1. Lecture 41 Statistical Mechanics and Boltzmann factor
What is statistical mechanics?
Microscopic state and ergodic principle
Boltzmann factor
Ideal gas in gravitational field
Equipartition of energy and heat capacity

2. Statistical mechanics
Statistical mechanics is a science that describes the properties of a thermal system starting from its microscopic structure.
It started from study of gases from molecular view.
The microscopic description of a system starts with the energy of the system in terms of microscopic variables.
𝐸 𝑟 𝑖 , 𝑝 𝑖 = 𝑖 𝑝 𝑖 2 2 𝑚 𝑖 + 𝑖𝑗 𝑉 𝑖𝑗

3. Ergodic principle
Microscopic states: defined by specifications of coordinates and momenta of all particles in the system.
Statistical hypothesis: in a thermal equilibrium with a fixed energy, all microscopic states can be accessed with equal probability.
This the fundamental principle (ergodic principle) of statistical mechanics, sometime it is called micro- canonical ensemble.

4. Practical example
If we have a fair coin, every time the probability of getting head or tail is the same, ½
If we throw coin 3 times, we have will 2^3=8 possible results (states):
HHH, HHT, HTH, THH, TTH, THT, HTT, TTT
Ergodic principle states that all results come with equal probability.

5. Microscopic state
For particles with position 𝑟 and momentum 𝑝 , the microscopic state is defined by 𝑟 and 𝑝
The combined 𝑟 and 𝑝 space is called phase space.
Any any time, the state of a particle is defined by a point in phase space.
The phase space volume element is
𝑑 3 𝑟 𝑑 3 𝑝

6. Boltzmann factor
Most of the time, we are interested in a system in contact with a thermal reservoir with temp 𝑇, in this case, the system can have various energy 𝐸.
The probability distribution for a microscopic state with energy E is
𝑷~ 𝒆 − 𝑬 𝒌𝑻
This is exponential, also called Boltzmann factor.
Taking an assumption, it is called canonical ensemble.

7. Applications: ideal gas
For ideal gas with 𝑁 particles,
𝐸= 𝑖 𝑚 𝑣 𝑖 2
thus we have
𝑃∼ Π 𝑖 𝑒 − 𝑚 𝑣 𝑖 2 2 𝑘 𝐵 𝑇
if we consider the velocity distribution of one particle, one has
𝑃∼ 𝑒 − 𝑚 𝑣 𝑘 𝐵 𝑇
this is the just the Maxwell-Boltzmann velocity distribution.

8. Ideal gas in gravitational field
Consider now an ideal gas in a gravitational field. The energy is
𝐸= 1 2 𝑚 𝑣 2 +𝑚𝑔𝑧
The Boltzmann factor becomes,
𝑝∼ 𝑒 − 𝑚𝑔𝑧 𝑘 𝐵 𝑇
Thus the density of gas decreases exponentially as 𝑧 gets large.
Since the pressure of gas depends on the density, it also decreases exponentially.

9. Gas with more complicated molecules
If we have a gas with molecules, the thermal property of the system is more complicated.
For example with a diatomic molecule, the energy consists of three parts,
Translational motion
Relative motion
Rotation.
𝐸= 𝐸 𝑡𝑟𝑎𝑛𝑠𝑙 + 𝐸 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 + 𝐸 𝑣𝑖𝑏

10. Equipartition of energy
In thermal equilibrium, energy is shared equally among all of its various forms;
for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that of its rotational motions
Any degree of freedom (such as a component of the position or velocity of a particle) has an average energy of 𝑘 𝐵 𝑇

11. Denote
𝑡: number of freedom degree for translation
𝑟: number of freedom degree for rotation
𝑠: number of freedom degree for vibration
The internal energy
𝑈 𝑚𝑜𝑙 = 𝑟+𝑡+2𝑠 2 𝑁 𝐴 𝑘 𝐵 𝑇= 𝑟+𝑡+2𝑠 2 𝑅𝑇
For each vibration, there should be a corresponding potential. Therefore one more 𝑠 is added.
For constant volume process, the heat capacity is
𝐶 𝑉 𝑚𝑜𝑙 = 𝑑 𝑈 𝑚𝑜𝑙 𝑑𝑇 = 𝑟+𝑡+2𝑠 2 𝑅

12. Monatomic and diatomic gases
Monatomic gases have only translational motion, 3 degrees of freedom
𝑈 𝑚𝑜𝑙 = 3 2 𝑁 𝐴 𝑘 𝐵 𝑇= 3 2 𝑅𝑇
Diatomic gases, 3 degrees of freedom for translational motion the center of mass, 2 degrees of freedom for rotation, 1 degree of freedom for oscillation and 1 degree of freedom for potential
𝑈 𝑚𝑜𝑙 = 7 2 𝑁 𝐴 𝑘 𝐵 𝑇= 7 2 𝑅𝑇