Category:Ensembles - Revision history

Huebsch at 06:28, 8 May 2024

2024-05-08T06:28:01Z

← Older revision		Revision as of 06:28, 8 May 2024
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	The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:		The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:
	{{Template:MDCOMBINATIONS}}		{{Template:MDCOMBINATIONS}}

			[[Category:Molecular dynamics]]

Singraber: /* Theory */

2023-11-21T10:05:21Z

Theory

← Older revision		Revision as of 10:05, 21 November 2023
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	</math>.		</math>.

	The canonical ensemble can be visualized as N particle system in a fixed volume box which is allowed to exchange thermal energy with a huge heat bath surrounding it. Therefore, the temperature is ~~constant and will always equal~~ the ~~heats baths~~ temperature. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NVT ensemble can be sampled from a [[NVT_ensemble\|molecular-dynamics]] run.		The canonical ensemble can be visualized as N particle system in a fixed volume box which is allowed to exchange thermal energy with a huge heat bath surrounding it. Therefore, in equilibrium the instantaneous temperature (temperature derived from kinetic energy) is fluctuating around the heat bath temperature. Temperature fluctuations are proportional to <math>N^{-1/2}</math>. Hence, in the thermodynamic limit (<math>N \rightarrow \infty</math>) temperature fluctuations vanish. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NVT ensemble can be sampled from a [[NVT_ensemble\|molecular-dynamics]] run.

	''' Isothermal–isobaric ensemble (NpT) '''		''' Isothermal–isobaric ensemble (NpT) '''
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	</math>.		</math>.

	The ~~canonical~~ ensemble can be visualized as N particle system in a box which is allowed		The isothermal–isobaric ensemble can be visualized as N particle system in a box without rigid boundaries which is allowed to exchange thermal energy with a huge heat bath surrounding it. Because the box has no rigid sides the volume and shape will change according to the pressure difference within and outside the box. In the isothermal–isobaric ensemble both the instantaneous temperature and pressure will fluctuate around the desired temperature (heat bath) and pressure (volume reservoir), respectively. As in the canonical ensemble these fluctuations vanish in the thermodynamic limit. There are several techniques to realize the isothermal–isobaric ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a [[NpT_ensemble\|molecular-dynamics]] run.
	to exchange thermal energy with a huge heat bath surrounding it. ~~Contrary to the NVT~~
	~~ensemble~~ the box has no rigid sides ~~and therefore~~ the volume ~~of the box~~ will change
	according to the pressure difference within and outside the box. ~~Therefore,~~ the temperature
	and pressure ~~are constant~~ and ~~will always equal~~ the ~~heats baths values~~. There are several techniques to realize the ~~canonical~~ ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a [[NpT_ensemble\|molecular-dynamics]] run.

	''' Isoenthalpic–isobaric ensemble (NpH)'''		''' Isoenthalpic–isobaric ensemble (NpH)'''
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	<math>		<math>
	X(N,p,H) = \sum_{H-\delta H<H(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})<H+\delta H}		X(N,p,H) = \sum_{H-\delta H<H(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})<H+\delta H} 1
	</math>,		</math>.

	The probability for a single microstate r is		The probability for a single microstate r is
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	</math>.		</math>.

	The isoenthalpic–isobaric can be visualized as N particle system in a box which is thermally isolated from ~~the~~ surroundings. ~~The~~ box has no rigid sides ~~and~~ the volume ~~of the box~~ will change		The isoenthalpic–isobaric ensemble can be visualized as N particle system in a box without rigid boundaries which is thermally isolated from its surroundings. Because the box has no rigid sides the volume and shape will change according to the pressure difference within and outside the box. In the isoenthalpic–isobaric ensemble the instantaneous pressure will fluctuate around the desired pressure value. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.
	according to the pressure difference within and outside the box. ~~Therefore,~~ the pressure ~~is constant and~~ will ~~always equal to~~ the ~~heats baths values~~. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.

	=== How To ===		=== How To ===
	The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:		The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:
	{{Template:MDCOMBINATIONS}}		{{Template:MDCOMBINATIONS}}

Singraber: /* Introduction */

2023-11-21T09:17:44Z

Introduction

← Older revision		Revision as of 09:17, 21 November 2023
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	These three variables determine the type of ensemble that is studied. Depending on		These three variables determine the type of ensemble that is studied. Depending on
	these three variables there is a thermodynamic potential associated with the		these three variables there is a thermodynamic potential associated with the
	ensemble, which would be the ~~Helmholtz~~ free energy in the case of N,T and p.		ensemble, which would be the Gibbs free energy in the case of N,T and p.
	Therefore, the concept of the ensemble gives access to any thermodynamic quantity.		Therefore, the concept of the ensemble gives access to any thermodynamic quantity.
	The configurations of your system building up the ensemble can be obtained from		The configurations of your system building up the ensemble can be obtained from
	molecular dynamics simulations. The molecular-dynamics approach generates the configurations for		molecular dynamics simulations. The molecular-dynamics approach generates the configurations for
	the ensemble by integrating Newton's equations of motion.		the ensemble by integrating Newton's equations of motion.

