Statistical mechanics links the microscopic properties of individual atoms and molecules to the macroscopic properties of materials.
A single macrostate can correspond to a vast number of different microstates. The central assumption of statistical mechanics is that all accessible microstates are equally probable.
Phase Space is an imaginary multi-dimensional space where every possible state of a system is represented by a single point.
The state of the entire system at any instant is represented by a Phase Point. As the system evolves over time, this point traces a trajectory in phase space.
The Thermodynamic Probability (W) of a macrostate is the total number of microstates that correspond to that macrostate.
Ludwig Boltzmann established the fundamental link between the microscopic disorder (W) and the macroscopic property of Entropy (S):
Where k is the Boltzmann constant (k = 1.38 x 10^-23 J/K). This relation shows that entropy is a measure of the statistical "randomness" or "disorder" of a system.
This law describes the distribution of particles among various energy states in a system of identical, distinguishable particles where there is no limit on the number of particles in a single state.
Where:
ni = Number of particles in energy level Ei
gi = Degeneracy (number of microstates) of the level Ei
beta = 1 / (k * T)
An Ensemble is a collection of a large number of independent systems that are macroscopically identical but in different microstates.
| Ensemble Type | Fixed Parameters | Description |
|---|---|---|
| Micro-canonical | N, V, E | Systems are isolated; total energy and particle number are constant. |
| Canonical | N, V, T | Systems can exchange energy with a heat reservoir to maintain constant temperature. |
| Grand canonical | V, T, mu | Systems can exchange both energy and particles with a reservoir. |
The Partition Function (Z) is the "bridge" between statistical mechanics and thermodynamics. It is the sum over all possible states of the Boltzmann factor:
Tip: When deriving thermodynamic quantities from the partition function, always start by defining F = -kT log Z. Almost all other variables (S, P, U) can be derived directly from the derivatives of F.