1. Classical Statistical Mechanics in the Canonical Ensemble

2. Classical Statistical Mechanics
1. The Equipartition Theorem
2. The Classical Ideal Gas
a. Kinetic Theory
b. Maxwell-Boltzmann Distribution

3. Classical Statistical Mechanics (ONLY!)
The Equipartition Theorem in
Classical Statistical Mechanics (ONLY!)

4. The Equipartition Theorem is Valid in Classical Stat. Mech. ONLY!!!
“Each degree of freedom in a system of particles contributes (½)kBT to the thermal average energy of the system.”
Note:
1. This theorem is valid only if each term in the
classical energy is proportional to a momentum (p) squared or to a coordinate (q) squared.
2. The degrees of freedom are associated with translation, rotation & vibration of the system’s
molecules.

5. Outline of a Proof Follows:
In the Classical Cannonical
Ensemble, it is straighforward to
show that
The average energy of a particle per independent degree of freedom  (½)kBT.
Outline of a Proof Follows:

6. System Total Energy  Sum of single particle energies:
Proof
System Total Energy 
Sum of single particle energies:
System Partition Function Z 
Z' , Z'' , etc. = Partition functions
for each particle.

7. System Partition Function Z  Z = Product of partition functions
Z' , Z'' , etc. of each particle
Canonical Ensemble “Recipe” for the Mean
(Thermal) Energy:
So, the Thermal Energy
per Particle is:

8.  = KEt + KEr + KEv + PEv + …. KEt = (½)mv2 = [(p2)/(2m)] KEr = (½)I2
Various contributions to the Classical Energy of each particle:
 = KEt + KEr + KEv + PEv + ….
Translational Kinetic Energy:
KEt = (½)mv2 = [(p2)/(2m)]
Rotational Kinetic Energy:
KEr = (½)I2
Vibrational Potential Energy:
PEv = (½)kx2
Assume that each degree of freedom has an energy
that is proportional to either a p2 or to a q2.

9. Proof Continued! Plus a similar sum of terms containing the (qi)2
With this assumption, the total energy  has the form:
Plus a similar sum of terms containing the (qi)2
For simplicity, focus on the p2 sum above:
For each particle, change the sum into an integral over
momentum, as below. It is a Gaussian & is tabulated.
Ki  Kinetic Energy
of particle i

10. Proof Continued! Finally, Z can be written: Ki  Kinetic Energy
of particle i
The system partition function Z is then proportional to the
product of integrals like above. Or, Z is proportional to P:
Finally, Z can be written:

11. For a Monatomic Ideal Gas: For a Diatomic Ideal Gas:
Use the Canonical Ensemble “Recipe” to get the
average energy per particle per independent degree of freedom:
Note! u  <>
For a Monatomic Ideal Gas:
For a Diatomic Ideal Gas: l
For a Polyatomic Ideal Gas
in which the molecules vibrate
with q different frequencies:

12. The Boltzmann Distribution
Canonical Probability Function P(E):
Defined so that P(E) dE  probability to find a
particular molecule between E & E + dE is: Z
Define: Energy Distribution Function
 Number Density  nV(E):
Defined  nV(E) dE  Number of molecules per
unit volume with energy between E & E + dE

13. Examples: Equipartition of Energy in Classical Statistical Mechanics
Free Particle: Z

14. Other Examples of the Equipartion Theorem
LC Circuit
Harmonic
Oscillator
Free Particle
in 3 D
Rotating
Rigid Body

15. Simple Harmonic Oscillator

16. Classical Ideal Monatomic Gas
For this system, it’s easy to show that The Temperature is related to the average kinetic energy. For one molecule moving with velocity v in 3 dimensions this takes the form:
Also, for each degree of freedom, it can be shown that

17. Classical Statistical Mechanics:
Canonical Ensemble Averages
Probability Function: Z
P(E) dE  probability to find a
particular molecule between E & E + dE
Normalization:

18. Z
Average Energy:
Average Velocity:

19. Classical Kinetic Theory Results
We just saw that, from the Equipartition Theorem, the kinetic energy of each particle in an ideal gas is related to the gas temperature as:
<E> = (½)mv2 = (3/2)kBT (1)
v is the thermal average velocity.
Canonical Ensemble Probability Function: Z
In this form, P(E) is known as the
Maxwell-Boltzmann Energy Distribution

20. Maxwell-Boltzmann EnergyDistribution
n(E,T) E

21. P(v) = C exp[- (½)m(v)2/(kT)] Maxwell-Boltzmann Velocity Distribution
Using <E> = (½)mv2 = (3/2)kBT along with P(E), the Probability Distribution of Energy E can be converted into a Probability Distribution of Velocity P(v).
This has the form:
P(v) = C exp[- (½)m(v)2/(kT)]
In this form, P(v) is known as the
Maxwell-Boltzmann Velocity Distribution

22. Kinetic Molecular Model for Ideal Gases
Assumptions
The gas consists of large number of individual point particles (zero size).
Particles are in constant random motion & collisions.
No forces are exerted between molecules.
From the Equipartition Theorem,
The Gas Kinetic Energy is Proportional to the Temperature in Kelvin.

23. Maxwell-Boltzmann Velocity Distribution
The Canonical Ensemble gives a distribution
of molecules in terms of Speed/Velocity or
Energy.
The 1-Dimensional Velocity Distribution
in the x-direction (ux) has the form:

24. High T
Low T

25. In Cartesian Coordinates:
3D Maxwell-Boltzmann Velocity
Distribution a  (½)[m/(kBT)]
In Cartesian Coordinates:

26. Maxwell-Boltzmann Speed Distribution
Change to spherical coordinates in Velocity
Space. Reshape the box into a sphere in
velocity space of the same volume with radius u .
V = (4/3) u3 with u2 = ux2 + uy2 + uz2
dV = dux duy duz = 4  u2 du

27. 3D Maxwell-Boltzmann Speed Distribution
Low T
High T

28. Maxwell-Boltzmann Speed Distribution
Convert the speed-distribution into an energy-distribution:  = (½)mu2, d = mu du

29. Some Velocity Values from the M-B Distribution
urms = root mean square (rms) velocity
uavg = average speed
ump = most probable velocity

30. Comparison of Velocity Values
Ratios in Terms of
urms
uavg
ump
1.73
1.60
1.41

31. Maxwell-Boltzmann Velocity Distribution

32. Maxwell-Boltzmann Speed Distribution

33. Maxwell-Boltzmann Speed Distribution

34. The Probability Density Function
The random motions of the molecules
can be characterized by a probability
distribution function.
Since the velocity directions are
uniformly distributed, we can reduce the
problem to a speed distribution function
f(v)dv which is isotropic.

35. The Probability Density Function
Let f(v)dv  fractional number of
molecules in the speed range from
v to v + dv.
A probability distribution function
has to satisfy the condition

36. The Probability Density Function
We can use the distribution function to
compute the average behavior of the
molecules: