PH503
Motivations
Classical Mechanics
- 6N DoF $(\vec{q},\vec{p})=\{q_1,\cdots,q_{3N},p_1,p_{3N}\}$
- Canonical eqs. of motion $$\dot{q}_i=\frac{\partial \mathcal{H}(\vec{p},\vec{q})}{\partial p_i},\quad \dot{p}_i=\frac{\partial \mathcal{H}}{\partial q_i},\quad \text{for}\ i=1,\cdots, 3N$$
time reversal $t\rightarrow -t$ $(\vec{q},\vec{p},t)\rightarrow (\vec{q},-\vec{p},-t)$
Thermodynamics
- a few DoF, $E$, $V$, $N$, $T$, $P$, $\cdots$
- phenomenological laws $\Delta S\ge 0,\cdots$ (time's direction)
- $S,E,W,Q$
How link these?
- Kinetic Theory $\rho (\vec{q},\vec{p},t)$ $\rightarrow \frac{\partial\rho}{\partial t}$ However, we don't learn time dependent, but only equilibrium.
1. Ensemble Theory
1.1 Statistical Ensemble
- consider an isolated microscopic system of $N$ particles contained in a volume $V$
at time $t$, $\{q_1,\cdots,q_{3N},p_1,\cdots,p_{3N}\}\equiv \vec{\mu}(t)=(\vec{q}(t),\vec{p}(t)$
phase space
- Macroscopic state
$E,V,N,T,P,\cdots$ small number of state functions
for a $M=(E,V,N)$, $\vec{\mu}_1,\vec{\mu}_2,\cdots,\vec{\mu}_m$ are same macroscopic state.
- Statistical ensemble
a collection of a large number of copies of identical systems, each constructed to have the given macroscopic state.
$\rightarrow$ a continuous distribution of function of representative points ($\vec{\mu}$) in Phase space.
- phase space density (distribution function) $\rho(\vec{q},\vec{p},t)d\vec{q}d\vec{p}$; number of RPs in an infinitesimal volume element around $(\vec{p},\vec{q})$
$$\rho(\vec{q},\vec{p},t)=\lim_{N\rightarrow\infty}dN(\vec{q},\vec{p},t)/N \Rightarrow \int \rho(\vec{q},\vec{p},t)d\vec{q}d\vec{p}=\prod^{3N}_{I=1}dq_i dp_i\equiv d\Gamma$$
- ensemble average $\mathcal{O}(\vec{p},\vec{q})$
$\langle \mathcal{O}\rangle =\int d\vec{q}d\vec{p}\mathcal{O}(\vec{p},\vec{q})\rho(\vec{q},\vec{p},t)$
1.2 Liouville theorem
$\vec{\mu}=(\vec{q}(t),\vec{p}(t))$ Canonical equations of motion $$\dot{q}_i=\frac{\partial\mathcal{H}(\vec{p},\vec{q})}{\partial p_i},\quad \dot{p}_i=-\frac{\partial\mathcal{H}(\vec{p},\vec{q})}{\partial q_i}$$
$\rho(\vec{q},\vec{p},t)$ phase space distribution function
$$\frac{\partial \rho}{\partial t}=?$$
from the continuity condition, $$\frac{d}{dt}\int_Vd\vec{p}d\vec{q}\rho(\vec{p},\vec{q},t)\\=\int_Vd\Gamma\frac{\partial\rho}{\partial t}=-\int_{\partial V}(\rho\vec{v})\cdot \hat{n}ds=-\int_Vd\Gamma\nabla\cdot(\rho\vec{v})$$
$$\int d\vec{p}d\vec{q}\left[ \frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\vec{v})\right]=0$$
$$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\cdot \vec{v})=0$$
$$\rho(\nabla\cdot\vec{v})+(\nabla\rho)\cdot\vec{v}\\ =\sum^{3N}_{i=1}\rho\left(\frac{\partial\dot{q}_i}{\partial q_i}+\frac{\partial \dot{p}_i}{\partial p_i}\right) + \sum^{3N}_{i=1}\left(\frac{\partial\rho}{\partial \dot{q}_i}+ \cdots\right)$$
$$\frac{\partial\rho}{\partial t}+\{ \rho,\mathcal{H}\}=0$$
$$\frac{d\rho}{dt}=\frac{\partial \rho}{\partial t}+\vec{v}\cdot \nabla\rho=0$$
$$\frac{d\rho}{dt}=\frac{\partial \rho}{\partial t}+\frac{\partial \rho}{\partial q}\frac{\partial \vec{q}}{\partial t}+\frac{\partial\rho}{\partial \vec{p}}\frac{\partial \vec{p}}{\partial t}=\frac{\partial \rho}{\partial t}+\vec{v}\cdot\nabla\rho=0$$
1.3 Statistical postulate
ensemble average of $\mathcal{O}(\vec{p},\vec{q})$
$$\frac{d}{dt}\langle \mathcal{O}\rangle (t)=\int d\vec{q}\vec{p}\frac{\partial \rho}{\partial t}(\vec{q},\vec{p},t)\mathcal{O})\vec{q},\vec{p})$$
$$=\sum^{3N}{i=1} \int d\vec{q}d\vec{p}\mathcal{O}\left( \frac{\partial \rho}{\partial q_i}\frac{\partial\mathcal{H}}{\partial p_i}-\frac{\partial \rho}{\partial p_i}\frac{\partial\mathcal{H}}{\partial q_i} \right)$$
$$\int dq_i\frac{\partial}{\partial q_i}\left(\mathcal{O}\rho\frac{\partial\mathcal{H}}{\partial p_i}\right)-\int dq_i\rho\frac{\partial\mathcal{O}}{\partial q_i}\frac{\partial\mathcal{H}}{\partial p_i}-\int dq_i\rho\mathcal{O}\frac{\partial^2\mathcal{H}}{\partial q_i\partial p_i}$$
$$-\int dp_i \frac{\partial}{\partial p_i}\left( \mathcal{O}\rho\frac{\partial\mathcal{H}}{\partial q_i}\right)+\int dp_i\rho \frac{\partial \mathcal{O}}{\partial p_i}\frac{\partial \mathcal{H}}{\partial q_i}+\int dp_i\rho\mathcal{O}\frac{\partial^2\mathcal{H}}{\partial p_i\partial q_i}$$
$$\sum_{i=1}\int d\Gamma \rho\left( \frac{\partial \mathcal{O}}{\partial q_i}\frac{\partial\mathcal{H}}{\partial p_i}-\frac{\partial\mathcal{O}}{\partial p_i}\frac{\partial\mathcal{H}}{\partial q_i}\right)$$
$$=\langle \{\mathcal{O},\mathcal{H}\}\rangle =\int d\Gamma \rho \{\mathcal{O},\mathcal{H}\}$$
$=0$ in equilibrium
for an equilibrium ensemble $$\frac{\partial\rho}{\partial t}=0$$
$\Rightarrow \{\rho,\mathcal{H}\}=0$
$\Rightarrow \rho_{eq} (\vec{q},\vec{p})=\rho(\mathcal{H}(\vec{q},\vec{p}))$
$\rho=\text{const}$ for $\mathcal{H}(\vec{q},\vec{p})=E$
postulate of equal a priori probability
in thermodynamic equlibrium, an isolated system (of fixed $E,V,N$) is equally likely to be any state satisfying the macroscopic condition of the system.
