Time-propagation algorithms in molecular dynamics - Revision history

Singraber at 21:33, 17 October 2025

2025-10-17T21:33:18Z

← Older revision		Revision as of 21:33, 17 October 2025
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	! MDALGO !! thermostat !! integration algorithm		! MDALGO !! thermostat !! integration algorithm
	\|-		\|-
	\|style="text-align:center;"\| 0 \|\| [[~~Nose~~-Hoover thermostat\|~~Nose~~-Hoover]] \|\| style="text-align:center;"\| [[#Velocity-Verlet integration scheme\|Velocity-Verlet]]		\|style="text-align:center;"\| 0 \|\| [[Nosé-Hoover thermostat\|Nosé-Hoover]] \|\| style="text-align:center;"\| [[#Velocity-Verlet integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 1 \|\| [[Andersen thermostat\|Andersen]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 1 \|\| [[Andersen thermostat\|Andersen]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 2 \|\| [[~~Nose~~-Hoover thermostat\|~~Nose~~-Hoover]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 2 \|\| [[Nosé-Hoover thermostat\|Nosé-Hoover]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 3 \|\| [[Langevin thermostat\|Langevin]] \|\| style="text-align:center;"\| [[#Velocity-Verlet Integration scheme\|Velocity-Verlet]]		\|style="text-align:center;"\| 3 \|\| [[Langevin thermostat\|Langevin]] \|\| style="text-align:center;"\| [[#Velocity-Verlet Integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 4 \|\| [[~~Nose~~-Hoover-chain thermostat\|~~NHC~~]] \|\| style="text-align:center;"\| [[#Velocity-Verlet integration scheme\|Velocity-Verlet]]		\|style="text-align:center;"\| 4 \|\| [[Nosé-Hoover chain thermostat\|Nosé-Hoover chain]] \|\| style="text-align:center;"\| [[#Velocity-Verlet integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 5 \|\| [[CSVR thermostat\|CSVR]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 5 \|\| [[CSVR thermostat\|CSVR]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]

Huebsch: /* Related tags and articles */

2024-10-18T10:28:08Z

Huebsch at 10:27, 18 October 2024

2024-10-18T10:27:49Z

Huebsch: Huebsch moved page Construction:Molecular dynamics propagation algorithms to Time-propagation algorithms in molecular dynamics without leaving a redirect

2024-10-18T10:21:06Z

Huebsch moved page Construction:Molecular dynamics propagation algorithms to Time-propagation algorithms in molecular dynamics without leaving a redirect

← Older revision		Revision as of 10:21, 18 October 2024
(No difference)

Singraber: /* Leap-Frog integration scheme */

2024-10-17T14:17:28Z

Leap-Frog integration scheme

← Older revision		Revision as of 14:17, 17 October 2024
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	<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>		<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
	</ol>		</ol>
	In this ~~from~~ the velocity and the position vectors are asynchronous in time.		In this form the velocity and the position vectors are asynchronous in time.

	=== Thermostats and used integrators ===		=== Thermostats and used integrators ===
	{\|class="wikitable" style="margin:aut		{\|class="wikitable" style="margin:aut

Singraber: /* Velocity-Verlet integration scheme */

2024-10-17T14:16:33Z

Velocity-Verlet integration scheme

← Older revision		Revision as of 14:16, 17 October 2024
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	<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>		<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
	<li>compute forces <math> \mathbf{F}_{i}(t)</math> from density functional theory or machine learning</li>		<li>compute forces <math> \mathbf{F}_{i}(t)</math> from density functional theory or machine learning</li>
	<li><math>\mathbf{v}_{i}(t + \Delta t)=\mathbf{v}_{i}(t+\frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t+~~\frac{1}{2}~~\Delta t)}{2m_{i}}\Delta t</math></li>		<li><math>\mathbf{v}_{i}(t + \Delta t)=\mathbf{v}_{i}(t+\frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t+\Delta t)}{2m_{i}}\Delta t</math></li>
	</ol>		</ol>
	From these equations it can be seen that the velocity and the position vectors are synchronous in time.		From these equations it can be seen that the velocity and the position vectors are synchronous in time.

	=== Leap-Frog integration scheme ===		=== Leap-Frog integration scheme ===
	Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:		Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:

Jona at 19:26, 16 October 2024

2024-10-16T19:26:21Z

← Older revision		Revision as of 19:26, 16 October 2024
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	\mathbf{r}_{i}(t)+ \mathbf{v}_{i}(t)\Delta t+\frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2}		\mathbf{r}_{i}(t)+ \mathbf{v}_{i}(t)\Delta t+\frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2}
	</math>		</math>
	=== Velocity-Verlet ~~Integration~~ scheme ===		=== Velocity-Verlet integration scheme ===
	The Velocity-Verlet algorithm can be decomposed into the following steps:		The Velocity-Verlet algorithm can be decomposed into the following steps:
	<ol>		<ol>
Line 26:		Line 26:
	</ol>		</ol>
	From these equations it can be seen that the velocity and the position vectors are synchronous in time.		From these equations it can be seen that the velocity and the position vectors are synchronous in time.
	=== Leap-Frog ~~Integration~~ scheme ===		=== Leap-Frog integration scheme ===
	Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:		Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:
	<ol>		<ol>
Line 34:		Line 34:
	</ol>		</ol>
	In this from the velocity and the position vectors are asynchronous in time.		In this from the velocity and the position vectors are asynchronous in time.
			=== Thermostats and used integrators ===
	{\|class="wikitable" style="margin:aut		{\|class="wikitable" style="margin:aut
	! MDALGO !! thermostat !! integration algorithm		! MDALGO !! thermostat !! integration algorithm
	\|-		\|-
	\|style="text-align:center;"\| 0 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| Velocity-Verlet		\|style="text-align:center;"\| 0 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Velocity-Verlet integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 1 \|\| [[Andersen thermostat\|Andersen]] \|\| style="text-align:center;"\| Leap-Frog		\|style="text-align:center;"\| 1 \|\| [[Andersen thermostat\|Andersen]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 2 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| Leap-Frog		\|style="text-align:center;"\| 2 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 3 \|\| [[Langevin thermostat\|Langevin]] \|\| style="text-align:center;"\| Velocity-Verlet		\|style="text-align:center;"\| 3 \|\| [[Langevin thermostat\|Langevin]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Velocity-Verlet Integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 4 \|\| [[Nose-Hoover-chain thermostat\|NHC]] \|\| style="text-align:center;"\| ~~Leap~~-~~Frog~~		\|style="text-align:center;"\| 4 \|\| [[Nose-Hoover-chain thermostat\|NHC]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Velocity-Verlet integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 5 \|\| [[CSVR thermostat\|CSVR]] \|\| style="text-align:center;"\| Leap-Frog		\|style="text-align:center;"\| 5 \|\| [[CSVR thermostat\|CSVR]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Leap-Frog integration scheme\|Leap-Frog]]
			\|-
			\|style="text-align:center;"\| 5 \|\| [[MDALGO#MDALGO=13: Multiple Andersen thermostats\|Multiple Andersen]] \|\| style="text-align:center;"\| [[Construction:Molecular dynamics propagation algorithms#Leap-Frog integration scheme\|Leap-Frog]]
	\|}		\|}

Jona: /* Leap-Frog Integration scheme */

2024-10-16T19:16:13Z

Leap-Frog Integration scheme

← Older revision		Revision as of 19:16, 16 October 2024
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	<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>		<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
	</ol>		</ol>
			In this from the velocity and the position vectors are asynchronous in time.
	{\|class="wikitable" style="margin:aut		{\|class="wikitable" style="margin:aut
	! MDALGO !! thermostat !! integration algorithm		! MDALGO !! thermostat !! integration algorithm

