1. Statistical Thermodynamics CHEN 689 Fall 2015 Perla B. Balbuena 240 JEB balbuena@tamu.edu 5-3375 1

2. Goals of Statistical Thermodynamics Based on a microscopic description, predict macroscopic thermodynamic properties Example: what is pressure? – Molecular simulations – Probabilistic description Objective: Determine probability distributions and average values of properties considering all possible states of molecules consistent with a set of constraints (an ensemble of states) 2

3. Connection between microscopic state and macroscopic state Example: NVE microcanonical ensemble – how to describe an “ideal gas” of identical indistinguishable particles; – And a real condensable gas?? 3

4. Quantum mechanical description of microstates Cubic box side L = V 1/3 h = Planck’s constant m = mass of the particle lx, ly, lz: quantum numbers (0,1, 2,…) 4

5. Consider a macroscopic system N particles, volume V, and given certain forces among the particles Schrödinger equation: For an ideal gas: Monoatomic gas: 5 J = 1, 2, 3,… Ignore electronic states and focus on translational energies Cubic container of side L 1/3

6. energy states and energy levels Energy statelxlylz A211 B121 C112 compute  each energy state gives the same energy level; this jth energy level has a degeneracy (  j ) of 3 6

7. distribution of non-interacting identical molecules (indistinguishable) in energy states we define a set of occupation numbers: n( n 1, n 2, n 3 …) each number is associated with the i th molecular state energy of i th microstate (all molecular energy states)  degeneracy is the number of microstates that have this energy level 7

8. Statistical Mechanics Postulates All microstates with the same E, V, N are equally probable The long time average of any mechanical property in a real macroscopic system is equal to the average value of that property over all the microscopic states of the system (ergodic hypothesis) – each state weighted with its probability of occurrence, provided that all the microstates reproduce the thermodynamic state and environment of the actual system 8

9. Boltzmann energy distribution how to assign probabilities to states of different energies? Assume a system at N, V, in contact with a very large heat bath at constant T 9

10. 10 Boltzmann energy distribution p A (E n ) p B (E m ) since A and B are totally independent from each other, p A (E n ) p B (E m ) is the probability of finding the complete system with A and B in the specified energy sub-states

11. Boltzmann energy distribution 11 how the energy of the composite system would change if we change E n without changing E m ? as a first approximation we assume that the energy levels are closely spaced (  continuous function)

12. The Boltzmann distribution of energy Consider two macroscopic subsystems (no need to be identical) inside an infinite thermostatic bath (i.e., their temperatures are perfectly controlled and fixed). If the energy in A fluctuates, this has no effect on B, and vice-versa

13. The Boltzmann distribution of energy Probability that subsystem A is in a microstate s A with energy E n This is NOT the probability of finding subsystem A with energy E n Several microstates of subsystem A may have the same energy These that have the same energy are equally probable (postulate 1)

14. The Boltzmann distribution of energy Probability that subsystem A is in an energy level of energy E n is the degeneracy of energy level E n in subsystem A

15. The Boltzmann distribution of energy Probability that subsystem B is in a microstate s B with energy E m This is NOT the probability of finding subsystem B with energy E m. Several microstates of subsystem B have the same energy. These that have the same energy are equally probable (postulate 1)

16. The Boltzmann distribution of energy Probability that subsystem B is in an energy level of energy E m is the degeneracy of energy level E m in subsystem B

17. The Boltzmann distribution of energy These probabilities in subsystems A and B are independent of each other. Remember our initial assumption: if the energy in A fluctuates, this has no effect on B, and vice-versa

18. The Boltzmann distribution of energy John enters a draw with 8% chance of winning. Mary enters another, independent draw with 5% chance of winning. What is the probability that John AND Mary win?

19. The Boltzmann distribution of energy John enters a draw with 8% chance of winning. Mary enters another, independent draw with 5% chance of winning. What is the probability that John AND Mary win?

20. The Boltzmann distribution of energy The probability of finding the composite system in a particular microstate s AB is only a function of the total energy of the composite system, E AB (because of postulate 1). Suppose:

21. The Boltzmann distribution of energy Other states with same energy of the composite system, E AB, will have the same probability (again because of postulate 1) For example:

22. The Boltzmann distribution of energy Let us now examine another issue: what is the effect of changing the value of E n without changing E m ?

23. The Boltzmann distribution of energy Using: We also have that:

24. The Boltzmann distribution of energy Combining the results of the two previous slides: An analogous development for the effect of E m gives:

25. The Boltzmann distribution of energy Then:

26. The Boltzmann distribution of energy This equation deserves careful examination: Its left hand side is independent of subsystem B Its right hand side is independent of subsystem A Then, each side of the equation should be independent of both subsystems and depend on a characteristic that they share

27. The Boltzmann distribution of energy The two subsystems share their contact with the thermal reservoir that keeps their temperature constant and equal The minus sign is introduced for convenience

28. The Boltzmann distribution of energy Integrating these two differential equations:

29. The Boltzmann distribution of energy But the result of adding the probabilities of each subsystem should be equal to 1 Canonical partition function

30. The Boltzmann distribution of energy But the result of adding the probabilities of each subsystem should be equal to 1 Canonical partition function

31. The Boltzmann distribution of energy The canonical partition function of a system is: The probability of a particular microstate i with energy E a is: