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\newcommand{\la}{\left<}			\newcommand{\ra}{\right>}			\newcommand{\bok}[3]{ \la #1 \right| #2 \left| #3 \ra }
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\author{pod}
\date{Primavera, 2002}
\title{Física estadística}

\pagestyle{empty}



\begin{document}
\section*{Física estadística}

\noindent
 $  S = k_B \ln \Omega(E, V, N) $
 $$ \frac{1}{T} = \parcte{S}{E}{V,N} \ , \quad p = \parcte{S}{V}{E,N} $$ 
 $$ \quad \mu = - T \parcte{S}{N}{E,V} $$
 $$ \Omega = \frac{1}{h^{3N}} \mathop{ \int \cdots  \int}_{E < H < E+\delta E} \dd^{3N}q \dd^{3N}p \quad , \ \lambda = \sqrt{\frac{h^2}{2\pi m k_B T}} $$
 $$ V_n = C_n r^n \ , \quad A_n = n C_n r^{n-1} \ , \quad \to \ C_n = \frac{2}{n} \frac{\pi^{n/2}}{\Gamma(n/2)} $$
 $$ \textrm{Gas ideal:}\ S \approx k_B N \left( \ln \frac{V}{N} + \frac{5}{2} \ln \frac{4m \pi E}{3h^2N}+\frac{3}{2} \right) $$

\hrule\sep
 $$ Z = \sum_{E_t} \Omega_1(E_r)\ \expon{-\beta E_r} = \sum_r \expon{-\beta E_r} $$
 $$ F(T,V,N) = U - T S= - k_B T \ln Z $$
 $$ p = - \left. \parcial[F]{V} \right)_{T,N} \ ,
    \quad S = - \left. \parcial[F]{T} \right)_{V,N} \ , \quad \mu = \left. \parcial[F]{N} \right)_{T,V}$$ $$ C_v = \left.
    \parcial[\la E \ra]{T} \right)_{V,N} = \left. T \parcial[S]{T} \right)_{V,N} \ , \quad \la E \ra = \left. - \parcial{\beta} \ln Z \right)_{V,N} $$
 $$ (\Delta E)^2 = \left(\parcial[^2]{\beta^2} \ln Z \right)_{N,V} = - k_B T^2 C_v $$
 $$ Z(T,V,N) = \frac{1}{N!} Z_1^N = \frac{1}{N! h^{3N}} \int \expon{-\beta H(p,q)}\ \dd^{3N}q\ \dd^{3N}p $$
 $$ Z_1(T,V) = \frac{1}{h^3} \int \expon{-\beta H(p,q) } \dd^{3}q\ \dd^{3}p $$
 $$ \la x_i \parcial[H]{x_j} \ra = k_B T \delta_{ij} $$
 $$ H = \sum_{k=1}^{N'} \alpha_k x_k^\eta \to \la E \ra = \frac{N'k_B T}{\eta} $$

\hrule\sep
 $$ \QQ(T,\mu,V) = \sum^\infty_{N_s=0} \sum_{E_r} \Omega(E_r,N_s) \expon{-\beta E_r + \beta \mu N_s} $$
 $$ z = \expon{\mu \nu} = \expon{-\alpha} :\quad  \QQ = \sum^\infty_{N_s=0} z^{N_s} Z(T,V,N_s) $$
 $$ \QQ = \sum_{N_s} \left(z Z_1(T,V)\right)^{N_s} = \frac1{1-z Z_1} $$
 $$ \QQ = \sum_{N_s} \frac1{N_s!}\left(z Z_1(T,V)\right)^{N_s} = \expon{z Z_1}$$
 $$ \la N \ra = \left. k_B T \parcial{\mu} \ln \QQ
    \right)_{T,V} = \left. - \parcial{\alpha} \ln \QQ \right)_{T,V} = \left. z \parcial{z} \ln \QQ \right)_{T,V} $$
 $$ (\Delta N)^2 = k_B T \left. \parcial{\mu} \la N \ra \right)_{T,V} = \left. - \parcial{\alpha}
    \la N \ra \right)_{T,V}  = \frac{\la N \ra^2}{\beta\ V} \kappa_T $$
 $$ \la U \ra = - \left( \parcial{\beta} \ln\QQ \right)_{z,V} = k_B T^2 \left( \parcial{T} \ln\QQ \right)_{z,V} $$
 $$ (\Delta U)^2 =  k_B T^2 C_v + \left( \parcial{\la N \ra} \la E \ra \right)^2_{T,V} (\Delta N)^2 $$
 $$ \Xi(T,\nu,V) = - k_B T \ln \QQ \quad,\quad  S = - \left( \parcial[\Xi]{T} \right)_{V,\mu} $$
 $$ \Xi(T,\mu,V) = U - T S - \mu N = - p V $$

