M10 Angular momentum is conserved or not - Revision history

Kwkang: /* MD simulations of NVT ensemble */

2009-08-15T00:28:54Z

MD simulations of NVT ensemble

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	\|}		\|}

	This relation explains the exponential behavior at the ~~early~~ ~~time. Later~~, ~~when the system reaches equilibrium state, temperature will fluctuate around the objective temperature and~~ <math>d\zeta~~/dt'~~</math> ~~keeps~~ ~~changing~~ ~~its sign to compensate the difference, ''T''-''T''<sub>OBJ</sub>. Thus, <math>\zeta</math> will fluctuate around zero in the long run and that explains the flat region~~ of ~~angular momentum curve~~. Linear momentum ~~would~~ show ~~similiar~~ behavior if its initial value is not zero.		This relation explains the exponential behavior of the angular momentum, though <math>\zeta</math> is a function of time. Linear momentum will show similar behavior when its initial value is not zero, because

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			\|<math> \frac{d}{dt'} \mathbf{P}' = - \zeta \mathbf{P}'</math>
			\|}

			from the 1st equation of NH equations of motion. It would be an interesting question how changes in ''Q'' will affect the behavior of momenta.

	== Notes ==		== Notes ==

Kwkang: /* MD simulations of NVT ensemble */

2009-08-14T23:54:50Z

MD simulations of NVT ensemble

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	\|<math> \begin{array}{rcl} \frac{d \mathbf{p}'_i}{dt'} &=& - \frac{\partial U(\left\{ \mathbf{r}'_i \right\})}{\partial \mathbf{r}'_i} - \zeta \mathbf{p}'_i \\		\|<math> \begin{array}{rcl} \frac{d \mathbf{p}'_i}{dt'} &=& - \frac{\partial U(\left\{ \mathbf{r}'_i \right\})}{\partial \mathbf{r}'_i} - \zeta \mathbf{p}'_i \\
	\frac{d \mathbf{r}'_i}{dt'} &=& \frac{\mathbf{p}'_i}{m_i} \\		\frac{d \mathbf{r}'_i}{dt'} &=& \frac{\mathbf{p}'_i}{m_i} \\
	\~~dot~~{\zeta}' &=& \frac{gk_B}{Q} \left( T - T_\mathsf{obj} \right) \end{array}</math>.		\frac{d\zeta}{dt'} &=& \frac{gk_B}{Q} \left( T - T_\mathsf{obj} \right) \end{array}</math>.
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Kwkang: /* MD simulations of NVT ensemble */

2009-08-14T23:41:38Z

MD simulations of NVT ensemble

← Older revision		Revision as of 23:41, 14 August 2009
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	== MD simulations of NVT ensemble ==		== MD simulations of NVT ensemble ==

	Upto here we have considered MD simulations of NVE ensemble. People might wonder whether the linear and the angular momenta will be conserved in case of canonical or NVT ensemble. This question is ~~important~~ because a system of NVT ensemble is not isolated system ~~because~~ it ~~has~~ ~~heat~~ ~~bath.~~ In ~~MD++,~~ ~~Nosé-Hoover~~ ~~thermostat~~ is ~~used~~ ~~for~~ ~~NVT~~ ~~ensemble~~. ~~Nosé's Hamiltonian is expressed in time scaling variable ''s'' as~~		Upto here we have considered MD simulations of NVE ensemble. People might wonder whether the linear and the angular momenta will be conserved in case of canonical or NVT ensemble. This question is not trivial because a system of NVT ensemble is not an isolated system in a sense that it is connected to the heat bath to control temperature.

			In MD++, Nosé-Hoover thermostat is used for NVT ensemble. Nosé's Hamiltonian is expressed in time scaling variable ''s'' as

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	~~According to~~ figure 5(c), angular momentum exponentially decays in the early time and fluctuates around some constant value later. The equations of motion of Nosé-Hoover thermostat are given as		In figure 5(c), it is interesting to observe that angular momentum exponentially decays in the early time and eventually fluctuates around some constant value later. This behavior can be understood if we take a closer look on Nosé-Hoover dynamics. The equations of motion of Nosé-Hoover thermostat are given as

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Kwkang: /* MD simulations of NVT ensemble */

2009-07-20T17:51:20Z

MD simulations of NVT ensemble

← Older revision		Revision as of 17:51, 20 July 2009
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	== MD simulations of NVT ensemble ==		== MD simulations of NVT ensemble ==