1.4 Ergodic hypothesis
Temporal average: $\bar{f}=\lim_{T\rightarrow\infty}\frac{1}{T}\int^T_0dt f(\vec{q},\vec{p},t)$
Ensemble average: $\langle f\rangle =\int d\vec{q}d\vec{p} f\rho_{eq}$
Ergodic hypothesis: after a sufficiently long time, the locus of RP of a system passes through every point inside allowed $P$-space with equal number of times
9/8은 안옴
9/10
1.6 Entropy
$S=k_0\ln\Omega$
$\Delta S=S(E-p\delta V,V+\delta V)-S(E,V) =-p(\frac{\partial S}{\partial E})_{V,N}\delta V+(\frac{\partial S}{\partial V})_{E,N}\delta V =[-\frac{P}{T}+(\frac{\partial S}{\partial V})_{E,N}]\delta V$
thermodynamic equilibrium
$=0\Rightarrow (\frac{\partial S}{\partial V})_{E,N}=\frac{P}{T}$
$dS(E,V) =(\frac{\partial S}{\partial E})_V dE+(\frac{\partial S}{\partial V})_EdV$
$=\frac{dE}{T}+\frac{P}{T}dV$
$dE=TdS-pdV$
2nd law
$\Omega=\Omega_1(E_1^*)\Omega_2(E-E_1^*)\ge \Omega_1(E_1^0)\Omega_2(E-E_1^0)$
$\Delta S=[S_1(E^*)1)+S_2(E_2^*)]-[S_1(E_1^0)+S_2(E_2^0)]\ge 0$
($E_2^*=E-E_1^*$)
$\delta S=(\frac{\partial S_1}{\partial E_1})\delta E_1+(\frac{\partial S_2}{\partial E_2})\delta E_2$
$=[\frac{1}{T_1}-\frac{1}{T_2}]\delta E_1\ge 0$
($T_1<T_2,\ \delta E_1\ge0$ $T_1> T_2,\ \delta E_1<0$)
1.7 The ideal gas
$\rho_{mc}=\frac{1}{\Omega}$
consider a classical ideal gas of $N$ particles confined in $V$.
$\mathcal{H}=\sum^N_{\alpha=1}\frac{\vec{p}_\alpha^2}{2m}$
phase space volume ($\mathcal{H}<E$)
$\sum(E)=\int d\vec{q}_1\cdots d\vec{q}_N\int d\vec{p}_1\cdots d\vec{p}_N =V^N\int_{\mathcal{H}\le E} d\vec{p}_1\cdots d\vec{p}_N$
Gaussian int. $I_d\int^\infty_{-\infty}dx_1\cdots\int^\infty_{-\infty} e^{-(x_1^2+\cdots+x^2_d)}=(\int^\infty_{-\infty}dx e^{-x^2})^d=(\pi^{1/2})^d$
$V_d=c_\alpha R^\alpha$, $c_\alpha$: volume of a unit sphere
$A_d=S_dR^{d-1}$, $S_d$: generalized solid angle
$dV_d=d C_d R^{d-1}dR=A_d dR$ finally, $S_d=dC_d$
$I_d=\int dV_d(R)e^{-R^2}=\int^\infty_{-\infty}dR\ A_d(R)e^{-R^2}=\int^\infty_0 dR\ S_dR^{d-1}e^{-R^2}=\frac{S_d}{2}\int^\infty_0dt t^{\frac{\alpha}{2}-1}e^{-t}$
Use integration as $\Gamma (d/2)$, $S_d=\frac{2\pi^{d/2}}{(\frac{d}{2}-1)!}$, $C_d=S_d/d=\frac{\pi^{d/2}}{(d/2)!}$
$\Sigma(E)=V^N\times V_{d=3N}(R=\sqrt{2mE})=\frac{V^N}{h^{3N}}\frac{\pi^{3N/2}}{(3N/2)!}(2mE)^{3N/2}$
$S(E,V,N)=k_B\ln \Omega=k_BN(\ln(\frac{V}{h^3})+\frac{3}{2}\ln(2\pi mE))-k_B\ln(\frac{3N}{2})!=k_BN(\ln[\frac{V}{h^3}\frac{(4\pi mE)^{3/2}}{(3N)^{3/2}}]+\frac{3/2})$
Thermodynamics
- E?
$E(S,V,N)=\frac{3h^2N}{4\pi mV^{2/3}}\exp(\frac{2S}{3Nk_B}-1)$
So, $S(E,V,N)=k_BN(\ln V+\ln(\frac{2\pi m k_BT}{h^2})^{3/2}+\frac{3/2})$
$\lambda_{th}^2=\frac{h^2}{2\pi m k_B T}$
- temp
$\frac{1}{T}=(\frac{\partial S}{\partial E})_{V,N}=\frac{3}{2}\frac{k_BN}{E}\ \Rightarrow E=\frac{3}{2}Nk_BT$
- pressure
$\frac{P}{T}=(\frac{\partial S}{\partial V})_{E,N}=\frac{k_BN}{V}\ \Rightarrow PV=Nk_BT$
- heat capacities
Use $dQ=dE-dW=dE+pdV=d(E+pV)$
$C_V=(\frac{dQ}{dT})_V=(\frac{dE}{dT})_V=\frac{3}{2}Nk_B$
$C_P=(\frac{dQ}{dT}_P=\frac{5}{2}Nk_B$
$\gamma=\frac{C_P}{C_V}=\frac{5}{3}$
Reversible adiabatic process ($dQ=0$)
$(dE)_{ada}=-pdV=\frac{-Nk_BT}{V}dV=-\frac{2}{3}\frac{E}{V}dV$
$E\propto V^{-2/3}$
9/15
1.8 The Gibbs Paradox
for a classical ideal gas $$S(E,V,N)=k_B\ln \Omega(E,V,N)=Nk_B\ln [\frac{V}{N^3}(\frac{4\pi emE}{3N})^{3/2}] =Nk_B [\ln V + \frac{3}{2}(1+\ln (\frac{2\pi mk_BT}{h^2}))]=Nk_B[\ln V+\sigma(T)]$$
mixing of two gases
$V_1,N_1+V_2,N_2$
$S_i=S_1+S_2=N_1k_B\ln V_1+N_1k_B\sigma(T)+N_2k_B\ln V_2+N_2 k_B\sigma(T)$
$S_f=(N_1+N_2)k_B \ln (V_1+V_2)+(N_1+N_2)\sigma(T)$
$\Delta S=S_f-S_i=N_1k_B\ln (\frac{V}{V_1})+N_2k_B\ln (\frac{V}{V_2})>0$
If two identical gases, $\Delta S=0$. However, above formula is not changed. Then What is difference?
We didn't consider indistinguishability. $\Omega\rightarrow \frac{\Omega}{N!}$
Change $V_1\rightarrow \frac{V_1}{N_1},V_2\rightarrow \frac{V_2}{N_2},V\rightarrow\frac{V}{N}$
Then, above formula becomes 0.
1.9 The classical Harmonic Oscillators
- Consider a 1d HO: $\mathcal{H}=\frac{P^2}{2m}+\frac{1}{2}m\omega^2q^2$
$q(t)=A\cos(\omega t+\phi)$
$p(t)=m\dot{q}=Am\omega \sin(wt+\phi)$
$$\frac{p^2}{2mE}+\frac{q^2}{2E/m\omega^2}=1=\pi\sqrt{\frac{2E}{m\omega+^2}}\sqrt{2mE}=\frac{2\pi E}{\omega}=2\pi K=h$$
$$\Gamma(E)\approx \frac{\partial \Sigma}{\partial E}\Delta=\frac{2\pi\Delta}{\omega}$$
??
$$\Sigma(E)=\int \vec{q}\vec{p}$$
- Consider classical N HO's. (Id).
$$\mathcal{H}=\sum^N_{I=1}(\frac{p_i^2}{2m}+\frac{1}{2}m\omega^2g^2_i)=E$$
$\Omega(E)=\frac{1}{h^N}\int_{x=E} dq_1\cdots dq_N=\frac{1}{h^3}(\frac{1}{m\omega})^N=C_{d-2N}(R)^{3N}=C_{2N}(\sqrt{2mE})^{2N}=\frac{\pi^N}{N!} (2mE)^N$
S(E)=k_B\ln \Omega(E,V,N)=k_BN[\ln (\frac{E}{Nh\omega}+1]$
$\frac{1}{T}=)(\frac{\partial S}{\partial E})_{V,N}=\frac{Nk_B}{E}$
$\frac{P}{T}=(\frac{\partial S}{\partial V})_{E,N}$
$C_V=(dQ/dT)=\frac{dE}{dT}=\frac{dE}{dT}=Nk_B$
$C_\gamma=\frac{\partial 5}{d Q}dT_P=\frac{d}{dt}(E+pV)=Nk_B$
(9/17)
2. Canonical Ensemlbe
2.1 Introduction
micro microcanonical ensemble
- (E,V,N)
- $\Omega(E,V,N) \rightarrow S=k_B\ln \Omega$
- $\frac{1}{T}=(\frac{\partial S}{\partial E})_{V,N}\ \frac{P}{T}=(\frac{\partial S}{\partial V})_{E,N}$
canonical emsemble
- (T,V,N)
$E_{tot}=\mathcal{H}_S(\vec{x}_S)+\mathcal{H}_R(\vec{x}_R)=const$ with $\vec{x}_S=(\vec{p}_S,\vec{q}_S)$
$\Omega_{tot}(E_{tot})=\sum_{E_S} \Omega_S(E_S)\Omega_R(E_{tot}-E_S)$
$p(\vec{x}_S,\vec{x}_R)=\frac{1}{\Omega_{tot}})\delta(E_{tot}-\mathcal{H}_S(\vec{x}_S)-\mathcal{H}_R(\vec{x}_R)$
$p(\vec{x}_S)=\sum_{\{\vec{x}_R\} }p(\vec{x}_S,\vec{x}_R)=\frac{1}{\Omega_tot}\sum_{\vec{x}_R}\delta(E_{tot}-\mathcal{H}_S-\mathcal{H}_R)$
$p(\vec{x}_S)=\frac{1}{\Omega}\Omega_R(E_{tot}-\mathcal{H}(\vec{x}_S))$
$p(\vec{x}_S)\propto \exp (-\frac{\mathcal{H}_S(\vec{x}_S)}{k_BT})$
$p(\vec{x}_S)=\frac{1}{Z}\exp (-\beta\mathcal{H}_S(\vec{x}_S))$ where $\beta=\frac{1}{k_BT}$, $Z=\sum_{\{\vec{x}_S\}} \exp (-\beta \mathcal{H}_S(\vec{x}_S))$
2.2 Partition function
$Z=\sum_{\{\vec{x}_S\}} \exp (-\beta \mathcal{H}_S(\vec{x}_S))=\frac{1}{N!}\frac{1}{h^{3N}}\int d^{3N}q d^{3N}p \exp(-\beta\mathcal{H}_S(\vec{p}_S,\vec{q}_S))$
two independent systems
$\mathcal{H}_{tot}=\mathcal{H}_1(\vec{x}_1)+\mathcal{H}_2(\vec{x}_2)$
$Z=\sum_{\vec{x}_1,\vec{x}_2}\exp (-\beta(\mathcal{H}_1(\vec{x}_1)+\mathcal{H}(\vec{x}_2)))=\sum_{\{\vec{x}_1\}}\exp(-\beta\mathcal{H}_1(\vec{x}_1)\sum_{\{\vec{x}_2\} }\exp(-\beta\mathcal{H}_2(\vec{x}_2))=Z_1Z_2$
$Z_N=Z_1^N$
$E\rightarrow$ a random variable
average energy $\langle E\rangle=\sum_{\vec{x}}\mathcal{H}(\vec{x})p(\vec{x})=\sum_\vec{x}\mathcal{H}(\vec{x})\frac{1}{Z}e^{-\beta\mathcal{H}(\vec{x})}=\frac{1}{Z}\sum_\vec{x}(-\frac{\partial}{\partial \beta})e^{-\beta\mathcal{H}(\vec{x})}=-\frac{\partial}{\partial \beta}\ln Z$
Change $S$ to $E$.
$F=E-TS$, $dF=dE-d(TS)=TdS-pdV+\mu dN-(TdS+SdT)$
$dF(T,V,N)=-SdT-pdV+\mu dN$
Helmholtz free energy
$S=-(\frac{\partial F}{\partial T})_{V,N}$, $p=-(\frac{\partial F}{\partial V})_{T,N}$, $\mu=(\frac{\partial F}{\partial N})_{T,V}$
$E=F+TS=F-T(\frac{\partial F}{\partial T})_{V,N}=-T^2\frac{\partial }{\partial T}(\frac{F}{T})=-\frac{\partial}{\partial \beta}(\beta F)$
$F=-k_BT\ln Z(T,V,N)$
analogy
ME
statistical mechanics: $\Omega(E,V,N)=\sum_\vec{x} \delta(E-\mathcal{H}(\vec{x}))$
thermodynamic potential: $S(E,V,N)=k_B\ln \Omega(E,V,N)$
CE
statistical mechanics: $Z(T,V,N)=\sum_\vec{x}e^{-\beta \mathcal{H}(\vec{x})}$
thermodynamic potential: $F(T,V,N)=-k_BT\ln Z(T,V,N)$
2.3 Ensemble equivalence
$\langle E\rangle $ vs $E_{mc}$
prob.distribution of $E$
$p(E)=\sum_\vec{x}p(\vec{x})\delta(E-\mathcal{H}(\vec{x}))=\sum_\vec{x}\frac{1}{Z}e^{-\beta\mathcal{H}(\vec{x})}\delta(E-\mathcal{H}(\vec{x}))=\frac{1}{Z}e^{-\beta E}\sum_\vec{x}\delta(E-\mathcal{H}(\vec{x}))=\frac{1}{Z}e^{-\beta E}\Omega(E)=\frac{1}{Z}e^{-\beta(E-TS)}=\frac{1}{Z}e^{-\beta F(E)}$
$Z=\sum_E e^{-\beta(E-TS)}\sim \sum_E e^{-N\phi_E}\simeq_{N\rightarrow \infty}e^{-\beta(E-TS)}|_{E=E^*}=e^{-\beta F(E^*)}$
$F=-k_BT\ln Z=-k_B\ln(\sum_\vec{x}e^{-\beta\mathcal{H}(\vec{x})})=-k_BT\ln(\sum_E\Omega(E)e^{-\beta E})=-k_BT\ln(\sum_E e^{-\beta(E-TS)})\simeq -k_BT\ln e^{-\beta(E^*-TS(E^*))}\simeq E^*-TS$
$E^*?$
$\frac{\partial}{\partial E}(\Omega(E)e^{-\beta E})|_{E=E^*}=0$
$\frac{\partial \Omega}{\partial E}e^{-\beta E}|_{E=E^*}-\beta \Omega e^{-\beta E}|_{E=E^*}=0$
$\frac{1}{\Omega}\frac{\partial\Omega}{\partial E}|_{E=E^*}=\beta $
$\frac{\partial}{\partial E}(k_B\ln \Omega)|_{E^*}=\frac{1}{T}$
$(\frac{\partial S}{\partial E})_{E=E^*}=\frac{1}{T}$
different from micocanonical ensemble, $T$ is given and $E$ is not given.
9/22
2.4 Moments & Cumulates
- consider a random variable $x$ a set of possible statistics, $\mathcal{S}=\{-\infty<x<\infty\}$
$$P(A)=\lim_{N\rightarrow \infty} \frac{N_A}{N}\ (\text{objective})$$
$$P(A) \ (\text{subjective})$$
- probability density function (PDF) $$P(x)dx\equiv \text{prob}[ X\in [x,x+dx]]$$ ($\int_{-\infty}^\infty dx\ P(x)=1$
- average of $f(x)$ $$\langle f\rangle =\int dx\ f(x)P(x)$$
- moments $$\mu_n\equiv \langle x^n\rangle =\int dx\ x^n P(x)$$
- characteristic function $$\hat{P}(k)=\langle e^{-ikx}\rangle =\int dx\ e^{-ikx}p(x)$$ ($P(x)=\frac{1}{2\pi}\int dx\ e^{ikx}\hat{P}(k)$)
$$\hat{P}(k)=\langle e^{-ikx}\rnagle =\langle \sum^\infty_{n=0}\frac{(-ik)^n}{n!}x^n\rangle =\sum^\infty_{n=0}\frac{(-ik)^n}{n!}\langle x^n\rangle $$
$$\rightarrow \ \langle x^n\rangle =\frac{\partial^n}{\partial (-ik)^n} \hat{P}(k)|_{k=0}$$
- cumulant generating function $$\ln \hat{P}(k)=\ln \langle e^{-ikx}\rangle =\sum^infty_{n=0}\frac{(-ik)^n}{n!}\langle x^n\rangle_c$$
$$\rightarrow\ \langle x^m\rangle_c=\frac{\partial^n}{\partial (-ik)^n}\ln \tilde{P}(k)|_{k=0}$$
$$\langle x^1\rangle_c=\frac{\partial}{|partial(-ik)}\ln \tilde{P}(k)|_{k=0}=\frac{1}{\tilde{P}(k)}\frac{\partial \tilde{P}}{\partial (-ik)}|_{k=0}=\langle x^1\rangle$$
$$\langle x^2\rangle_c=\frac{\partial}{\partial (-ik)} \frac{\partial}{\partial (-ik)} \ln \tilde{P}(k)|_{k=0}=\frac{\partial}{\partial (-ik)} [\frac{1}{\tilde{P}}\frac{\partial \tilde{P}}{\partial (-ik)}]|_{k=0}=\frac{1}{\tilde{P}}\frac{\partial^2\tilde{P}}{\partial(-ik)^2}|_{k=0}-\frac{1}{\tilde{P}^2}(\frac{\partial\tilde{P}}{\partial(-ik)})^2|_{k=0}=\langle x^2\rangle -\langle x\rangle^2$$
$$\langle x^3\rangle_c=\langle x^3\rangle -3\langle x^2\rangle\langle x\rangle + 2\langle x\rangle^3$$
- Gaussian pdf $$P(x)=\frac{1}{\sqrt{2\pi \sigma^2}}\exp(-\frac{(x-\mu)^2}{2\sigma^2})$$
$$\tilde{P}(k)=\int dx e^{-ikx}P(x)=\frac{1}{\sqrt{2\pi \sigma^2}}\int dx e^{-ikx-\frac{(x-\mu)^2}{2\sigma^2}}$$ set $x-\mu=y$.
$$=\int dy e^{-\frac{1}{2\sigma^2}(y+i\sigma^2 k)^2-\frac{\sigma^2k^2}{2}}=e^{-ik\mu -\frac{1}{2}\sigma^2k^2}$$
$\langle x^1\rangle_c=\mu$, $\langle x^2\rangle_c=\sigma^2$, $\langle x^3\rangle_c =0$
- several random variables $x_1,\cdots,x_N$
joint pdf $p(x_1,\cdots x_N)dx_1\cdots dx_N=\text{prob}[\vec{X}\in [vec{x},\vec{x}+d^N\vec{x}]]$
joint characteristic function $$\tilde{P}(k_1,\cdots,k_N)=\langle e^{-i(k_1x_1+\cdots+k_Nx_N)}\rangle \equiv e^{-i\vec{k}\cdot\vec{x}}\rangle=\int dx_1\cdots dx_N e^{-ik(k_1x_1+\cdots k_Nx_N)}P(x_1,\cdots,x_N)$$
$$\frac{\partial}{\partial (-ik)}\tilde{P}=\int dx_1\cdots dx_N $$
joint moments $$\langle x_1^{m_1}\cdots x_N^{m_N}\rangle =\int dx_1\cdots dx_N\ x_1^{m_1}\cdots x_N^{m_N}\ p(x_1,\cdots,x_N)=[\frac{\partial}{\partial (-ik_1)}]^{m_1}\cdots [\frac{\partial}{\partial (-ik_N)}]^{m_N}\tilde{P}(k_1,\cdots,k_N)|_{\vec{k}=0}$$
joint cumulants $$\langle x^{m_1}_1\cdots x^{m_N}_N\rangle_c=[\frac{\partial}{\partial (-ik_1)}]^{m_1}\cdots [\frac{\partial}{\partial(-ik_N)}]^{m_N}\ln \tilde{P}(k_1,\cdots,k_N)|_{\vec{k}=0}$$
- independent random variables $$P(x_1,\cdots,x_N)=P_1(x_1)\cdots P_N(x_N)=\prod^N_{i=1}P_i(x_i)$$
2.5 Energy Fluctuations
$$Z=\sum_\vec{x}e^{-\beta \mathcal{H}(\vec{x})}$$
$$\langle \mathcal{H}^n\rangle=\sum_\vec{x}\mathcal{H}^n(\vec{x}P(\vec{x})=\frac{1}{Z}\sum_\vec{x}\mathcal{H}^n(\vec{x})e^{-\beta\mathcal{H}(\vec{x})}=\frac{1}{Z}(-\frac{\partial}{\partial \beta})^n\sum_\vec{x}e^{-\beta \mathcal{H}(\vec{x})}=\frac{1}{Z}(-\frac{\partial}{\partial\beta})^nZ$$
$$\langle \mathcal{H}^n\rangle_c=(-\frac{\partial}{\partial\beta})^n\ln Z$$
$\langle \mathcal{H}^1\rangle_c=\langle \mathcal{H}\rangle$, $\langle\mathcal{H}^2\rangle_c=k_BT^2C_V$.
