1. Bernoulli's choice: Heads or Tails?
N = # of slots, # of macrostates
 = multiplicity, # of microstates
Pascal’s triangle N 1 2 3 4 5
=2N 20 21 22 23 24 25 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 4Cn occurrences of n heads with a total # of microstates equal to the multiplicity .

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

3. (50h)(50t): P=(100!/50!50!)/2100 = 0.079 for 100 tosses
Most likely macrostate the system will find itself in is the one with the maximum number of microstates out of a total of 2100 = 1.27E30 configurations
(50h)(50t): P=(100!/50!50!)/2100 = for 100 tosses
(57h)(43t): P=(100!/57!43!)/2100 = for 100 tosses
(60h)(40t): P=(100!/60!40!)/2100 = for 100 tosses
(90h)(10t): P=(100!/90!10!)/2100 = 1.34E-17 for 100 tosses
(100h)(0t): P=(100!/100!0!)/2100 = 7.87E-31 for 100 tosses
Number of Microstates ()
Macrostate

4. 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 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 1 1 1 1 1 1 1

5.  4 quanta, 16 microstates, 5 macrostates Increasing energy 4C4=1
For N = 4 quanta in N+1=5 energy states (macrostates) with total energies E = 0, 1, 2, 3, 4. Each macrostate has 4Cn occurrences of equal energy states E= n (n=0,1,2,3,4) with a total # of microstates equal to the multiplicity

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

8. 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)

9. The most likely macrostate the system will find itself in is the one with the maximum number of microstates. E1
1(E1) E2
2(E2)

10. 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’

12. 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()

13. Canonical ensemble: P()  (E-)1  exp[-/kBT]
Log10 (P())
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 log10 of corresponding multiplicity of reservoir ‘2’