Jona at 19:14, 16 October 2024

2024-10-16T19:14:39Z

← Older revision		Revision as of 19:14, 16 October 2024
Line 1:		Line 1:
	In molecular dynamics simulations, the positions <math>\mathbf{r}_{i}(t)</math> and velocities <math>\mathbf{v}_{i}(t)</math> are monitored as functions of time <math>t</math>. This time dependence is obtained by integrating Newton's equations of motion. To solve the equations of motion, various integration algorithms have been developed. The time dependence of a particle can be expressed in a Taylor expansion		In molecular dynamics simulations, the positions <math>\mathbf{r}_{i}(t)</math> and velocities <math>\mathbf{v}_{i}(t)</math> are monitored as functions of time <math>t</math>. This time dependence is obtained by integrating Newton's equations of motion. When integrating the equations of motions it is important to use symplectic algorithms which conserve phase space volume. To solve the equations of motion under symplectic conditions, various integration algorithms have been developed. The time dependence of a particle can be expressed in a Taylor expansion
	::<math>		::<math>
	\mathbf{r}_{i}(t+\Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t)\Delta t + \frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2} + \frac{\partial^{3} \mathbf{r}_{i}(t)}{\partial t^{3}}\Delta t^{3} + \mathcal{O}(\Delta t^{4})		\mathbf{r}_{i}(t+\Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t)\Delta t + \frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2} + \frac{\partial^{3} \mathbf{r}_{i}(t)}{\partial t^{3}}\Delta t^{3} + \mathcal{O}(\Delta t^{4})

Jona: /* Leap-Frog Integration scheme */

2024-10-16T19:09:54Z

Leap-Frog Integration scheme

← Older revision		Revision as of 19:09, 16 October 2024
Line 30:		Line 30:
	<ol>		<ol>
	<li> compute forces <math> \mathbf{F}_{i}(t)</math> from density functional theory or machine learning </li>		<li> compute forces <math> \mathbf{F}_{i}(t)</math> from density functional theory or machine learning </li>
	<li><math>\mathbf{v}_{i}(t + \frac{1}{2}\Delta t)=\mathbf{v}_{i}(t-+ \frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t)}{m_{i}}\Delta t</math></li>		<li><math>\mathbf{v}_{i}(t + \frac{1}{2}\Delta t)=\mathbf{v}_{i}(t- \frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t)}{m_{i}}\Delta t</math></li>
	<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>		<li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
	</ol>		</ol>

← Older revision		Revision as of 10:28, 18 October 2024
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	{{TAG\|IBRION}}, {{TAG\|MDALGO}}, [[Thermostats]]		{{TAG\|IBRION}}, {{TAG\|MDALGO}}, [[Thermostats]]

	[[Category:Theory]][[Category:~~Molacular~~ dynamics]][[Category:Thermostats]]		[[Category:Theory]][[Category:Molecular dynamics]][[Category:Thermostats]]

