1. Classical Statistical Mechanics in the Canonical Ensemble

2. The Equipartition Theorem is Valid in Classical Stat. Mech. ONLY!!!

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

4. Classical Statistical Mechanics (ONLY!) It states:
The Equipartition Theorem in
Classical Statistical Mechanics (ONLY!) It states:
“Each degree of freedom in a system of particles contributes (½)kBT to the thermal average energy Ē of the system.”

5. 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 Ē of the system.” Note:
1. This is strictly 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.

6. We just finished an outline of the proof
In the Classical Cannonical
Ensemble, it is straighforward to
show that
The average energy of a particle per independent degree of freedom  (½)kBT.
We just finished an outline of the proof

7. The Boltzmann (or Maxwell-Boltzmann) Distribution
Start with the Canonical Ensemble Probability Function P(E):
This is defined so that P(E) dE 
probability to find a particular molecule
between E & E + dE has the form:
Z  Partition Function Z

8. The Boltzmann Distribution Define: Energy Distribution Function
Canonical Ensemble Probability
Function P(E): Z
Define: Energy Distribution Function
 Number Density  nV(E):
Defined so that  nV(E) dE  Number of
molecules per unit volume with energy
between E & E + dE

9. Examples: Equipartition of Energy in Classical Statistical Mechanics
Free Particle
(One dimension): Z

10. Equipartition Theorem Examples 1 d Harmonic Oscillator:
LC Circuit:
1 d Harmonic Oscillator:

11. Equipartition Theorem Examples Free Particle in 3 Dimensions:
Rotating Rigid Body:

12. 1d Simple Harmonic Oscillator

13. Classical Ideal Monatomic Gas
For this system, it’s easy to show
that the Temperature T is related
to the average kinetic energy. For
1 molecule moving with velocity v
in 3 d, equipartition takes the form:
For each degree of freedom, it’s easy to show:

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

15. So: Z
Average Energy:
Average Velocity:

16. 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

17. 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

18. Equipartition Theorem: 
Kinetic Molecular Model for Ideal Gases Due originally to Maxwell & Boltzmann
Assumptions
The gas consists of large number of individual point particles (zero volume).
Particles are in constant random motion & collisions.
No forces are exerted between molecules.
Equipartition Theorem: 
Gas Average Kinetic Energy is Proportional to the Temperature in Kelvin.

19. 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:

20. Maxwell-Boltzmann Velocity Distribution
High T
Low T

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

22. 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

23. 3D Maxwell-Boltzmann Speed Distribution
Low T
High T

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

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

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

27. Maxwell-Boltzmann Velocity Distribution

28. Maxwell-Boltzmann Speed Distribution

29. Maxwell-Boltzmann Speed Distribution

30. The Probability Density Function
Random motions of the molecules can be
characterized by a probability distribution
function. Since the velocity directions are
uniformly distributed, the problem reduces to a
speed distribution.
The function f(v)dv is isotropic. f(v)dv 
fractional number of mol ecules in the speed range
from v to v + dv. Of course, a probability
distribution function has to satisfy the condition:

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