Fluctuation $\langle\mathcal{H}^2\rangle$ is equilibrium, but dissipation $C_V$ is non-equilibrium. (Fluctuation-dissipation theorem)
$$\frac{\langle \Delta E\rangle}{\langle E\rangle}\sim \frac{N^{1/2}}{N}\rightarrow 0$$
$$P(E)\propto \Omega(E)e^{-\beta E}=e^{\beta(TS(E)-E)}$$
$$TS(E)-E\simeq TS(E^*)-E^*+(T\frac{\partial S}{\partial E}-1)|_{E=E^*}(E-E^*)+\frac{1}{2}(T\frac{\partial^2S}{\partial E^2})_{E=E^*}(E-E^*)^2+\cdots$$
$$T(\frac{\partial^2S}{\partial E^2})=T(\frac{\partial}{\partial E}(\frac{1}{T}))=-\frac{1}{T}\frac{\partial T}{\partial E}=-\frac{1}{TC_V}<0$$
$$P(E)\propto \Omega(E)e^{-\beta E}=e^{\beta(TS(E)-E)}\sim e^{\beta(TS(E^*)-E^*)}e^{-\frac{E-E^*)^2}{2k_BT^2C_V}}\propto e^{-\frac{(E-\langle E\rangle)^2}{2k_BT^2C_V}}$$
9/24
2.6 Unbiased estimates
- $P(X)$?
- consider a $\mu E$ of $N$ identical systems
$N_i$: numbers of systems $n$ the $i$th state
$M$ distinct staes.
$$\sum^M_{I=1}N_i=N$$
$$\Omega=W[\{N_i\}^*]=\frac{N!}{\prod_{i=1}^M N_i!}$$
$$\ln W[\{N_i\}]=\ln N!-\sum_{i=1}^M\ln N_i!\simeq N\ln N-N -(\sum_{i=1}^MN_i\ln N_i-N)=-N\sum_{i=1}^M (\frac{N_i}{N})\ln (\frac{N_i}{N})=-N\sum_{i=1}^MP_i\ln P_i$$
* $P_i$ that maximizes $S$ or $W$
$$S=-\sum_{I=1}P_i\ln P_i-\alpha(\sum_{I=1}P_i-1)$$
$$\frac{\partial S}{\partial P_i}=0=-\ln P_i-1-\alpha$$
$$\frac{\partial S}{\partial \alpha}=0=\sum_{i=1}P_i-1=Me^{-(1+\alpha)}-1$$
$$P_i=e^{-(1+\alpha)}=\frac{1}{M}$$
* $P_i+(\sum_{I=1}P_i=1)+(\sum_{I=1}x_iP_i=\langle x\rangle)$
$$S=-\sum_i P_i\ln P_i-\alpha(\sum_iP_i-1)-\beta(\sum_ix_iP_i-\langle x\rangle)$$
$$\frac{\partial S}{\partial P_i}=0=-\ln P_i-1-\alpha-\beta x_i\quad \Rightarrow P_i=e^{-(1+\alpha)}e^{\beta x_i}$$
$$\frac{\partial S}{\partial \alpha}=0=\sum_{i=1}P_i-1=e^{-(1+\alpha)}\sum_{i=1}e^{-\beta x_i}-1$$
$$P_i=\frac{e^{-\beta x_i}}{\sum_{i=1}e^{-\beta x_i}}$$
2.7 Equipartition & virai theorem
* let $x_i$ be either $q_i$ or $p_i$ $(I=1,\cdots,3N)$
* in the CE, $$\langle x_i,\frac{\partial \mathcal{H}}{\partial x_j}\rangle =\sum_\vec{\mu} x_i\frac{\partial \mathcal{H}}{\partial x_j}P(\vec{\mu})=\frac{\int d\vec{p}d\vec{q}\ x_i\frac{\partial \mathcal{H}}{\partial x_j}e^{-\beta x(\vec{p},\vec{q})}}{\int d\vec{p}d\vec{q}e^{-\beta x(\vec{p},\vec{q})}}$$
Use $\frac{\partial }{\partial x_j}(x_ie^{-\beta\mathcal{H}})=(\frac{\partial x_i}{\partial x_j}e^{-\beta\mathcal{H}}-\beta x_i \frac{\partial \mathcal{H}}{\partial x_j}e^{-\beta\mathcal{H}}$
$$=\frac{\int d\vec{p}d\vec{q}\frac{1}{\beta}\delta_{ij}e^{-\beta\mathcal{H}}-\int d\vec{p}d\vec{q}\frac{1}{\beta}\frac{\partial}{\partial x_j}(x_ie^{-\beta\mathcal{H}})}{Z}$$
second term is surface integral, so blow up
$$=\frac{\frac{\delta_{ij}}{\beta}\int d\vec{p}d\vec{q}e^{-\beta\mathcal{H}}}{\int d\vec{p}d\vec{q}e^{-\beta\mathcal{H}}}=\delta_{ij}k_BT$$
'generalized equipartition theorem'
* in the ME, $$\langle x_i\frac{\partial\mathcal{H}}{\partial x_j}\rangle=\frac{1}{\Gamma(E)}\int_{E\le \mathcal{H}\le E+\Delta}d\vec{p}d\vec{q}(x_i\frac{\partial \mathcal{H}}{\partial x_j})=\frac{\Delta}{\Gamma(E)}\frac{\partial }{\partial E}\int_{\mathcal{H}\le E}d\vec{p}d\vec{q}(x_i\frac{\partial \mathcal{H}}{\partial x_j})$$
$\mathcal{H}$ change $\mathcal{H}-E$
$I=\int d\vec{p}d\vec{q}\frac{\partial}{\partial x_j}(x_i(\mathcal{H}-E))-\int d\vec{p}d\vec{q}(\frac{\partial x_i}{\partial x_j})(\mathcal{H}-E)$
$$=\delta_{ij}\frac{\Delta}{\Gamma(E)}\frac{\partial}{\partial E}\int_{\mathcal{H}\le E}d\vec{p}d\vec{q}(E-\mathcal{H})=\delta_{ij}\frac{\Delta}{\Gamma(E)}\int_{\mathcal{H}\le E}d\vec{p}d\vec{q} =\delta_{ij}\frac{\Sigma(E)}{(\frac{\partial\Sigma}{\partial E})}=\frac{\delta_{ij}}{\frac{1}{\Sigma}(\frac{\partial \Sigma}{\partial E})}=\frac{\delta_{ij}}{\frac{\partial}{\partial E}\ln\Sigma (E)}=\frac{\delta_{ij}}{\frac{1}{k_B}\frac{\partial S}{\partial E}}=\frac{\delta_{ij}}{\frac{1}{k_B}\frac{1}{T}}$$
* $\mathcal{H}=\sum_{I=1}A_iP_i^2+\sum_{I=1}B_iQ_i^2$
$$\sum_i(P_i\frac{\partial \mathcal{H}}{\partial P_i}+Q_i\frac{\partial\mathcal{H}}{\partial Q_i})=2\sum_i(A_iP_i^2+B_iQ_i^2)=2\mathcal{H}$$
Take average
$$2 \langle\mathcal{H}\rangle=Nk_BT+Nk_BT$$
$$\langle \mathcal{H}\rangle=\frac{k_BT}{2}\times f$$
* $i=j$, $x_i=p_i$
$$\langle p_i\frac{\partial \mathcal{H}}{\partial p_i}\rangle =\langle p_i\dot{q}_i\rangle=\frac{1}{m}\langle p_i^2\rangle=k_BT$$
$$\frac{1}{2}m\langle v_i^2\rangle=\frac{1}{2}k_BT$$