	=== Theory ===		=== Theory ===

Singraber: /* Theory */

2023-11-20T16:33:20Z

Theory

← Older revision		Revision as of 16:33, 20 November 2023
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	and pressure are constant and will always equal the heats baths values. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a [[NpT_ensemble\|molecular-dynamics]] run.		and pressure are constant and will always equal the heats baths values. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a [[NpT_ensemble\|molecular-dynamics]] run.

	''' Isoenthalpic–isobaric ~~ensmble~~ (NpH)'''		''' Isoenthalpic–isobaric ensemble (NpH)'''

	In the isoenthalpic–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure and the enthalpy H. The ensemble consists of all configurations accessible to the system at the given (N,p,H) conditions. The isoenthalpic–isobaric partition function can be derived to be		In the isoenthalpic–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure and the enthalpy H. The ensemble consists of all configurations accessible to the system at the given (N,p,H) conditions. The isoenthalpic–isobaric partition function can be derived to be

Huebsch: Huebsch moved page Construction:Category:Ensembles to Category:Ensembles without leaving a redirect

2023-05-03T13:04:24Z

Huebsch moved page Construction:Category:Ensembles to Category:Ensembles without leaving a redirect

← Older revision		Revision as of 13:04, 3 May 2023
(No difference)

Jona at 10:18, 3 May 2023

2023-05-03T10:18:40Z

← Older revision		Revision as of 10:18, 3 May 2023
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	=== How To ===		=== How To ===
	The following table gives an overview of the possible ~~combination~~ of ensembles and thermostats in VASP:		The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:
	{{Template:MDCOMBINATIONS}}		{{Template:MDCOMBINATIONS}}

Jona at 20:36, 12 August 2022

2022-08-12T20:36:55Z

← Older revision		Revision as of 20:36, 12 August 2022
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	\end{cases}		\end{cases}
	</math>.		</math>.

	~~The average internal energy of the system under isothermal–isobaric conditions can be computed as~~

	~~<math>~~
	~~\langle E \rangle = -k_{B}T\frac{\partial G(N,p,T)}{\partial \beta}=~~
	~~\sum_{r}~~
	~~\frac{e^{-\beta \left(E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)}E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}{X}~~

	~~</math>.~~

	The isoenthalpic–isobaric can be visualized as N particle system in a box which is thermally isolated from the surroundings. The box has no rigid sides and the volume of the box will change		The isoenthalpic–isobaric can be visualized as N particle system in a box which is thermally isolated from the surroundings. The box has no rigid sides and the volume of the box will change
	according to the pressure difference within and outside the box. Therefore, the pressure is constant and will always equal to the heats baths values. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.		according to the pressure difference within and outside the box. Therefore, the pressure is constant and will always equal to the heats baths values. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.





	=== How To ===		=== How To ===
	The following table gives an overview of the possible combination of ensembles and thermostats in VASP:		The following table gives an overview of the possible combination of ensembles and thermostats in VASP:
	{{Template:MDCOMBINATIONS}}		{{Template:MDCOMBINATIONS}}

Jona at 20:35, 12 August 2022

2022-08-12T20:35:36Z

← Older revision		Revision as of 20:35, 12 August 2022
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	P_{r}(N,V,E) =		P_{r}(N,V,E) =
	\begin{cases} \frac{1}{\Omega(N,V,E)}\ for\ E-\delta E < E~~(N,V)~~ < E + \delta E \\		\begin{cases} \frac{1}{\Omega(N,V,E)}\ for\ E-\delta E < E < E + \delta E \\
	0\ else		0\ else
	\end{cases}		\end{cases}
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	<math>		<math>
	Z(N,V,T) = \sum_{r}e^{-\beta E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}		Z(N,V,T) = \sum_{r}e^{-\beta E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})} 1
	</math>,		</math>,

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	<math>		<math>
	X(N,p,H) = \sum_{~~r}e^{~~-\~~beta\left( E~~(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+~~pV_{r}~~\~~right)~~}		X(N,p,H) = \sum_{H-\delta H<H(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})<H+\delta H}
	</math>,		</math>,

