> \lambda </math>:		When the simulation for a particular stage have finished a new stage is initiated, with a smaller value of <math> \lambda </math>:

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

	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>		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 ==		== References ==