	Upto here we have considered MD simulations of NVE ensemble. People might wonder whether the linear and the angular momenta will be conserved in case of canonical or NVT ensemble. This question is important because a system of NVT ensemble is not isolated system because of ~~its~~ heat bath. In MD++, Nosé-Hoover thermostat is used for NVT ensemble. Nosé's Hamiltonian is expressed in time scaling variable ''s'' as		Upto here we have considered MD simulations of NVE ensemble. People might wonder whether the linear and the angular momenta will be conserved in case of canonical or NVT ensemble. This question is important because a system of NVT ensemble is not isolated system because it has heat bath. In MD++, Nosé-Hoover thermostat is used for NVT ensemble. Nosé's Hamiltonian is expressed in time scaling variable ''s'' as

	{\|border="0" align="center"		{\|border="0" align="center"

Kwkang: /* MD simulations of NVT ensemble */

2009-07-20T07:05:45Z

MD simulations of NVT ensemble

← Older revision		Revision as of 07:05, 20 July 2009
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	== MD simulations of NVT ensemble ==		== MD simulations of NVT ensemble ==

	Upto here we have considered MD simulations of NVE ensemble. People might wonder whether the linear and the angular momenta will be conserved in case of canonical or NVT ensemble. In MD++, Nosé-Hoover thermostat is used for NVT ensemble. Nosé's Hamiltonian is expressed in time scaling variable ''s'' as		Upto here we have considered MD simulations of NVE ensemble. People might wonder whether the linear and the angular momenta will be conserved in case of canonical or NVT ensemble. This question is important because a system of NVT ensemble is not isolated system because of its heat bath. In MD++, Nosé-Hoover thermostat is used for NVT ensemble. Nosé's Hamiltonian is expressed in time scaling variable ''s'' as

	{\|border="0" align="center"		{\|border="0" align="center"

Kwkang: /* Si NW */

2009-07-20T07:00:43Z

Si NW

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	Since the angular momentum is not conserved in MD simulations under periodic boundary conditions, it is usually unnecessary to make angular momentum be zero when the velocities of atoms are initialized. However, when we perform simulations of nanowires or rods, it is desirable to prevent the rotation around wire axis. For example, if a nanowire is aligned along the ''z''-direction, we want to let the ''z''-component of angular momentum be zero, or '''L'''<sub>''z''</sub> = 0.		Since the angular momentum is not conserved in MD simulations under periodic boundary conditions, it is usually unnecessary to make angular momentum be zero when the velocities of atoms are initialized. However, when we perform simulations of nanowires or rods, it is desirable to prevent the rotation around wire axis. For example, if a nanowire is aligned along the ''z''-direction, we want to let the ''z''-component of angular momentum be zero, or '''L'''<sub>''z''</sub> = 0.

	In MD++ there is a variable '''zerorot''', depending on whose value components of the angular momentum can be ~~selletively~~ zero. If '''zerorot = "x"''', ''x''-component of angular momentum will be zero. If '''zerorot = "all"''', all three components of the angular momentum will be zero. The default value of '''zerorot''' is "none". The other possible values are "y" and "z".		In MD++ there is a variable '''zerorot''', depending on whose value components of the angular momentum can be selectively zero. If '''zerorot = "x"''', ''x''-component of angular momentum will be zero. If '''zerorot = "all"''', all three components of the angular momentum will be zero. The default value of '''zerorot''' is "none". The other possible values are "y" and "z".

	In the next test simulation, a Si NW is aligned along the ''z''-direction and we expect '''L'''<sub>''z''</sub> = 0.		In the next test simulation, a Si NW is aligned along the ''z''-direction and we expect '''L'''<sub>''z''</sub> = 0.

Kwkang: /* Si cluster */

2009-07-20T06:57:40Z

Si cluster

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	In figure 2(b) and 2(c), we can see that linear and angular momenta are conserved within the machine precision in this test simulation.		In figure 2(b) and 2(c), we can see that linear and angular momenta are conserved within the machine precision in this test simulation.
	<HR><BR>

	=== Si NW ===		=== Si NW ===

Kwkang: /* Bulk Si */

2009-07-20T06:56:19Z

Bulk Si

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	<blockquote>If the system is invariant to rotation about an axis, then the corresponding angular momentum component is conserved. ... In the case of the periodic boundary conditions, ... none of them were spherically symmetrical; in fact it is impossible (in Euclidean space) to construct a spherically symmetric periodic system. Hence, ... '''total angular momentum is not conserved in most molecular dynamics simulations'''.</blockquote>		<blockquote>If the system is invariant to rotation about an axis, then the corresponding angular momentum component is conserved. ... In the case of the periodic boundary conditions, ... none of them were spherically symmetrical; in fact it is impossible (in Euclidean space) to construct a spherically symmetric periodic system. Hence, ... '''total angular momentum is not conserved in most molecular dynamics simulations'''.</blockquote>

