Quick Ref

Ensembles

Super helpful guide here

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

$$Z=\sum _{microstates(k)}{e}^{-\beta E_k}=\sum _{energies(i)}{g}_i{e}^{-\beta E_i},$$

$g$ 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:
  
  ${\displaystyle f(v)d^{3}v={\left(\frac{m}{2\pi kT}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2kT}}{d}^{3}v}$
• Expectation value
• The expected number of particles with energy
  ${\epsilon }_i$ for Maxwell–Boltzmann statistics is
  ${\displaystyle ⟨N_i⟩=\frac{g_i}{e^{({\epsilon }_i-\mu )/kT}}=\frac{N}{Z}{g}_i{e}^{-{\epsilon }_i/kT},}$
• 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
  $A/h^2\ast \frac{1}{1/\sqrt{\pi 2mkT}}=A/{\lambda }^2$
• Classical, discrete partition function (say, vibrational modes
  ${ϵ}_s=\hslash \omega (s+1/2)$)
  $$Z={\sum }_{n=0}^{\infty }{e}^{-\beta (n+12)\hslash \omega }=\frac{1}{e^{\beta \hslash \omega /2}-e^{-\beta \hslash \omega /2}}$$
• $dE=TdS-PdV+\beta dN$
• 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
• Stirling approximation:
  $ln(N!)=NlnN-N$
• $\mathrm{\Omega }$ Multiplicity = # ways to have a certain state
• $\mathrm{\Omega }$ for two state (a,b) system = N!(N_a!N_b!)
• Entropy S =
  $kln(\mathrm{\Omega })$
• 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'
    ${\displaystyle S_s=k\log \frac{dv}{dE}=S_{\text{B}}-k\log \omega }$
  • ${\displaystyle 1/T_v=d{S}_v/dE}$
  • ${\displaystyle 1/T_s=d{S}_s/dE=d{S}_{\text{B}}/dE}$
• Microcanonical pressure and chemical potential are given by:
  ${\displaystyle \frac{p}{T}=\frac{\partial S}{\partial V};\frac{\mu }{T}=-\frac{\partial S}{\partial N}}$
• Ideal gas if λ(𝑇) ≪ a
• det(
  $\left[\begin{array}{cc}A& 0\\ C& D\end{array}\right]$) = det(A)det(D)
• Lagrangian = KE - PE
• conjugated momenta
  $p_a=\partial L/\partial \dot{q_a}$
• $\overrightarrow{r}=rcos\phi \hat{x}+rsin\phi \hat{y}$
• $\hat{r}=cos\phi \hat{x}+sin\phi \hat{y}$
• $\hat{\phi }=-sin\phi \hat{x}+cos\phi \hat{y}$
• $KE=1/2m|\dot{r}{|}^2=1/2m{\dot{r}}^2+1/2m{r}^2{\dot{\phi }}^2$
• Width of distribution - probably means variance:
• Moving from binomial to gaussian:
  ${\mathrm{\Omega }}_1{\mathrm{\Omega }}_2=A{e}^{-(x-\mu {)}^2/2{\sigma }_{\mu }^2}$
• $\sum _{n=0}^{N-1}{x}^n=\frac{x^N-1}{x-1}$
• $\sum _{n=0}^{N-1}n{x}^n=x\frac{d}{dx}\sum _{n=0}^{N-1}{x}^n$
• Expected values using partition function:
  $<A>=\sum _{k}A{e}^{-\beta {ϵ}_k}/Z$
• 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