	~~where <math> \beta </math> is the inverse temperature <math>\frac{1}{k_{B}T}</math> and pV~~
	~~is the contribution to the energy due to volume work.~~
	The probability for a single microstate r is		The probability for a single microstate r is

	<math>		<math>
	P_{r}(N,p,H) = \frac{1}{X(Np,H)}~~e^{~~-\~~beta \left(~~E~~(N,p,~~\~~{\mathbf{r}_{i}~~\},\{\~~mathbf{p}_{i}~~\~~})+pV_~~{~~r}\right)~~}		P_{r}(N,p,H) =
	</math>.		\begin{cases} \frac{1}{X(N,p,H)}\ for\ H-\delta H < H < E + \delta H \\
			0\ else
			\end{cases}
			</math>.

	The average internal energy of the system under isothermal–isobaric conditions can be computed as		The average internal energy of the system under isothermal–isobaric conditions can be computed as

Jona at 19:29, 12 August 2022

2022-08-12T19:29:48Z

← Older revision		Revision as of 19:29, 12 August 2022
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	The isoenthalpic–isobaric can be visualized as N particle system in a box which is thermally isolated from the surroundings. The box has no rigid sides and the volume of the box will change		The isoenthalpic–isobaric can be visualized as N particle system in a box which is thermally isolated from the surroundings. The box has no rigid sides and the volume of the box will change
	according to the pressure difference within and outside the box. Therefore, the pressure is constant and will always equal the heats baths values. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.		according to the pressure difference within and outside the box. Therefore, the pressure is constant and will always equal to the heats baths values. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.

Jona at 19:18, 12 August 2022

2022-08-12T19:18:00Z

← Older revision		Revision as of 19:18, 12 August 2022
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	</math>.		</math>.

	The canonical ensemble can be visualized as N particle system in a fixed volume box which is allowed to exchange thermal energy with a huge heat bath surrounding it. Therefore, the temperature is constant and will always equal the heats baths temperature. There are several techniques to realize the canonical ensemble in a computer simulation. ~~A guide to sample~~ configurations		The canonical ensemble can be visualized as N particle system in a fixed volume box which is allowed to exchange thermal energy with a huge heat bath surrounding it. Therefore, the temperature is constant and will always equal the heats baths temperature. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NVT ensemble can be sampled from a [[NVT_ensemble\|molecular-dynamics]] run.
	in a NVT ensemble from a [[NVT_ensemble\|molecular-dynamics]]

	''' Isothermal–isobaric ensemble (NpT) '''		''' Isothermal–isobaric ensemble (NpT) '''
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	ensemble the box has no rigid sides and therefore the volume of the box will change		ensemble the box has no rigid sides and therefore the volume of the box will change
	according to the pressure difference within and outside the box. Therefore, the temperature		according to the pressure difference within and outside the box. Therefore, the temperature
	and pressure are constant and will always equal the heats baths values. There are several techniques to realize the canonical ensemble in a computer simulation. ~~A guide to sample~~ configurations in a NpT ensemble from a [[NpT_ensemble\|molecular-dynamics]]		and pressure are constant and will always equal the heats baths values. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a [[NpT_ensemble\|molecular-dynamics]] run.

	''' NpH ~~ensmble~~ '''		''' Isoenthalpic–isobaric ensmble (NpH)'''

			In the isoenthalpic–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure and the enthalpy H. The ensemble consists of all configurations accessible to the system at the given (N,p,H) conditions. The isoenthalpic–isobaric partition function can be derived to be

			<math>
			X(N,p,H) = \sum_{r}e^{-\beta\left( E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)}
			</math>,

			where <math> \beta </math> is the inverse temperature <math>\frac{1}{k_{B}T}</math> and pV
			is the contribution to the energy due to volume work.
			The probability for a single microstate r is

			<math>
			P_{r}(N,p,H) = \frac{1}{X(Np,H)}e^{-\beta \left(E(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})+pV_{r}\right)}
			</math>.

			The average internal energy of the system under isothermal–isobaric conditions can be computed as

			<math>
			\langle E \rangle = -k_{B}T\frac{\partial G(N,p,T)}{\partial \beta}=
			\sum_{r}
			\frac{e^{-\beta \left(E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)}E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}{X}

			</math>.

			The isoenthalpic–isobaric can be visualized as N particle system in a box which is thermally isolated from the surroundings. The box has no rigid sides and the volume of the box will change
			according to the pressure difference within and outside the box. Therefore, the pressure is constant and will always equal the heats baths values. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble\|molecular-dynamics]] run.