2.8 Classical ideal gas
* $$Z=\sum_\vec{x}e^{-\beta\mathcal{H}(\vec{x})}=\frac{1}{N!h^{3N}}\int d\vec{p}d\vec{q}e^{-\frac{\beta \vec{p}^2}{2m}}=\frac{1}{N!}(\frac{V}{h^3}(\frac{2\pi m}{\beta})^{3/2})^N=$$
$$F=-k_BT\ln Z=Nk_BT(\ln[\frac{Nh^3}{V}(\frac{\beta}{2\pi m})^{3/2}]+1)$$
안보임ㅋㅋ
$$F=E-TS$$
$$dF=dE-d(TS)=TdS-pdV+\mu dN-(TdS+SdT)=-SdT-pdV+\mu dN$$
$$S=-\frac{\partial F}{\partial T})_{V,N}$$
9/29
$\langle x_i\frac{\partial \mathcal{H}}{\partial x_j}\rangle=\delta_{ij}k_BT,$, $x_i\rightarrow (q_i,p_i)$
$I=j$, $x_i=p_i$: $$\langle p_i\frac{\partial\mathcal{H}}{\partial p_i}\rangle= \langle p_i\dot{q}_i\rangle=\frac{1}{2}m\langle v_i^2\rangle=\frac{1}{2}k_BT$$
$x_i=q_i$: $$\langle q_i\frac{\partial\mathcal{H}}{\partial q_i}\rangle=-\langle q_i\dot{p}_i\rangle=k_BT=\langle q_i\nabla_i V_i\rangle=\alpha\langle V_i\rangle$$
$\langle V_i\rangle=\frac{1}{2}k_BT$ Virial theorem
$x_i=p_i,\ x_j=q_i$: $$\langle p_i\frac{\partial\mathcal{H}}{\partial q_i}\rangle=-\langle p_i\dot{p}_i\rangle=-\frac{\partia}{\partial t}\langle p_i^2\rangle=0$$
2.8 The classical ideal gas
$$Z_N=\sum_\vec{x}e^{-\beta\mathcal{H}(\vec{x})}=\frac{1}{N!h^{3N}}\int d^{3N}pd^{3N}q\ e^{-\beta\sum^N_{I=1}\frac{\vec{p}_i^2}{2m}}=\frac{1}{N!}(\frac{1}{h^3}\int d^3\vec{q}d^3\vec{p}e^{-\beta\frac{\vec{p}^2}{2m}})^N=\frac{1}{N!}[\frac{V}{h^3}(\frac{2\pi m}{\beta})^{3/2}]^N=\frac{1}{N!}Z^N$$
$F=-k_BT\ln Z_N$
2.9 Harmonic Oscillations
1d $N$ classical Harmonic Oscillator $$\mathcal{H}=\sum^N_{I=1}(\frac{p^2_i}{2m}+\frac{1}{2}m\omega^2q_i^2)$$
$$Z_N=\frac{1}{h^N}\int d^Npd^Nq\ e^{-\beta\sum^N_{i=1}(\frac{p_i^2}{2m}+\frac{1}{2}m\omega^2q_i^2)}=(\frac{1}{h}\int dp dq\ e^{-\frac{\beta p^2}{2m}}e^{-\frac{\beta}{2}m\omega^2q^2})^N =(\frac{1}{h}\sqrt{\frac{2\pi m}{\beta}}\sqrt{\frac{2\pi}{\beta m\omega^2}})^N=(\frac{1}{\beta\hbar\omega})^N$$
$$-F_N=-k_BT\ln Z_N=Nk_BT\ln (\frac{\hbar\omega}{k_BT})$$
$$-F=E-TS$$
$$dF=dE-d(TS)=\Gamma dS-pdV+\mu dN-(TdS+SdT)=-SdT-pdV+\mu dN$$
$$p=-(\frac{\partial F}{\partial V})_{T,N}=0$$
$$\mu=(\frac{\partial F}{\partial N})_{T,V}=k_BT\ln (\frac{\hbar\omega}{k_BT})$$
$$S=-(\frac{\partial F}{\partial T})_{V,N}=-Nk_B\ln (\frac{\hbar\omega}{k_BT})+Nk_B=Nk_B[\ln \frac{k_BT}{\hbar \omega})+1]$$
$$U=\langle E\rangle=-\frac{\partial}{\partial\beta}\ln Z_N=\frac{\partial}{\partial \beta}(N\ln (\beta\hbar\omega))=\frac{N}{\beta}=Nk_BT$$
$$\frac{1}{T}=(\frac{\partial S}{\partial E})=\frac{Nk_B}{E}$$
1d N quantum HO's
$$q_n=\hbar \omega(n+\frac{1}{2})$$
$$Z_N=\sum_\vec{x}e^{-\beta \mathcal{H}(\vec{x})}=(Z_1)^N$$
$$Z_1=\sum^\infty_{k=0}e^{-\beta\hbar\omega(n+\frac{1}{2})}=e^{-\frac{\beta\hbar\omega}{2}}\sum^\infty_{n=0}e^{-\beta\hbar\omega n}=\frac{1}{1-e^{-\beta\hbar\omega}}$$
$$Z_N=\frac{e^{-\frac{1}{2}\beta\hbar\omega N}}{(1-e^{-\beta\hbar\omega})^N}$$
thermodynamic
$$F_N=-k_BT\ln Z_N=Nk_BT\frac{\beta}{2}\hbar\omega+Nk_BT\ln (1-e^{-\beta\hbar\omega})=N[\frac{\hbar\omega}{2}+k_BT\ln(1-e^{-\beta\hbar\omega})]=\frac{1}{1-e^{-\beta\hbar\omega}}$$
$$p=-(\frac{\partial F}{\partial V})_{T,N}=0$$
$$S=-(\frac{\partial F}{\partial T})_{V,N}=-[Nk_B\ln (1-e^{-\beta\hbar\omega})+Nk_BT\frac{\hbar\omega e^{-\beta\hbar\omega}}{1-e^{-\beta\hbar\omega}}(-\frac{1}{k_BT^2})]=Nk_B[\frac{\beta\hbar\omega}{e^{\beta\hbar\omega}-1}-\ln(1-e^{-\beta\hbar\omega})]$$
$$U=-\frac{\partial}{\partial\beta}\ln Z_N=\frac{\partial}{\partial\beta}[\frac{\beta\hbar\omega}{2}N+N\ln (1-e^{-\beta\omega})]=\frac{N\hbar\omega}{2}+N\frac{\hbar\omega e^{-\beta\hbar\omega}}{1-e^{-\beta\hbar\omega}}=N\hbar\omega[\frac{1}{2}+\frac{1}{e^{\beta\hbar\omega}-1}]$$
$T\rightarrow \infty$, $\beta\rightarrow 0$, $U\rightarrow Nk_BT$, Classic
$T\rightarrow 0$, $\frac{N}{2}\hbar\omega$
$$C_P=C_V=(\frac{\partial U}{\partial T}=Nk_B\frac{(\beta\hbar\omega)^2e^{\beta\hbar\omega}}{(e^{\beta\hbar\omega}-1)^2}$$
$g(E)$ from $Z(\beta)$
$$Z(T)=\sum_E\Omega(E)e^{-\beta E}$$
$$Z_N=\frac{e^{-\frac{N}{2}\beta\hbar\omega}}{(1-e^{-\beta\hbar\omega})^N}=e^{-\frac{N}{2}\beta\hbar\omega}\sum^\infty_{l=0}\quad _{-N}C_l (-1)^l (e^{-\beta\hbar\omega})^l=\sum^\infty_{l=0}\quad _{N+l-1}C_l e^{-\frac{\beta}{2}\hbar\omega(N+l)}$$
$E_N=\sum^N_{I=1}\hbar\omega(\frac{1}{2}+n_i)=\hbar\omega(\frac{N}{2}+l)$
$$Z_N=\sum_{l=0}\Omega(l)e^{-\beta E(l)}$$
$\Omega(E)=2.718\ _{N+l-1}C_l$
10/13
3. Grand Canonical Ensemble
3.1 Intro
$$p(\vec{x}_S)=\sum_{\{\vec{x}_r\} }p(\vec{x}_s,\vec_r)$$
