1. Lecture 7: Distribution functions in classical monatomic liquids
Statistical Thermodynamics
Lecture 7: Distribution functions in classical monatomic liquids
Dr. Ronald M. Levy

2. 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

3. 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

4. n particle correlation function
n particle correlation function 4 3

5. 5
This equation defines the Conditional probability 4

6. N 1 6 5

7. 6

8. Mean kinetic energy
Mean potential energy 7

9. 7
Expression for average internal energy in terms of pair correlation function 8

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

11. 9 10 The factor V comes from the integration of dr9
The factor V comes from the integration of dr
coupling parameter integration 10 10

12. 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

13. 12 13 12

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

15. 15 Direct+indirect indirect15
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

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

17. 16

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