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13. Molecular Dynamics
	13.a Applications in
			materials science
			bio-chemistry
			astro-physics		
	13.b Types of problems:
		(1) classical/quantum mechanical
		(2) short/long-ranged interactions
	Particles move under mutual influence and determining
	the trajectories of these particles and some bulk
	physical quantaties is the main point of the study.
		x1'' = f1(t, x1, x2, ..., xn, x1', x2', ..., xn')
		x2'' = f2(t, x1, x2, ..., xn, x1', x2', ..., xn')
		... ...
		xn'' = fn(t, x1, x2, ..., xn, x1', x2', ..., xn')
	How to solve this problem?
	Now, we take astrophysical N-body problem as one example.
	After you have computed the forces, or the acceleration a,
	Fn =m_n a_n = -G \sum_(m=/=n) m_m (x_n-x_m)/||x_n-x_m||^3
	(1). Using Euler's method, we get the velocity for the next step
	,  
	.  
	From here, we can compute the velocity and coordinates
	of the particles.
	Apparently, this method is only of 1st order
	and may easily blow up the calculation. We need to use
	better method:
	
	(2). Leap-frog method (2nd order)
	,  
	.
			|<--velocity--->|
	------------------------------------------------------------->
		|	|	|	|	|
		|		|		|
		t-dt	t-dt/2	t	t+dt/2	t+dt
				|		|
				|<--position--->|

	(3) The Verlet method---velocity doesn't appear
	
	How to prove? Very easy using Tayor expansion:
	
	
	
	Summing over the above, we get the formula.
	Force calculation takes 90% of the cpu time and the solution
	of the ODE system takes much less.
	We will return to this problem soon when we discuss parallel computing.

Last Modified: 10/08/00