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CM_Part_2

# CM_Part_2 - Monte Carlo evaluation of statistical-mechanics...

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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Monte Carlo evaluation of statistical-mechanics integrals Why are we interested in integration? N N N N N N N N p d r d kT p r E p r A Z p r A r r r r r r r r = ) , ( exp ) , ( 1 ) , ( Ensemble average of quantity A( r , p ) can be calculated for a given distribution function. For NVT ensemble we have distribution function ρ ( r , p ), N N N N N N p d r d kT ) p , r ( E exp Z 1 ) p , r ( r r r r r r = ρ Energy can always be expressed as a sum of kinetic and potential contributions. The contribution of the kinetic part is trivial and we can consider integral in only configurational 3N dimensional space, where Z is configurational integral. N N N N r d kT r U r A Z r A r r r r = ) ( exp ) ( 1 ) ( = N N r d kT ) r ( U exp Z r r

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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Monte Carlo evaluation of statistical-mechanics integrals Statistical-mechanics integrals typically have significant contributions only from very small fractions of the 3N space. For example for hard spheres contributions are coming from the areas
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CM_Part_2 - Monte Carlo evaluation of statistical-mechanics...

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