Statistical Mechanics

Common 'Energies'

• Internal Energy: U + Q + W
  Checks the gain/loss from any change in the system's internal state.
• Helmholtz Energy: A = U - TS
  Thermodynamic potential, measure of useful work. Obtainable from a closed thermodynamic system at constant T & V.
• Gibbs Energy: G = U + PV - TS = A + PV
  Thermodynamic potential giving the maximum amount of reversible work doable by a system at constant temperature and pressure.
  

Strictly speaking, all of these should probably factor in the particle number and chemical potential, $+\mu N$.

Ensembles:

• Canonical Ensemble:
  T=const, sytem in contact with thermal bath, energy flows between but is in thermal equilibrium; particles are stuck but heat flows.
  

  $$P = \frac{1}{\mathcal{Z}}e^{-E\beta}$$ (138)

  $$e^{-F\beta} = \int \frac{1}{h^n c}e^{-E\beta} dp_1...dq_n = \sum_i e^{-E_i\beta} = \mathcal{Z}$$ (139)

  
  
• Microcanonical Ensemble:
  Isolated system - statistical ensemble representing possible states of equal energy, multiplicity $\Omega$
  
  $$S = k\ln{\Omega}$$

• Grand Canonical Ensemble:
  System in contact with a heat bath and particle bath. Particles and energy can be swapped, with an equilibrium T.

  $$\boxed{\text{fugacity } f = e^{\mu \beta}}$$ (140)

  $$\boxed{\mathcal{Z} = \sum_i f_i e^{-\epsilon_i\beta}}$$ (141)

  
  

Equipartition Theorem
At temperature T, the average energy of a quadratic degree of freedom $=\frac{1}{2}kT$

$$U_{thermal} = N\cdot f \cdot \frac{1}{2} kT$$ (142)

$$\frac{1}{2}mv_i^2, \frac{1}{2}I\omega_i^2, \frac{1}{2}k_sx^2, .$$ (143)

Entropy, Fluctuations, Heat Capacity

$$S = k\ln{\Omega}$$ (144)

$$\Delta S = \int \frac{dQ}{T}$$ (145)

$$S = \beta(\langle E\rangle -\langle F\rangle )$$ (146)

$$S = -\sum_i \mathcal{P}_i\ln\mathcal{P}_i$$ (147)

$$S = -k Tr(\rho \ln\rho)$$ (148)

$$\langle(\Delta E)^2\rangle = \frac{\partial^2\ln \mathcal{Z}}{\partial\beta^2}$$ (149)

$$C_v = \frac{\partial\langle E\rangle }{\partial T} = \frac{\beta}{T}\langle(\Delta E )^2\rangle$$ (150)

Blackbody Radiation

$$\mathcal{Z} = \sum_n e^{-\beta \hbar \omega n} = \frac{1}{1 - e^{-\beta \hbar \omega}}$$ (151)

$$d^3n = \frac{V}{h^3} \int d^3p = \frac{4\pi V}{h^3}\hbar^3 \int k^2 dk$$ (152)

$$(p=\hbar k, \omega=ck) \hbar = c = 1$$

$$u = -\frac{\partial\ln{\mathcal{Z}}}{\partial\beta}$$

$$\rightarrow U = (2) \frac{4\pi V}{(2\pi)^3}\int \frac{\omega k^2}{1 - e^{-\beta\omega}}dke^{-\beta\omega}$$

$$\frac{U}{V} = \frac{8\pi}{(2\pi)^3}\int \frac{k^3dk}{1-e^{-\beta k}}e^{-\beta k}$$ (153)

$$\beta k = x, \beta dk = dx;$$

$$\int \frac{x^3dx}{e^x - 1} = ...$$

$$\boxed{\frac{U}{V} = \frac{8\pi}{(2\pi)^3}(kT)^4\int\frac{x^3dx}{e^x-1}}$$ (154)

$$\boxed{ = \frac{\pi^2(kT)^4}{15\hbar^3 c^3} \equiv \sigma T^4}$$ (155)

With a 2 added somewhere along the way to properly count polarization modes.

Fugacity, q-potential, average particle number:

$$f = e^{\mu\beta}$$ (156)

$$\mathcal{Z}=\sum_{N=0}^{\infty} \Omega \big[ f e^{- \beta E} \big]^N$$ (157)

$$\boxed{\langle n\rangle = \frac{1}{\beta} \frac{\partial\ln{\mathcal{Z}}}{\partial\mu}}$$ (158)