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md2011-05-note2_Part_9

md2011-05-note2_Part_9 - Thermal Conductivity The thermal...

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Thermal Conductivity I The thermal conductivity λ T is λ T V k B T 2 Z 0 d t h j ε α ( t ) j ε α ( 0 ) i , (52) with the corresponding relation 2 t λ T = V k B T 2 Z 0 d t D ( δε α ( t ) - δε α ( 0 )) 2 E . (53) I Here j ε α is a component of an energy current ( j ε α = ∂δε α /∂ t ), which in turn is δε α = 1 V X i r i α ( ε i - h ε i i ) , ε i = p 2 i 2 m i + 1 2 X i 6 = j U ( r ij ) . (54) Notes
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S Q I Pair correlation function (radial distribution function) g ( r ) tells at which distance the atoms (on average) are from each other in the system. I It is defined as g ( r ) ρ - 2 * X ij δ ( r i ) δ ( r j - r ) + = V N 2 * X i X j δ ( r - r ij ) + , (55) and it basically counts the number of bonds at each interatomic distance scaled by the corresponding atomic density. I In practice, we count the number of atoms within a certain Δ r , divided by the average number of atoms at the same distance in an ideal gas. Notes
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I Pair correlation function gives valuable information on the state of the system, f.ex. on melting a crystal. I g can be used to calculate averages for any pair functions h A i = * X i X j a ( r ij ) + = 1 2 N ρ Z 0 a ( r ) g ( r ) 4 π r 2 d r , (56) but the direct evaluation is usually more accurate. I For example the energy: E = 3 2 Nk B T + 2 π N ρ Z 0 U ( r ) g ( r ) r 2 d r , (57) I and the pressure:
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