Statistical Mechanics and Canonical Ensemble Lecture 6 Statistical Mechanics and Canonical Ensemble Time average and ensemble average Postulates Problem 6.1 Probabilities for microcanonical and canonical ensembles

Ensemble Collection of configurations of the system Configurations are labeled according to quantum energy state Quantum states energy levels

Postulates Microcanonical ensemble - energy is the same for all states Probability of being in a given state i (pi) is the same for every state Over a long time the physical system spends equal amounts of time in every state  From 2 average of any quantity (A) over ensemble is equal to the average over time

Problem 6.1 For distinguishable particles, Three possible single particle states with energies, E1, E2, E3. a, b) List and group four particle states according to energy levels c) What are available states with energy E1 +2E2 + E3 d) If number of particles is allowed to change, how does this affect the number of available energy levels.

Probabilities in Microcanonical Ensemble From the postulate pi is the same for each state, i. Remember each state has the same energy Of course and the average value of any property, eg., pressure and energy is given by Thermodynamic pressure and energy

Probabilities in Canonical Ensemble Rigorous derivation (see Hill chapter 1) Create an ensemble of constant temperature states in thermal contact, but isolated from the surroundings - we can consider this ensemble system as super - system keep at constant energy i represents system in a given quantum energy state Ei ni is a number of systems in state i We denote “n” as a given distribution n = n1, n2, n3…. 1 2 1 5 4 i 7

Distribution Weight The “super system” is isolated, has a total energy Et and consists of M “members” nj denotes a number of members in state j with energy Ej Total number of possible states of the super system, Ω (n) with distribution n = (n1, n2 …..) is given by

Probability and Most Probable Distribution Since the “super system” each state in it has the same probability, thus probability is observing a given distribution (n) is proportional to Ω (n). This leads to a formula for the probability of having a given energy Ej associated with quantum state nj is given by As we increase the number of members M to infinity, the most probable distribution (n*), will have Ω (n*) much larger than any other Ω (n), consequently

Most Probable Distribution We need to find what distribution has a maximum possible number of states, i.e., For convenience we look for maximum lnΩ and use Stirling approximation, ln x! = xlnx - x

Conditional maximum We need to find maximum of a function Under two constrains, fixed number of canonical systems and fixed total energy (of the super system) According to the method of Lagrange multipliers Where α and β are not yet determined constants

Probability formula Carrying the differentiation One obtains From which within a constant Q

Probability formula and the Partition Function Sum of probabilities pj has to be equal to one From which Q, which is called the canonical ensemble partition function is given by We will see shortly that knowledge of Q allows us to completely describe the thermodynamic of the system

Probabilities in Canonical Ensemble - Less Rigorous Derivation pi for to a system in contact with a thermostat is a function of energy only pi = f (Ei) Consider combined system consisting of two identical systems. One original system, i, has pi = f (Ei) and the second has pj = f (Ej). Probability of the combined states pi,j = f (Ei +Ej). Since the probability of a state is (i) a function of energy of this state only and (ii) does not depend on other states pi,j = pi pj  f(Ei +Ej) = f (Ei) f (Ej)

Exponential Function Consider f(x+y) = f (x) f (y) Therefore Where β and A are constants

Probabilities in Canonical Ensemble - Less Rigorous Derivation Substituting Ei for x Since the sum of all pis =1 Q is the canonical partition function that we en With this

Evaluating Averages in Canonical Ensemble Once the partition function is known the energy can be evaluated as prove Pressure of a given state is And the thermodynamic pressure In general any other quantity, A,