md2011-05-note_Part_3

# md2011-05-note_Part_3 - I On the other hand some processes...

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I Pressure is less straightforward than the total energy or temperature of the system, but still accessible during MD calculations. I Now, we start with the generalized equipartition theorem for the particle positions q k : ± q k d H d q k ² = k B T . (13) I In Cartesian coordinates, this leads to - 1 3 X i r i · ∇ r i U ( r ) = 1 3 X i r i · f tot i = - Nk B T . (14) I We can divide the force acting on particle i in external and internal parts (with respect to our system) as f tot i = f ext i + f i . (15) Notes

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I Then the external pressure as the eﬀect of container walls to our system is - 1 3 X i r i · f ext i = PV , (16) and we can deﬁne an ’ internal virial W as - 1 3 X i r i · ∇ r i U = 1 3 X i r i · f i = W , (17) where we now limit the interaction to forces acting between the particles. Notes
I Hence, we can write h W i - PV = - Nk B T PV = Nk B T + h W i . (18) I For a pair interaction we would have W = - 1 3 X i , j w ( r ij ) , w ( r ij ) = r ij d U 2 ( r ij ) d r ij . (19) Notes

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T±²³±´µ¶·´± µ¸¹ P´±ºº·´± C»¸¶´»¼ I As said before, plain MD is an NVE method. I However, real simulations still sometimes need guidance to keep the temperature in the desired value.
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Unformatted text preview: I On the other hand, some processes can bring so much heat to the system that it has to be artiﬁcially cooled due to the computationally eﬃcient system sizes. I When this is done, one must pay attention not to destroy the physics. I With applied temperature control, MD transforms into a sort of an NVE – NVT hybrid. Notes I The simplest way to apply T-control is to force T → T after each time step. v new = v ( t ) s T T ( t ) . (20) I Although this method keeps the temperature at T , it is completely unphysical and usually inappropriate for simulating NVT conditions. I An alternative approach is the Berendsen method [Berendsen, J.Chem.Phys. 81, 3684 (1984)] , which can be derived based on a rate equation (although the original approach was diﬀerent): d T ( t ) d t = T-T ( t ) τ T . (21) Notes...
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md2011-05-note_Part_3 - I On the other hand some processes...

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