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# lecture9 - Heat capacity C = H 2-H 2 k B T 2 • Angle...

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Chapter 9: Molecular-dynamics Integrate equations of motion-- classical! Discrete form of Newton’s second law Forces from interaction potential For a simple pair potential, we get ρ F i = - r Ñ i U r r 1 , r r 2 ,..., r r N ( ) U r r 1 ,..., r r N ( ) = 1 2 u ( ij å r ij )

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Integrating equations of motion F i , x = m i dv i , x dt F i , y = m i dv i , y dt F i , z = m i dv i , z dt dv i , x dt = d 2 x i dt 2 » x i ( n +1) - 2 x i ( n ) + x i ( n - 1) D t 2 x i ( n +1) = 2 x i ( n ) - x i ( n - 1) + F i , x m i D t 2 This works out to give the Verlet algorithm,
Lennard-Jones potential for noble gas V r ( ) = 4 e s r ae è ç ö ø ÷ 12 - s r ae è ç ö ø ÷ 6 é ë ê ù û ú QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture.

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Unformatted text preview: Heat capacity C = H 2-H 2 k B T 2 • Angle brackets thermal average • In MD, thermal averages done by time average • Heat capacity given by fluctuations in total energy Radial distribution function g ( r ) = 1 r N d r-r ij ( ) i ¹ j å • ρ is the density N/ Ω 2200 Dirac-delta defined numerically (not infinitely sharp) • In an ideal gas, g(r) =1 • In a liquid, g(r) =1 at long ranges, short range structure • In a crystal, g( r) has sharp peaks, long-range order...
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lecture9 - Heat capacity C = H 2-H 2 k B T 2 • Angle...

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