\hrule\sep Vibr. sol. $Z_1(\omega,T) = \expon{- \beta \hbar \omega /2} \big/ (1 - \expon{-\beta\hbar\omega})$
 $$ \ln Z = \int^{\omega_M}_{\omega_m}\!\!\! \dd\omega\ g(\omega) \left(  - \beta\omega\hbar/2 - \ln(1-\expon{-\beta\hbar\omega}) \right) $$
 $$ U = \int^{\omega_M}_{\omega_m}\!\!\! \dd\omega\ g(\omega) \left(  \omega\hbar/2 + \frac{\hbar\omega \expon{-\beta\hbar\omega}}{1-\expon{-\beta\hbar
  \omega}} \right) $$
 $$ C_v = k_B \int^{\omega_M}_{\omega_m}\!\!\! \dd\omega\ g(\omega) (\beta\hbar\omega)^2 \frac{\expon{\beta\hbar\omega}}{(\expon{\beta\hbar\omega}-1)^2} $$
 $$ \int^{\omega_M}_{\omega_m}\!\!\! \dd\omega\ g(\omega) = d N $$
 $$ \textrm{Einstein}\ g(\omega) = 3N\ \delta( \omega - \omega_E) $$
 $$ \textrm{Debye}\  p = \frac{h \omega}{2\pi c} \ , \ g(\omega) \dd\omega = \frac1{h^d} \dd^d q\, \dd^d p \ , \ \omega\in (0,\omega_D)$$


\hrule\sep Eq. sólido-vapor $ \QQ_v = \expon{z_v Z_v^1} $ , $ \QQ_s = 1/(1-z_s Z_s^1) $
 $$ \la N_v \ra = \frac{Z_v^1}{Z_s^1} \quad , \quad  N_s = N_T - N_v $$
 $$ N_v(T_c, V) = N_T \ , \quad N_s(T_c, V) = 0 \ , \quad pV = k_B T\frac{Z_v^1}{Z_s^1} $$

\hrule\sep
 Poliatòmicas\\$ \displaystyle H = \frac12 m \dot R^2 + \frac{p_r^2}{2\mu} + \frac{L^2}{2\mu r^2} + V(r) $ ,  $ \displaystyle L^2 = p_\theta^2 +\frac{ p_\varphi^2}{\sin\theta} $
 $$ \textrm{Rot.} \ H = \frac{L^2}{2I} \ ,\quad I = \mu r_0^2 \ ,\quad \theta_r =\frac{\hbar^2}{2 I k_B} \ ,\quad \varepsilon_r =\frac{l(l+1)\hbar^2}{2I}$$
 $$ g_l = 2 l + 1 \ , \quad Z_{1,r} = \sum_l (2l+1) \expon{-l(l+1) \frac{\theta_r}{T}} $$
 $$ \textrm{cont.}\ \theta_r \ll T \  \quad Z_{1,r} = \int_0^\infty \dd l \, (2l+1) \expon{-l(l+1) \frac{\theta_r}{T}} = T / \theta_r $$
 $$ \textrm{Euler-Mc Laurin},T^2\ C_v = N k_B \left( 1 + \frac1{45} \left( \frac{\theta_r}{T} \right)^2 + \cdots \right) $$
 $$ \textrm{Vibr.}\ H_v = \frac{p_r^2}{2\mu} + \frac1 \mu\omega^2 (r-r_0)^2 \ , \quad \theta_v = \frac{\hbar\omega}{k_B} \approx 10^3 K $$
 $$ Z_{1,v} = \sum \expon{-\beta\hbar\omega(n+1/2)} = \frac{\expon{-\theta_v / T}}{1- \expon{-\theta_v / T}} $$

\hrule\sep
 Gas fotones $ \varepsilon = c p = \hbar \omega $ , $ \omega = k c $ , (3D) $g(\omega) = \frac{V \omega^2}{\pi^2 c^3}$ ,
 $\not\exists \mu ( = 0 ) $ $$ u(\omega, T) = \frac{\la n(\omega) \ra}{V} g(\omega) = \frac{\hbar \omega}{\expon{\beta\hbar\omega} - 1}
 \frac{\omega^2}{\pi^2 c^3} $$
 $$ u(T) = \frac{\pi^2}{15} \frac{k_B^4}{c^3 \hbar^3} T^4 = \frac4c \sigma T^4 $$
 $$ \frac{pV}{k_B T} = \ln \QQ = - \int_0^\infty \ln(1-\expon{-\beta\hbar\omega})\,g(\omega)\dd\omega = \frac13 u(T) \frac{V}{k_B T} $$

