Bernoullis choice Heads or Tails N of slots

Bernoulli's choice: Heads or Tails? N = # of slots, # of macrostates = multiplicity, # of microstates N =2 N 0 1 2 3 4 5 20 21 22 23 24 25 Pascal’s triangle 1 1 1 2 3 4 5 1 1 3 6 10 1 4 10 1 5 1 Example: For N=4 fair coin tosses there are N+1=5 macrostates each containing n heads where n = 0, 1, 2, 3, 4. Each macrostate has 4 Cn occurrences of n heads with a total # of microstates equal to the multiplicity .

microstate Prob. (microstate) Macrostates: n, m Macrostate: n-m hhhh 1/16 4, 0 4 thhh 1/16 3, 1 2 hthh 1/16 3, 1 2 hhth 1/16 3, 1 2 hhht 1/16 3, 1 2 tthh 1/16 2, 2 0 thth 1/16 2, 2 0 htht 1/16 2, 2 0 hhtt 1/16 2, 2 0 htth 1/16 2, 2 0 thht 1/16 2, 2 0 httt 1/16 1, 3 -2 thtt 1/16 1, 3 -2 ttht 1/16 1, 3 -2 ttth 1/16 1, 3 -2 tttt 1/16 0, 4 -4 16 different configurations (microstates), 5 different macrostates

Number of Microstates ( ) Most likely macrostate the system will find itself in is the one with the maximum number of microstates out of a total of 2100 = 1. 27 E 30 configurations (50 h)(50 t): P=(100!/50!50!)/2100 = 0. 079 for 100 tosses (57 h)(43 t): P=(100!/57!43!)/2100 = 0. 066 for 100 tosses (60 h)(40 t): P=(100!/60!40!)/2100 = 0. 011 for 100 tosses (90 h)(10 t): P=(100!/90!10!)/2100 = 1. 34 E-17 for 100 tosses (100 h)(0 t): P=(100!/100!0!)/2100 = 7. 87 E-31 for 100 tosses Macrostate

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

4 quanta, 16 microstates, 5 macrostates Increasing energy 4 C 4 C E = 4 4=1 3=4 E = 3 4 C 2=6 E = 2 4 C 1=4 E = 1 4 C 0=1 E = 0 For N = 4 quanta in N+1=5 energy states (macrostates) with total energies E = 0 , 1 , 2 , 3 , 4. Each macrostate has 4 Cn occurrences of equal energy states E= n (n=0, 1, 2, 3, 4) with a total # of microstates equal to the multiplicity

Ensemble: All the parts of a thing taken together, so that each part is considered only in relation to the whole.

Microcanonical ensemble: An ensemble of snapshots of a system with the same N, V, and E A collection of systems that each have the same fixed energy. E (E)

The most likely macrostate the system will find itself in is the one with the maximum number of microstates. E 1 E 2 1(E 1) 2(E 2)

Microcanonical ensemble: • Total system ‘ 1+2’ contains 20 energy quanta and 100 levels. • Subsystem ‘ 1’ containing 60 levels with total energy x is in equilibrium with subsystem ‘ 2’ containing 40 levels with total energy 20 -x. • At equilibrium (max), x=12 energy quanta in ‘ 1’ and 8 energy quanta in ‘ 2’

Canonical ensemble: An ensemble of snapshots of a system with the same N, V, and T (red box with energy << E. Exchange of energy with reservoir. E- (E- ) I ( )

Log 10 (P( )) Canonical ensemble: P( ) (E- ) 1 exp[- /k. BT] • Total system ‘ 1+2’ contains 20 energy quanta and 100 levels. • x-axis is # of energy quanta in subsystem ‘ 1’ in equilibrium with ‘ 2’ • y-axis is log 10 of corresponding multiplicity of reservoir ‘ 2’