Lecture 10—Ideas of Statistical Mechanics
Chapter 4, Wednesday January 30th
•Finish Ch. 3 - Statistical distributions
•Statistical mechanics - ideas and definitions
•Quantum states, classical probability,
ensembles, macrostates...
•Entropy
•Definition of a quantum state
Reading:
Exam 1:
All of chapter 4 (pages 67 - 88)
***Homework 3 due Fri. Feb. 1st****
Assigned problems, Ch. 3: 8, 10, 16, 18, 20
Homework 4 due next Thu. Feb. 7th
Assigned problems, Ch. 4: 2, 8, 10, 12, 14
Fri. Feb. 8th (in class), chapters 1-4
Statistical distributions
16
ni
xi
Mean:
nx

x
,
i
N
i i
where N   i ni
Statistical distributions
16
ni
xi
Mean:
nx

x
,
i
N
i i
where N   i ni
Statistical distributions
16
ni N  
xi
Mean:
x   i pi xi , where
ni
pi  lim
N  N
Statistical distributions
16
ni
xi
Standard
deviation

 x 
2

 p x  x
i
i
i
2
Statistical distributions
64
Gaussian distribution
(Bell curve)
2

1
  x  x  
p( x ) 
exp  

2
2
 2


Statistical Mechanics (Chapter 4)
•What is the physical basis for the 2nd law?
•What is the microscopic basis for entropy?
Boltzmann hypothesis: the entropy of a system is related to
the probability of its state; the basis of entropy is statistical.
Statistics + Mechanics
Statistical Mechanics
Thermal Properties
Statistical Mechanics
•Use classical probability to make predictions.
•Use statistical probability to test predictions.
Note: statistical probability has no basis if a system is out of
equilibrium (repeat tests, get different results).
How on earth is this possible?
•How do we define simple events?
•How do we count them?
•How can we be sure they have equal probabilities?
REQUIRES AN IMMENSE LEAP OF FAITH
Statistical Mechanics – ideas and definitions
A quantum state, or microstate
• A unique configuration.
• To know that it is unique, we must specify it as
completely as possible...
e.g. Determine:
Position
Momentum
Energy
Spin
............
of every particle, all at once!!!!!
THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEM
Statistical Mechanics – ideas and definitions
A quantum state, or microstate
• A unique configuration.
• To know that it is unique, we must specify it as
completely as possible...
Classical probability
• Cannot use statistical probability.
• Thus, we are forced to use classical probability.
An ensemble
• A collection of separate systems prepared in
precisely the same way.
Statistical Mechanics – ideas and definitions
The microcanonical ensemble:
Each system has same:
# of particles
Total energy
Volume
Shape
Magnetic field
Electric field
............
and so on....
These variables (parameters) specify the ‘macrostate’
of the ensemble. A macrostate is specified by ‘an
equation of state’. Many, many different microstates
might correspond to the same macrostate.
Statistical Mechanics – ideas and definitions
64
An example:
Coin toss again!!
width
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Volume V
10 particles, 36 cells
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
Ensembles and quantum states (microstates)
Many more states look like this, but no more probable than the last one
Volume V
10
 1 
pi   
 36 
16
 3  10
Cell volume, V
There’s a major flaw in this calculation.
Can anyone see it?
It turns out that we get away with it.
Entropy
Boltzmann hypothesis: the entropy of a system is related to
the probability of its being in a state.
1
p
W

S  f n W    W 
S  kB ln W 