\hrule\sep
 $\la \hat A \ra = \tr( \hat\rho \hat A ) = \sum \bok{\psi_i}{\hat A}{\psi_i} $ , $ \hat \rho =\sum \omega_i \ketbra{\psi_i}{\psi_i} $ , $\sum
 \omega_i = 1$
 $$ \omega_i = \frac1{\Omega} \quad , \quad Z_N = \tr \left( \expon{-\beta H} \right) = \sum_i \expon{-\beta E_i} $$
 $$ \QQ = \tr \left( \expon{-\beta(\hat H - \mu \hat N)} \right) = \sum_i \expon{-\beta(H_i - \mu N_i)} $$
 $$ \psi^A_\textrm{ferm} = \frac1{\sqrt{N!}} \sum_{P \in S_N} (-1)^{\sigma(P)}\ \varphi_{\varepsilon_1} (\xi_{P(1)}) \cdots \varphi_{\varepsilon_N}
  (\xi_{P(N)}) $$
 $$ \psi^S_\textrm{bos} = \sqrt{ \frac1{N! \cdot n_1! n_2! \cdots n_N!} } \sum_{P \in S_N} \varphi_{\varepsilon_1} (\xi_{P(1)}) \cdots
 \varphi_{\varepsilon_N} (\xi_{P(N)}) $$

\sep
 $$ \QQ (T, \mu, V) = \prod_j \QQ_j \quad \to \quad \QQ_j = \sum_{n_j}\left( \expon{-\beta \varepsilon_j} z \right)^{n_j} $$
 $$ \QQ_j^\textrm{B-E} = \frac1{1-z \expon{-\beta \varepsilon_j}} \ , \ \QQ_j^\textrm{F-D} = 1 + z \expon{- \beta \varepsilon_j} \ , \ \QQ^\textrm{M-B}
  = \expon{ z Z_1 } $$
 $$ \ln \QQ = \frac 1a \sum_j \ln \left( 1 + a z\ \expon{- \beta \varepsilon_j} \right) \ , \quad Q_j = \left( 1 + a z\,\expon{-\beta \varepsilon_j}
  \right)^{\frac 1 a} $$
 $$ \la n_j \ra = \frac1{\inv z \expon{\beta \varepsilon_j} + a} \ , \quad \left( \Delta n_j \right)^2 =
  \frac{\la n_j \ra}{1 + a z\ \expon{ -\beta \varepsilon_j} } $$

\sep
 $$ q = \ln \QQ = \frac 1a \int^\infty_{0^+} \ln \left( 1 + az\ \expon{-\beta\varepsilon} \right)\ g(\varepsilon) \dd\varepsilon + \frac1a \ln
  \left( 1 + a z \right) $$
 $$ \la E \ra = \int^\infty_0 \frac{\varepsilon\ g(\varepsilon) \dd\varepsilon}{\inv z \expon{\beta \varepsilon} + a} $$
 $$ \la N \ra = \int^\infty_{0^+} \frac{g(\varepsilon) \dd\varepsilon}{\inv z \expon{\beta \varepsilon} + a} + \frac z {1+az} $$
 $$ \left( \Delta N \right)^2 = \int_0^\infty \frac{\inv z \expon{\beta \varepsilon}\ g(\varepsilon) \dd\varepsilon}{ \left( \inv z
  \expon{\beta \varepsilon} + a \right)^2} $$

\hrule\sep
 Bosones ($a = -1$, $d=2$) , $ g(\varepsilon) = A 2\pi m / h^2 = g $
 $$ \mu_{B-E} = - \frac g\beta \ln( 1-\expon{-\la N \ra \beta / g} ) \ , \quad z = 1 - \expon{-\beta \la N \ra  / g} $$
 $$ \la E \ra = g \int_0^\infty \frac{\varepsilon\dd\varepsilon}{ \inv z \expon{\beta\varepsilon} - 1} = \frac g\beta L_2 ( 1 - \expon{-\beta \la N \ra
  / g} ) $$
 $$ L_2(|z|<1) = \sum_{n=1} \frac{z^n}{n^2} \ , \quad L_2(-z) = - L_2\left(-\frac1z\right)-\frac12\ln^2z - \frac{\pi^2}6 $$