	Though the angular momentum is not conserved, its average value is close to zero and we do not observe the simulation system rotating during the simulation. The only possible explanation is that there exists time-varying external torque that makes the angular momentum be zero in average sense. Periodic boundary conditions are interesting constraints because they break the angular momentum conservation but they enforce the system not to rotate.		Though the angular momentum is not conserved, its average value is close to zero and we do not observe the simulation system rotating during the simulation. The only possible explanation is that there exists time-varying external torque that makes the angular momentum be zero in average sense. Periodic boundary conditions are interesting constraints because they break the angular momentum conservation but they enforce the system not to rotate in a such way that the image cells apply external torques to the primary cell to compensate the rotation of the primary cell.

	=== Si cluster ===		=== Si cluster ===

Kwkang: /* Bulk Si */

2009-07-20T06:32:28Z

Bulk Si

← Older revision		Revision as of 06:32, 20 July 2009
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	=== Bulk Si ===		=== Bulk Si ===
	For the test simulation, a diamond cubic structure of 216 silicon atoms are equilbriated under periodic boundary conditions for 10,000 steps using NVE ensemble. Linear and angular momenta are initially zeroed out as explained above.		For the 1st test simulation, a diamond cubic structure of 216 silicon atoms are equilbriated under periodic boundary conditions for 10,000 steps using NVE ensemble. Linear and angular momenta are initially zeroed out as explained above.

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	During 10,000 steps, the linear momentum is conserved within the machine precision in figure 1(b), ~~while~~ the angular momentum is not as ~~shown~~ in ~~figure~~ ~~1(c)~~. What is interesting is that the angular momentum is fluctuating around zero. Even if the angular momentum is not initially zeroed out, its behavior is very much similar in terms of fluctuation maginutude and frequency, though it is not shown here. ~~According~~ ~~to Allen and Tildesley<ref>M. P. Allen and D. J. Tildesley, ''Computer Simulations of Liquids'', Oxford Science Publications (2004) pp.72-73</ref>, the reason is~~ ~~told~~ as		During 10,000 steps, the linear momentum is conserved within the machine precision in figure 1(b). On the other hand, figure 1(c) shows that the angular momentum is not conserved within the machine precision. What is interesting is that the angular momentum is fluctuating around zero. Even if the angular momentum is not initially zeroed out, its behavior is very much similar in terms of fluctuation maginutude and frequency, though it is not shown here. What does this mean?

			Allen and Tildesley<ref>M. P. Allen and D. J. Tildesley, ''Computer Simulations of Liquids'', Oxford Science Publications (2004) pp.72-73</ref> clearly stated that the angular momentum is not conserved in MD simulations with periodic boundary conditions as quoted below.

	<blockquote>If the system is invariant to rotation about an axis, then the corresponding angular momentum component is conserved. ... In the case of the periodic boundary conditions, ... none of them were spherically symmetrical; in fact it is impossible (in Euclidean space) to construct a spherically symmetric periodic system. Hence, ... '''total angular momentum is not conserved in most molecular dynamics simulations'''.</blockquote>		<blockquote>If the system is invariant to rotation about an axis, then the corresponding angular momentum component is conserved. ... In the case of the periodic boundary conditions, ... none of them were spherically symmetrical; in fact it is impossible (in Euclidean space) to construct a spherically symmetric periodic system. Hence, ... '''total angular momentum is not conserved in most molecular dynamics simulations'''.</blockquote>

			Though the angular momentum is not conserved, its average value is close to zero and we do not observe the simulation system rotating during the simulation. The only possible explanation is that there exists time-varying external torque that makes the angular momentum be zero in average sense. Periodic boundary conditions are interesting constraints because they break the angular momentum conservation but they enforce the system not to rotate.
	Basically, what it says is that in most MD simulations angular momentum is no longer conserved if periodic boudary conditions are used and atoms move across the simulation cell boundary.

	=== Si cluster ===		=== Si cluster ===

Kwkang: /* Zeroing out linear/angular momentum */

2009-07-20T05:44:00Z

Zeroing out linear/angular momentum

← Older revision		Revision as of 05:44, 20 July 2009
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	where '''E'''<sub>3</sub> is a 3 by 3 identity matrix.		where '''E'''<sub>3</sub> is a 3 by 3 identity matrix.

	However, it is questionable that a system under periodic boundary conditions can indeed maintain constant angular momentum. Let's check the conservations of both linear and angular momentum in ~~a few~~ test MD simulations below.		However, it is questionable that a system under periodic boundary conditions can indeed maintain constant angular momentum. Let's check the conservations of both linear and angular momentum in test MD simulations below.
	<HR><BR>		<HR><BR>