Thermal Physics
Class Information
The three laws of thermodynamics; thermodynamic potentials; ideal and non-ideal gases; phase transitions; chemical equilibrium. Introduction to equilibrium statistical mechanics; statistical definition of entropy; applications to fluids, magnetic systems, the ideal quantum gas. Prior to Fall 2017, this course was named “Thermodynamics and Kinetic Theory”.
Textbook: Schroeder
Probability
Permutation: Choose
- Note that permutation also refers to a different but related concept where
- The number of arrangements that a set can be placed in
- Note that this is just a specific case of the above where
Permutation: Choose
Product Rule
Find the probability that
Total number of configurations of the system where
For one single particle
For the system to have
The probability for a system to have
Alternatively,
Boltzmann Statistics
Partition Function
-
At any given time,
, a system of particles can be described by 6 variables: -
These 6 variables form a 6-D phase space
-
Microcanonical Ensemble: Each microstate with energy between
and are equally probable -
Canonical Ensemble: Each microstate with a particular energy is not equally probable, but follows Boltzmann-Statistics
- Heat can be exchanged
- Temperature are constant since there is a heat and particle resevoir connected to the system
-
Grand Canonical Ensemble: Each microstate with a particular energy is not equally probable
- Chemical potential,
, tells how much energy is needed to add/create a particle to the system - Heat and particles can be exchanged
- Temperature and
are constant since there is a heat and particle resevoir connected to the system
- Chemical potential,
-
The partition function gives the number of available system states
- https://physics.stackexchange.com/questions/203697/what-is-the-physical-meaning-of-the-partition-function-in-statistical-physics
- “The partition function is a measure of the volume occupied by the system in phase space. Basically, it tells you how many microstates are accessible to your system in a given ensemble”
- https://physics.stackexchange.com/questions/203697/what-is-the-physical-meaning-of-the-partition-function-in-statistical-physics
Canonical Partition Function:
The probability:
is the number of particles in state
Paramagnet
Average magnetic dipole moment,
Total Magnetization
Maxwell Speed Distribution
The partition function can be rewritten as an integral over the phase space as follows:
where the
Then the probability as a function of momentum,
The distribution of momentum,
where
Partition Function for Composite Systems
For
where
For
- where
is the single particle partition function - Only applies when
» N- When the number of available single-particle states is much greater than the number of particles
- Equivalently this formula only applies when
Quantum Statistics
Grand Canonical Partition Function
Ensemble- Thermal resevoir and particle resevoir
- Heat and particles can be exchanged
is constant
alternatively
Probability
Average Number of Particles in a State (Distribution Functions)
Let
For quantum particles
where
Let
or alternatively:
- you can use these formulas to find the distribution function for Fermions or Bosons
- Fermions: Particles cannot occupy the same state
- For some fermions, like electrons, two particles can have the energy, but they still have different states (spin up and spin down)
- Bosons: Any number of particles can occupy the same state/energy
- Fermions: Particles cannot occupy the same state
Compare with Boltzmann:
Probablity for single particle
If we have
Fermi-Dirac Statistics
- Degenerate Gas: All states below
are occupied and all states above are unoccupied- Occurs when
- Note that degeneracy here is completely different than the same word used to describe the number of states with the same energy (similar to Density of State)
- Occurs when
Density of States, Derivation for Fermi Energy, and Derivation for Number of Particles
Number of particles at a given energy,
where
The
Convert to an integral by integrating over all possible states (over the 6D phase space variables):
We divide by
Convert from momentum
- We also have
in terms of the density of states:- The density of states gives the number of single-particle states per energy
This implies
We arrive at the same density of states (DOS) from before in the Fermi Dirac Statistics Section