← Older revision		Revision as of 10:27, 18 October 2024
Line 1:		Line 1:
	In molecular dynamics simulations, the positions <math>\mathbf{r}_{i}(t)</math> and velocities <math>\mathbf{v}_{i}(t)</math> are monitored as functions of time <math>t</math>. This time dependence is obtained by integrating Newton's equations of motion. When integrating the equations of motions it is important to use symplectic algorithms which conserve phase space volume. To solve the equations of motion under symplectic conditions, various integration algorithms have been developed. The time dependence of a particle can be expressed in a Taylor expansion		In [[MD\|molecular dynamics simulations]], the ionic positions <math>\mathbf{r}_{i}(t)</math> and velocities <math>\mathbf{v}_{i}(t)</math> are monitored as functions of time <math>t</math>. This time dependence is obtained by integrating Newton's equations of motion. When integrating the equations of motions it is important to use symplectic algorithms which conserve the phase-space volume. To solve the equations of motion under symplectic conditions, various integration algorithms have been developed. The time dependence of a particle can be expressed in a Taylor expansion
	::<math>		::<math>
	\mathbf{r}_{i}(t+\Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t)\Delta t + \frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2} + \frac{\partial^{3} \mathbf{r}_{i}(t)}{\partial t^{3}}\Delta t^{3} + \mathcal{O}(\Delta t^{4})		\mathbf{r}_{i}(t+\Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t)\Delta t + \frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2} + \frac{\partial^{3} \mathbf{r}_{i}(t)}{\partial t^{3}}\Delta t^{3} + \mathcal{O}(\Delta t^{4})
Line 15:		Line 15:
	::<math>		::<math>
	\mathbf{r}_{i}(t+\Delta t) =		\mathbf{r}_{i}(t+\Delta t) =
	\mathbf{r}_{i}(t)+ \mathbf{v}_{i}(t)\Delta t+\frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2}		\mathbf{r}_{i}(t)+ \mathbf{v}_{i}(t)\Delta t+\frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2}.
	</math>		</math>
	=== Velocity-Verlet integration scheme ===		== Velocity-Verlet integration scheme ==
	The Velocity-Verlet algorithm can be decomposed into the following steps:		The Velocity-Verlet algorithm can be decomposed into the following steps:
	<ol>		<ol>
Line 27:		Line 27:
	From these equations it can be seen that the velocity and the position vectors are synchronous in time.		From these equations it can be seen that the velocity and the position vectors are synchronous in time.

	=== Leap-Frog integration scheme ===		== Leap-Frog integration scheme ==
	Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:		Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:
	<ol>		<ol>
Line 36:		Line 36:
	In this form the velocity and the position vectors are asynchronous in time.		In this form the velocity and the position vectors are asynchronous in time.

	=== Thermostats and used integrators ===		== Thermostats and used integrators ==
	{\|class="wikitable" style="margin:aut		{\|class="wikitable" style="margin:aut
	! MDALGO !! thermostat !! integration algorithm		! MDALGO !! thermostat !! integration algorithm
	\|-		\|-
	\|style="text-align:center;"\| 0 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Velocity-Verlet integration scheme\|Velocity-Verlet]]		\|style="text-align:center;"\| 0 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| [[#Velocity-Verlet integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 1 \|\| [[Andersen thermostat\|Andersen]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 1 \|\| [[Andersen thermostat\|Andersen]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 2 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 2 \|\| [[Nose-Hoover thermostat\|Nose-Hoover]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 3 \|\| [[Langevin thermostat\|Langevin]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Velocity-Verlet Integration scheme\|Velocity-Verlet]]		\|style="text-align:center;"\| 3 \|\| [[Langevin thermostat\|Langevin]] \|\| style="text-align:center;"\| [[#Velocity-Verlet Integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 4 \|\| [[Nose-Hoover-chain thermostat\|NHC]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Velocity-Verlet integration scheme\|Velocity-Verlet]]		\|style="text-align:center;"\| 4 \|\| [[Nose-Hoover-chain thermostat\|NHC]] \|\| style="text-align:center;"\| [[#Velocity-Verlet integration scheme\|Velocity-Verlet]]
	\|-		\|-
	\|style="text-align:center;"\| 5 \|\| [[CSVR thermostat\|CSVR]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 5 \|\| [[CSVR thermostat\|CSVR]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]
	\|-		\|-
	\|style="text-align:center;"\| 5 \|\| [[MDALGO#MDALGO=13: Multiple Andersen thermostats\|Multiple Andersen]] \|\| style="text-align:center;"\| [[~~Construction:Molecular dynamics propagation algorithms~~#Leap-Frog integration scheme\|Leap-Frog]]		\|style="text-align:center;"\| 5 \|\| [[MDALGO#MDALGO=13: Multiple Andersen thermostats\|Multiple Andersen]] \|\| style="text-align:center;"\| [[#Leap-Frog integration scheme\|Leap-Frog]]
	\|}		\|}

			== Related tags and articles ==

			{{TAG\|IBRION}}, {{TAG\|MDALGO}}, [[Thermostats]]

			[[Category:Theory]][[Category:Molacular dynamics]][[Category:Thermostats]]