---
tags: stat mech
---
# Quick Ref

## Ensembles
[Super helpful guide here](https://scholar.harvard.edu/files/schwartz/files/7-ensembles.pdf)

| Name | Fixed | Free | Z | Notes
| ----------- | ----------- | ----------- | ----------- | ----------- |
| Canonical | $N,V,T$ | $E_t$| $\displaystyle Z=\sum _{i}e^{-\beta E_{i}}$ | Can exchange w/ reservoir
| Micro-canonical | $N,V,E_t$ | $T$ | | Isolated/Closed System, No reservoir
| Gibbs | $N, P,$ ($H$ or $T$) | $V$ | $\displaystyle Z=\sum _{\mu ,\mathbf {x} }\exp[\beta \mathbf {J} \cdot \mathbf {x} -\beta {\mathcal {H}}(\mu )]$
| Grand canonical | $\mu,V,T$ | $N$ | $\displaystyle Z = \sum_{microstates (k)}{e^{\beta N_k \mu-\beta E_k}}$ | - in thermodynamic equilibrium (thermal and chemical) with a reservoir - system can exchange energy and particles with a reservoir - SOMETIMES this partition fxn is called the "GIBBS SUM"

## put this somewhere else
$$Z = \sum_{microstates (k)}{e^{-\beta E_k}} = \sum_{energies (i)}{g_ie^{-\beta E_i}},$$ $g$ is degeneracy of energy i

$Z_m = Z_R Z_T Z_v$ if rotational, vibrational and translational motions do not mix and are independent

$Z_N = (Z_1)^N$ if molecules don't interact with eachother and exchange energy w/ reservoir independently

$Z_N = \frac{1}{N!}(Z_1)^N$ introduce factor if regarding external degrees of freedom.... e.g. don't use if only inernal D.O.F. like vibration do use if external D.O.F. like translation?

Continuous, classical partition function:
$\displaystyle Z={\frac {1}{h^{3}}}\int e^{-\beta H(q,p)}\,\mathrm {d} ^{3}q\,\mathrm {d} ^{3}p$

## idec anymore
- PDF change of variables 
- Continuous PDF - 
- Both PDF and Partition function: if variables are independent, the joint value is product of the individual parts
- Maxwell-Boltzmann PDF:
$\displaystyle f(v)~d^{3}v=\left({\frac {m}{2\pi kT}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2kT}}}~d^{3}v$
- Expectation value 
- The expected number of particles with energy $ε_i$ for Maxwell–Boltzmann statistics is $\displaystyle \langle N_{i}\rangle ={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/kT}}}={\frac {N}{Z}}\,g_{i}e^{-\varepsilon _{i}/kT},$
- partition function is the normalization factor for a Boltzmann distribution: $P(s) = \frac{1}{Z}e^{−E(s)/kT}$, s = state (see https://qzhu2017.github.io/assets/pdfs/courses/phys-467-667/Lec21.pdf)
- Z = integral(rho dx dy dp_x dp_y) etc
- Classical, continuous partition function (translational only) Z = sum(exp(KE/kT)) -> continuous = integral 1/h^(# dimensions) * exp(- KE/kT) dxdydp_xdp_y = simplifies to $A/h^2*\frac{1}{1/\sqrt{\pi 2 m k T}} = A/\lambda^2$
- Classical, discrete partition function (say, vibrational modes $\epsilon_s = \hbar \omega (s+1/2)$) $$Z=∑_{n=0}^{∞}e^{−β(n+12)ℏω} = \frac{1}{e^{\beta ℏω/2}-e^{-\beta ℏω/2}}$$
- $dE = TdS - PdV + \beta dN$
- Helmholtz Free Energy F = E - TS = -kTln(Z) (for constant T (and V))
- Entropy = $-(\partial F/\partial T)|_{V,N}$
- E = F + TS
- Chemical Potential = $\mu$ = $-(\partial F/\partial N)|_{V,T}$
- $<E> = \partial/\partial \beta ln(Z)$
- Specific Heat dQ/dT = TdS/dT ( = dE/dT if constant N,V - see above for dE)
- Summation simplification (geometric series): $\sum_{x=0}^{oo}x^n = \frac{1}{1-x}$
[see example](https://blog.cupcakephysics.com/thermodynamics%20and%20statistical%20physics/2015/10/04/thermodynamic-properties-of-the-quantum-harmonic-oscillator.html)
- Stirling approximation: $ln(N!) = NlnN -N$
- $\Omega$ Multiplicity = # ways to have a certain state
- $\Omega$ for two state (a,b) system = N!\(N_a!N_b!)
- Entropy S = $kln(\Omega)$
- Gibbs Free Energy G = H - TS = (E - $\mathcal{J}x$) - TS = -kTln(Z) (for constant P and T)
- S = $-(\partial G/\partial T)|_{f,N}$
- Microcanonical wont have Z so much as S:
- the Boltzmann entropy $\displaystyle S_{\text{B}}=k\log W=k\log \left(\omega {\frac {dv}{dE}}\right)$
- the 'volume entropy' $\displaystyle S_{v}=k\log v$
- the 'surface entropy' $\displaystyle S_{s}=k\log {\frac {dv}{dE}}=S_{\text{B}}-k\log \omega$
- $\displaystyle 1/T_{v}=dS_{v}/dE$
- $\displaystyle 1/T_{s}=dS_{s}/dE=dS_{\text{B}}/dE$
- Microcanonical pressure and chemical potential are given by: $\displaystyle {\frac {p}{T}}={\frac {\partial S}{\partial V}};\qquad {\frac {\mu }{T}}=-{\frac {\partial S}{\partial N}}$
- Ideal gas if λ(𝑇) ≪ a
- det($\begin{bmatrix}
A & 0 \\
C & D
\end{bmatrix}$) = det(A)det(D)
- Lagrangian = KE - PE
- conjugated momenta $p_a = \partial L/\partial\dot{q_a}$
- $\vec{r} = rcos\varphi\hat{x}+rsin\varphi\hat{y}$
- $\hat{r} = cos\varphi\hat{x}+sin\varphi\hat{y}$
- $\hat{\varphi} = -sin\varphi\hat{x}+cos\varphi\hat{y}$
- $KE = 1/2m|\dot{r}|^2 = 1/2m\dot{r}^2+1/2mr^2\dot{\varphi}^2$
- Width of distribution - probably means variance:
- Moving from binomial to gaussian: $\Omega_1\Omega_2 = Ae^{-(x-\mu)^2/2\sigma_{\mu}^2}$
- $\sum_{n=0}^{N-1}x^n = \frac{x^N-1}{x-1}$
- $\sum_{n=0}^{N-1}nx^n = x\frac{d}{dx}\sum_{n=0}^{N-1}x^n$
- Expected values using partition function: $<A> = \sum_kAe^{-\beta \epsilon_k}/Z$
- Starting from 
we can deduce that 
- Taylor expansions:
- Partition function (canonical) for ideal gas
- Effective mass particle in liquid $m_e = V_{particle}(\rho_{particle}-\rho_{liquid}) = m -V_{particle} \rho_{liquid}$
- heat capacity
- Hydrogen energy levels 
- degeneracy hydrogen energy levels