Lecture 7: Distribution functions in classical monatomic liquids Statistical Thermodynamics Lecture 7: Distribution functions in classical monatomic liquids Dr. Ronald M. Levy ronlevy@temple.edu

Liquids behaves different from gas For imperfect gases, virial equation of state could be written as a sum of configurational integrals, involving 1 body, 2 body, 3 body, … For liquids, this expansion gives poor results. For liquids, it is not a good approximation to treat the system as a sequence of 1 body, 2 body, 3 body … etc. Hard sphere equation of state liquid solid   MD simulation - liquid MD simulation - solid   V0/V   1

1 2 3 N particle potential Conformational partition function   N particle potential 1 Conformational partition function Integrated over n+1 to N 2 3 2

n particle correlation function   n particle correlation function 4 3

  5     This equation defines the Conditional probability 4

  N 1 6   5

          6

      Mean kinetic energy Mean potential energy 7

  7 Expression for average internal energy in terms of pair correlation function 8

    Double count Averge potential energy between molecule1 and molecules locates between r and r+dr N identical terms 8 9

9 10 The factor V comes from the integration of dr   9 The factor V comes from the integration of dr coupling parameter integration 10 10

11 There are N-1 identical terms leading to N-1 identical integrals   There are N-1 identical terms leading to N-1 identical integrals     11 Depends on coupling parameter 11

  12 13 12

  Potential of mean force(potential which gives the mean force) 14   13

15 Direct+indirect indirect   15 Measure of the total influence of mol1 and mol2 Indirect part is the direct influence of particle 1 on 3. which in turn exerts its influence on 2 directly or indirectly The direct correlation function is defined by this equation Direct+indirect indirect Can solve 2 equation interatively 14

Set of solute coordinates   Set of solute coordinates Set of solvent coordinates Potential energy of the system     Defines W (PMF) 15

      16

Solute charge distribution   Solvent induced influence Intramolecular solute potential Solute charge distribution   Poisson equation 17