| Binomial probability eqn | p_N (k)=N!/k!(N−k)! p^k (1−p)^(N−k) |
| Macrostate definition | Specification of the state of a system based only on bulk properties containing minimal information to describe the system
Hides a lot of the underlying physics, as usually, we have averaged away much of the detail |
| Microstate definition | Complete specification of the state of the system, which is consistent with the theory
Things usually change very quickly, switching between microstates
The amount of information needed to describe the microstates of a system is huge |
| 3 Types of Ensembles (macrostates with fixed bulk properties) | a. Microcanonical ensemble - macrostate where N, U and V are unchanging (not very easy = number of particles, energy, and volume unchanging)
b. Canonical ensemble - macrostate where N, T and V are unchanging (not quite MTV as it sounds least posh and expensive)
c. Grand Canonical ensemble - macrostate where µ, T and V are unchanging (µTV as it sounds most posh and expensive - a boujee MTV) - in this µ is the chemical potential or Fermi level |
| Distributions with total energy U and N particles key things to remember | Number of microstates in a distribution option Di:
- Ω_i=N!∏1/(n_j !)
(To find the total number of microstates sum the Ω_i values)
You can then get the probability of a particle being in D_i using Ω_i/Ω
You can also find the average number of particles in a state ni using ⟨n_i ⟩=(∑n_(D_i ) Ω_i)/Ω, where n_(D_i ) is the ni in distribution Di |
| Entropy from microstates | S = k_b ln(Ω) |
| 1. Partition function | sum of exp(-beta*E) , where beta is 1/(KT). epsilon is energy |
| 10. Stirling approximation | In N!= NInN-N |
| Partition function overall dimensions | Dimensionless |
| Z for a particle system with degeneracy | Z = sum of the product of degeneracy and normal exponential term |
| Characteristic Rotational and Vibrational Temperatures | Rotational - k_b T = hbar^2/(2*I)
Vibrational - k_b T = hbar*omega |
| Approximating Partition Functions at high and low T justification | High T - sum turns to integral because high energy values are occupied (so sum becomes infinitessimal)
Low T - only need to consider first two terms of series as at low T, only low energy levels will be occupied |
| Internal Energy vs Free energy (equations) | U = Nkd/db ln(Z)
F = -NkT ln(Z) |
