Blue moon ensemble - Revision history

Csheldon at 13:01, 23 October 2024

2024-10-23T13:01:01Z

← Older revision		Revision as of 13:01, 23 October 2024
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	In general, [[:Category:Constrained molecular dynamics\|constrained molecular dynamics]] generates biased statistical averages. The blue moon ensemble average, also known as constrained-reaction-coordinate-dynamic (CRCD) ensemble, connects constrained and unconstrained molecular dynamics, cf. [[Blue_moon_ensemble_calculations\|blue moon ensemble ~~calcuations~~]]. It shows that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:		In general, [[:Category:Constrained molecular dynamics\|constrained molecular dynamics]] generates biased statistical averages. The blue moon ensemble average, also known as constrained-reaction-coordinate-dynamic (CRCD) ensemble, connects constrained and unconstrained molecular dynamics, cf. [[Blue_moon_ensemble_calculations\|blue moon ensemble calculations]]. It shows that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:
	:<math>		:<math>
	a(\xi)=\frac{\langle \|\mathbf{Z}\|^{-1/2} a(\xi^) \rangle_{\xi^}}{\langle \|\mathbf{Z}\|^{-1/2}\rangle_{\xi^*}},		a(\xi)=\frac{\langle \|\mathbf{Z}\|^{-1/2} a(\xi^) \rangle_{\xi^}}{\langle \|\mathbf{Z}\|^{-1/2}\rangle_{\xi^*}},

Csheldon at 09:59, 18 October 2024

2024-10-18T09:59:43Z

← Older revision		Revision as of 09:59, 18 October 2024
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	In general, [[:Category:Constrained molecular dynamics\|constrained molecular dynamics]] generates biased statistical averages. The blue-moon ensemble average, also known as constrained-reaction-coordinate-dynamic (CRCD) ensemble, connects ~~constained~~ and unconstrained molecular dynamics. It shows that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:		In general, [[:Category:Constrained molecular dynamics\|constrained molecular dynamics]] generates biased statistical averages. The blue moon ensemble average, also known as constrained-reaction-coordinate-dynamic (CRCD) ensemble, connects constrained and unconstrained molecular dynamics, cf. [[Blue_moon_ensemble_calculations\|blue moon ensemble calcuations]]. It shows that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:
	:<math>		:<math>
	a(\xi)=\frac{\langle \|\mathbf{Z}\|^{-1/2} a(\xi^) \rangle_{\xi^}}{\langle \|\mathbf{Z}\|^{-1/2}\rangle_{\xi^*}},		a(\xi)=\frac{\langle \|\mathbf{Z}\|^{-1/2} a(\xi^) \rangle_{\xi^}}{\langle \|\mathbf{Z}\|^{-1/2}\rangle_{\xi^*}},

Karsai at 13:11, 16 October 2024

2024-10-16T13:11:37Z

← Older revision		Revision as of 13:11, 16 October 2024
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	</references>		</references>

	[[Category:~~[[Category:Molecular~~ dynamics]][[Category:Theory]]]]		[[Category:Advanced molecular-dynamics sampling]][[Category:Theory]]

Karsai: Removed redirect to Blue-moon ensemble

2024-10-16T12:05:19Z

Removed redirect to Blue-moon ensemble

← Older revision		Revision as of 12:05, 16 October 2024
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	~~#REDIRECT~~ [[~~Blue~~-moon ensemble]]		In general, [[:Category:Constrained molecular dynamics\|constrained molecular dynamics]] generates biased statistical averages. The blue-moon ensemble average, also known as constrained-reaction-coordinate-dynamic (CRCD) ensemble, connects constained and unconstrained molecular dynamics. It shows that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:
			:<math>
			a(\xi)=\frac{\langle \|\mathbf{Z}\|^{-1/2} a(\xi^) \rangle_{\xi^}}{\langle \|\mathbf{Z}\|^{-1/2}\rangle_{\xi^*}},
			</math>
			where <math>{\xi}</math> is the reaction coordinate, <math>\xi^</math> restrains the reference coordinate, e.g. to a transition state, where the associate velocity is <math>\dot{\xi^} = 0</math>, the <math>\langle ... \rangle_{\xi^*}</math> stands for the statistical average of the quantity enclosed in angle brackets computed for a constrained ensemble, and <math>Z</math> is a mass metric tensor defined as:
			:<math>
			Z_{\alpha,\beta}={\sum}_{i=1}^{3N} m_i^{-1} \nabla_i \xi_\alpha \cdot \nabla_i \xi_\beta, \, \alpha=1,...,r, \, \beta=1,...,r,
			</math>
			It can be shown that the free energy gradient can be computed using the equation:<ref name="Carter89" /><ref name="Otter00" /><ref name="Darve02" /><ref name="Fleurat05" />
			:<math>
			\Bigl(\frac{\partial A}{\partial \xi_k}\Bigr)_{\xi^}=\frac{1}{\langle\|Z\|^{-1/2}\rangle_{\xi^}}\langle \|Z\|^{-1/2} [\lambda_k +\frac{k_B T}{2 \|Z\|} \sum_{j=1}^{r}(Z^{-1})_{kj} \sum_{i=1}^{3N} m_i^{-1}\nabla_i \xi_j \cdot \nabla_i \|Z\|]\rangle_{\xi^*},
			</math>
			where <math>A</math> is the free energy, <math>k_B</math> is the Boltzmann constant, <math>T</math> is the temperature, and <math>\lambda_{\xi_k}</math> is the Lagrange multiplier associated with the parameter <math>{\xi_k}</math> used in the [[Constrained molecular dynamics\|SHAKE algorithm]].<ref name="Ryckaert77" />

			The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path, e.g. using the Simpson method{{cite\|simpson:web}}:
			:<math>
			{\Delta}A_{1 \rightarrow 2} = \int_{{\xi(1)}}^{{\xi(2)}}\Bigl( \frac{\partial {A}} {\partial \xi} \Bigr)_{\xi^*} \cdot d{\xi}.
			</math>
			Note that as the free energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant. As an example, when calculating the transition state, if (1) were set to the reactant and (2) to the transition state, then <math>{\Delta}A_{1 \rightarrow 2}</math> would be the activation free energy for the reaction.

			==References==
			<references>
			<ref name="Ryckaert77">[http://dx.doi.org/10.1016/0021-9991(77)90098-5 J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).]</ref>
			<ref name="Carter89">[http://dx.doi.org/10.1016/S0009-2614(89)87314-2 E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Chem. Phys. Lett. 156, 472 (1989).]</ref>
			<ref name="Otter00">[http://dx.doi.org/10.1080/00268970009483348 W. K. Den Otter and W. J. Briels, Mol. Phys. 98, 773 (2000).]</ref>
			<ref name="Darve02">[http://dx.doi.org/10.1080/08927020211975 E. Darve, M. A. Wilson, and A. Pohorille, Mol. Simul. 28, 113 (2002).]</ref>
			<ref name="Fleurat05">[http://dx.doi.org/10.1063/1.1948367 P. Fleurat-Lessard and T. Ziegler, J. Chem. Phys. 123, 084101 (2005).]</ref>
			<ref name="Ryckaert77">[http://dx.doi.org/10.1016/0021-9991(77)90098-5 J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).]</ref>
			</references>

			[[Category:[[Category:Molecular dynamics]][[Category:Theory]]]]

Karsai: Karsai moved page Blue moon ensemble to Blue-moon ensemble

2022-07-19T08:18:09Z

Karsai moved page Blue moon ensemble to Blue-moon ensemble

New page

#REDIRECT [[Blue-moon ensemble]]