M10 Angular momentum is conserved or not

Manual 10 for MD++
Angular momentum is conserved or not?

Keonwook Kang and Wei Cai

Jul 06 , 2009

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In molecular dynamics simulations, positions and velocities of atoms are updated following Newton's equations of motion. Naturally, we would expect that linear and angular momenta are conserved during the time integration, or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \mathbf{P} & \equiv & \sum_i m_i\mathbf{v}_ i = \mathrm{Const.} \\ \mathbf{L} & \equiv & \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i = \mathrm{Const.} \end{array}, }

where P is linear momentum and L is angular mumentum. Usually, we subtract ceter-of-mass velocity from the velocity of each atom so that the whole system can not drift and the linear momentume becomes zero.

${\begin{array}{rcl}\mathbf {v} _{i}&:=&\mathbf {v} _{i}-\mathbf {v} _{\mathrm {CM} }\\\mathbf {P} &=&\sum _{i}m_{i}\mathbf {v} _{i}-\mathbf {v} _{\mathrm {CM} }\sum _{i}m_{i}=0\end{array}},$

where v_CM is defined as

$\mathbf {v} _{\mathrm {CM} }\equiv {\frac {\sum _{i}m_{i}\mathbf {v} _{i}}{\sum _{i}m_{i}}}$.

Similarly, we can additionaly subtract velocity component contributing rotation so that the whole system can not rotate and the angular momentum becomes zero as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_i := \mathbf{v}_i - \mathbf{\Omega}\times\mathbf{r}_i } .

The angular velocity Ω is obtained from

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\Omega} \equiv \mathbf{I}^{-1} \mathbf{L} }

and moment-of-inertia matrix I is expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I} \equiv \sum_i m_i(\mathbf{E}_3|\mathbf{r}_i|^2 - \mathbf{r}_i\otimes\mathbf{r}_i)} ,

where E₃ is a 3 by 3 identity matrix.

Let's see whether MD simulation results conserve both linear and angular momenta. For the test simulation, a diamomd cubic structure of 216 silicon atoms are equilbriated under periodic boundary conditions for 10,000 steps using NVE ensemble.

	Figure 1
(a)	(b)	(c)

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). According to Allen and Tildesley