$$=\frac{1}{\Omega_{tot}(E_{tot},N_{tot})}\times \Omega_r(E_{tot}-\mathcal{H}_s(\vec{x}_s),N_{tot}-N_S(\vec{x}_s)))$$
$$\propto \exp (\frac{1}{k_B}S_r(E_{tot}-\mathcal{H}_s(\vec{x}_s),N_s(\vec{x}_s))$$
$E_{tot}\gg \mathcal{H}_S(\vec{x}_S)$, $N_{tot}\gg N_s(\vec{x}_s)$
$$\exp (\frac{1}{k_B}(S_r(E_{tot},N_{tot})-(\frac{\partial S_r}{\partial E})\mathcal{H}_s(\vec{x}_s)-(\frac{\partial S_r}{\partial N})\mathcal{H_S(\vec{x}_S))$$
$$=\exp (-\frac{\mathcal{H}_s(\vec{x}_s)}{k_BT}+\frac{\mu}{k_BT}N_S(\vec{x}_S))$$
$$p(\vec{x}_S)\propto e^{-\beta(\mathcal{H}(\vec{x}_s)-\mu N(\vec{x}_s))}$$
$$p(\vec{x}_s)=\frac{1}{Q}e^{-\beta(E-\mu N)}$$
$$Q=\sum_{\{\vec{x}_s\}}e^{-\beta(E(\vec{x}_s)-\mu N (\vec{x}_s))}=\sum^\infty_{N=0}e^{\beta\mu N}e^{-\beta\mathcal{H}_N(\vec{x}_S)}$$
$$=\sum^\infty_{N=0}z^NZ(T,V,N)$$
Thermodynamics
$$S=-k_B\langle \ln p\rangle$$
$$=-k_B\langle \ln \frac{e^{-\beta(E-\mu N)}}{Q}\rangle$$
$$\frac{S}{k_B}=\beta\langle E\rangle -\beta \mu \langle N\rangle +\ln Q$$
$$-k_BT\ln Q=\langle E\rangle -TS -\mu \langle N\rangle =U-TS-\mu N$$
Euler's equation
$$U(\alpha S,\alpha V,\alpha N)=\alpha U(S,V,N)$$
Let $\alpha=1+\epsilon$
$$U((1+\epsilon)S,(1+\epsilon)V,(1+\epsilon)N)\simeq U(S,V,N)+(\frac{\partial U}{\partial S})\epsilon S + (\frac{\partial U}{\partial N})\epsilon V+(\frac{\partial U}{\partial N})\epsilon N$$
$$=U(S,V,N)+\epsilon [TS-pV+\mu N]$$
$$=U(S,V,N)+\epsilon U(S,V,N)$$
$$U=TS-pV+\mu N$$
$$-k_BT\ln Q=-pV\equiv \Phi$$
$$d\Phi =dU-d(TS)-d(\mu N)$$
$$=TdS-pdV+\mu dN - (TdS+SdT)-(\mu dN+Nd\mu)$$
$$=-SdT-pdV-Nd\mu$$
$$\equiv d\Phi (T,B,\mu)$$
$$S=-(\frac{\partial \Phi}{\partial T})_{V,\mu},\ p=-(\frac{\partial \Phi}{\partial V})_{T,\mu},\ N=-(\frac{\partial \Phi}{\partial \mu})_{T,V}$$
$$Z=\sum_E \Omega(E)e^{-\beta E}$$
$$Q=\sum_{N,E}e^{\beta\mu N}\Omega_N(E)e^{-\beta E_N}=\sum_Ne^{\beta\mu N}Z$$
$$\langle N\rangle =\sum_{N,E}NP(N,E)$$
$$=\sum_{N,E}N\frac{1}{Q}\Omega_N(E)e^{-\beta(E-\mu N)}$$
$$=\frac{1}{Q}\sum_{N,E}\frac{\partial }{\partial (\beta \mu)}\Omega_N(E)e^{-\beta(E-\mu N)}$$
$$=\frac{1}{Q}\frac{\partial }{\partial (\beta \mu)}Q|_{T,V}$$
$$=\frac{\partial }{\partial (\beta \mu)}\ln Q|_{T,V}$$
$$\langle E\rangle=-\frac{\partial}{\partial \beta}\ln Q|_{Z,V}$$
$$F=E-TS=\mu N-pV=Nk_BT\ln Z-k_BT\ln Q=-k_BT\ln (\frac{Q}{Z^N})$$
3.2 Number Fluctuations
$$Z(T,V,N)=\sum_E\Omega(E)e^{-\beta E}=\sum_Ee^{-\beta(E-TS)}=\sum_Ee^{-\beta F(E)}\simeq e^{-\beta F_{min}(E^*)}$$
$$Q(T,V,\mu)=\sum_{N,E}e^{\beta \mu N}\Omega_N(E)e^{-\beta E_N}=\sum_{N,E}e^{-\beta (E-TS-\mu N)}=\sum_{N,E}e^{-\beta \Phi(E,N)}\simeq e^{-\beta \Phi_{min}(E^*,N^*)}$$
most probable $(E^*,N^*)$
$$\frac{\partial P(E,N)}{\partial E}|_{E=E^*}=0\ \Rightarrow (\frac{\partial \Omega}{\partial E}-\beta\Omega)=0$$
$$\Rightarrow \frac{\partial }{\partial E}\ln Q=\beta\ \Rightarrow (\frac{\partial S}{\partial E})_{E=E^*}=\frac{1}{T}$$
$$\frac{\partial P(E,N)}{\partial N}|_{N=N^*}=0$$
$$\frac{\partial \Omega_N}{\partial N}|_{N=N^*}+\beta \mu \Omega_N|{N=N^*}=0$$
$$(\frac{\partial S}{\partial N})_{N=N^*}=-\frac{\mu}{T}$$
$E^*\simeq E_{mc}$, $N^*\simeq N_{mc}$
number fluctuation
$$\langle N\rangle =\frac{\partial}{\partial (\beta\mu)}\ln Q|_{T,V}$$
$$\langle N^2\rangle_c=\langle N^2\rangle -\langle N\rangle^2= \frac{1}{Q}\frac{\partial^2}{\partial (\beta\mu)^2}Q-(\frac{1}{Q}\frac{\partial}{\partial (\beta\mu)}Q)^2 $$
$$=\frac{\partial^2}{\partial(\beta\mu)^2}\ln Q|_{T,V}=\frac{\partial \langle N\rangle}{\partial (\beta\mu)}|_{T,V}\propto \langle N\rangle$$
$$\frac{\langle N^2\rangle_c}{\langle N\rangle^2}=\frac{k_BT}{\langle N\rangle^2}\frac{\partial \langle N\rangle}{\partial \mu}|_{T,V}=\frac{k_BT}{N^2}\frac{\partial N}{\partial \mu}|_{T,V}$$
Let $v=\frac{V}{N}$
$$=\frac{k_BT}{V^2}v^2\frac{\partial (V/v)}{\partial \mu}|_{T,V}=\frac{k_BT}{V}v^2(-\frac{1}{v^2})(\frac{\partial v}{\partial \mu})_T=-\frac{k_BT}{V}(\frac{\partial v}{\partial \mu})_T$$
$U=TS-pV+\mu N$ $0=SdT-VdP+Nd\mu$ Gibbs-Du어쩌구 relation
$$=-\frac{k_BT}{V}\frac{1}{v}(\frac{\partial v}{\partial p})_T=\frac{k_BT}{V}\kappa_T$$
10/15
3.3 Central limit theorem
$p(x), \hat{p}(k),\ln \hat{p}(k)$
$p(x_1,\cdots,x_N)dx_1\cdots dx_N$
$$\hat{[p}(k_1,\cdots, k_N)=\langle e^{-I(k_1x_1+\cdots k_Nx_N)}\rangle$$
$$=\int dx_1,\cdots dx_N\ e^{-i(k_1x_1+\cdots+k_Nx_N)}p(x_1,\cdots,x_N)$$
If statistical independent.
$$=\prod_{\alpha=1}\tilde{p}_\alpha(k_\alpha)$$
joint moments
$$\langle x^{m_1}_1\cdots x^{m_N}_N\rangle =[-(\frac{\partial}{\partial (ik_1)}]^{m_1}\cdots [-(\frac{\partial}{\partial (ik_N)}]^{m_N}\hat{p}|_{\vec{k}=0}$$
joint cumulents
$$\langle x^{m_1}_1\cdots x^{m_N}_N\rangle_c =[-(\frac{\partial}{\partial (ik_1)}]^{m_1}\cdots [-(\frac{\partial}{\partial (ik_N)}]^{m_N}\ln\hat{p}|_{\vec{k}=0}$$
consider a sum of $N$ random variables
$$X=\sum^\infty_{i=1}x_i,\quad \{x_i\}\sim P(x_1,\cdots,x_N)$$
$$\tilde{p}_X(k)=\langle e^{-ikX}\rangle=\int dx_1\cdots dx_N e^{-ik(\sum^N_{I=1}x_i)}p(x_1,\cdots,x_N)=\tilde{p}(k_1=k,\cdots,k_N=k)$$
$$\ln \tilde{p}_X(k)=\ln\langle e^{-ikx}\rangle=\sum^\infty_{n=1}\frac{(-ik)^n}{n!}\langle X^n\rangle_c=\sum^infty_{n=1}\frac{(-ik)^n}{n!}\langle (\sum^n__{I=1}x_i)^n\rangle_c$$
$$\langle X^m\rangle_c=\langle \sum^N_{i_1=1}\cdots\sum^N_{i_m=1}x_{i_1}\cdots x_{i_m}\rangle_C$$
$x_i$: statistically independent ($\{x_i\}\sim p_1(x_1)\cdots p_N(x_N)$, identically distributed $\{x_i\}\sim p(x_1)\cdots p(x_N)$ (IID)
$$\langle X^m\rangle_c=\sum^N_{i=1}\langle x_i^m\rangle_c=N\langle x^m\rangle_c$$
introduce $y=\frac{1}{N^{1/2}}(X-N\langle x\rangle_c)$
$\langle y\rangle_c=\frac{1}{N^{1/2}}(\langle X\rangle_c-N\langle x\rangle_c)=0$
$\langle y^2\rangle_c=\langle y^2\rangle-\langle y \rangle^2=\frac{1}{N}\langle (x_N\langle x\rangle_c)^2\rangle-\frac{1}{N}\langle X-N\langle x\rangle_c\rangle^2=\frac{\langle X^2\rangle_c}{N}=\langle x^2\rangle_c$
$$\langle y^m\rangle_c=\frac{\langle X^m\rangle_c}{N^{m/2}}=\frac{N\langle x^m\rangle_c}{N^{m/2}}\propto N^{1-\frac{m}{2}}\rightarrow 0$$
$$y\sim N(0,\langle x^2\rangle_c)$$
$$p(y)=\frac{1}{\sqrt{2\pi \langle x^2\rangle_c}}e^{-\frac{y^2}{2\langle x^2\rangle_c}}$$
3.4 The Ideal Gas
$\mathcal{H}=\sum_{i=1}\frac{\vec{p}^2_i}{2m}$
$$Q(T,V,\mu)=\sum^\infty_{N=0}e^{\beta \mu N} Z_N=\sum^\infty_{N=0}e^{\beta\mu N}(\frac{1}{N!}\int \frac{1}{h^{3N}}d^{3N}\vec{p} d^{3N}\vec{q} e^{-\beta \mathcal{H}(\vec{p},\vec{q})})$$
$$=\sum^\infty_{N=0}e^{\beta\mu N}\frac{1}{N!}(\frac{V}{h^3}\int 4\pi p^2e^{-\frac{\beta p^2}{2m}}dp)^N=\sum^\infty_{N=0}e^{\beta\mu N}\frac{V^N}{N!}(\frac{2\pi m k_BT}{h^2})^{3N/2}$$
$\lambda\equiv \sqrt{\frac{h^2}{2\pi m k_BT}}$
$$Q=\sum^\infty_{N=0}\frac{1}{N!}(e^{\beta\mu}\frac{V}{\lambda^3})^N=\exp (e^{\beta\mu}\frac{V}{\lambda})$$
$$\Phi=-k_BT\ln Q=-e^{\beta\mu}\frac{V}{\lambda^3}k_BT$$
$$\Phi=E-TS-\mu N$$
$$d\Phi=dE-d(TS)-d(\mu N)$$
$dE=TdS-pdV+\mu dN$
$$d\Phi -SdT-pdV-Nd\mu$$
$$S=-(\frac{\partial \Phi}{\partial T})_{V,\mu},\ p=-(\frac{\partial \Phi}{\partial V})_{T,\mu},\ N=-(\frac{\partial \Phi}{\partial \mu})_{T,V}$$
$$p=-\frac{\partial}{\partial V}(-e^{\beta\mu}\frac{V}{\lambda^3}k_BT)|_{T,\mu}=k_BT\frac{e^{\beta\mu}}{\lambda^3}=k_BT\frac{N}{V}$$
$$N=-\frac{\partial}{\partial \mu}(-e^{\beta\mu}\frac{V}{\lambda^3}k_BT)|_{T,V}=Te^{\beta\mu}\frac{V}{\lambda^3}$$
$$\mu=k_BT\ln (\frac{\lambda^3N}{V})=k_BT\ln (\frac{\lambda^3p}{k_BT})$$
$$\langle N^m\rangle_c=\frac{\partial^m}{\partial (\beta\mu)^m}\ln Q|_{T,V}$$
$$\langle N^2\rangle_c=\frac{\partial^2}{\partial (\beta\mu)^2}\ln Q|_{T,V}=\frac{\partial}{\partial (\beta\mu)}\ln \langle N\rangle|_{T,V}=k_BT\frac{\partial \langle N\rangle}{\partial \mu}=\frac{PV}{k_BT}=\langle N\rangle$$