\sep
 Fermiones ($a = 1$) $\mu_{F-D}(T\to0) = \varepsilon_F = \la N \ra / g $
 $$ \mu_{F-D} = k_B T \ln ( \expon{\la N \ra \beta / g} - 1 ) \ , \quad z_{F-D} = -1 + \expon{\la N \ra \beta / g} $$
 $$ \frac1{\expon{\beta(\varepsilon - \mu)} + 1} \mathop{\longrightarrow}_{T\to0} \theta( \varepsilon - \varepsilon_F) $$
 $$ \la E(T\to0) \ra = g \varepsilon_F^2 / 2 = \frac{\la N \ra^2}{2 g} \ , \quad \la N(T\to0) \ra = g\, \varepsilon_F $$
 $$ \textrm{Sommerfeld}\ T\downarrow \ \la E \ra = \frac {g \mu^2(0)}2 + \frac{\pi^2 g}{6 \beta^2} + o(\expon{-\beta\mu(T=0)} $$
 $$ \la E \ra = - \frac{g}{\beta^2} L_2 (-z) $$

\sep
 $$ \frac{p A}{k_B T} = \frac1a \int_0^\infty g \ln ( 1 + a z \expon{-\beta\varepsilon}) \dd\varepsilon \to p A = \la E \ra = \frac2d \la E \ra $$
 $ E_r = N_A^2 / 2g $ , $ T_r = N_A / 2 k_B g $ , $ p_r = E_r / A $ \\
 $ \Pi = p / p_r $ , $\omega = (A/\la N \ra) / (A / N_A) $ , $\tau = T/T_r $
 $$ \Pi_{M-B} = \tau / \omega \ , \quad \Pi_a = - \frac{\tau^2}{2a} L_2 \left( 1-\varepsilon{2a / \tau\omega} \right) $$

\hrule\sep
  Condensado B-E $s=0$
 $$ N_0 = \frac z {1-z} \ , \quad g(\varepsilon) = \frac{2\pi V}{h^3} (2m)^{3/2} \varepsilon^{1/2} $$
 $$ N_e = \int_0^\infty \frac{\varepsilon g(\varepsilon)}{\inv z \expon{\beta\varepsilon}-1}\dd\varepsilon = \frac V{\lambda^3} g_{3/2}(z) $$
 $$ \la E \ra = \frac32 k_B T \frac{V}{\lambda^3} g_{5/2}(z) \ , \quad p \beta = g_{5/2}(z) / \lambda^3 \leftarrow p V = \frac23 \la E \ra $$
 $$ T\to 0 \ , \ z \to 1 \ , \ N_e = \frac V{\lambda^3} 2.612\cdots $$
 $ \Pi = p / p_r $ , $\omega = V / V_r $ , $ \tau = T / T_r $ , $ T_r = T_c(N_A, V_r) $
 $$ \textrm{Coex.} \  \Pi_c \omega_c^{5/3} = 1 \quad , \quad \lambda^3(T_c) = v_r g_{3/2}(1) \tau_c^{-3/2} $$
 $$ \textrm{Vapor} \ \Pi = \tau^{5/2} \frac{g_{5/2}(z)}{g_{5/2}(1)} \ , \quad \omega = \tau^{-3/2} \frac{g_{5/2}(z)}{g_{5/2}(1)} $$
 $$ \textrm{v-cond}\ \Pi = \tau^{5/2} \ , \quad \deriv[\Pi_c]{\tau_c} = \frac{\bar l}{\tau_c \omega_c} \ , \quad \bar l = \frac52 \tau_c $$
 $$ N_0 / N = 1 - \frac{\lambda^3(T_c)}{\lambda^3(T)} = \left( 1 - \left( \frac{T}{T_c} \right)^{3/2} \right) \theta(T_c-T) $$
 $$ g_n(z) = \frac1{\Gamma(n)} \int_0^\infty \frac{x^{n-1}}{\inv z \expon z - 1} \dx $$

\hrule\sep
 $$ E_m = - \mu_B g_L H m \ , \quad m = -s,\cdots, s \ , \quad \mu_B = \frac{ e \hbar}{2 m_e} $$
 $$ x = \beta \mu_B g_L H s \ , \quad Z_1 = \sum_{m=-s}^s \expon{-x m /s} = \frac{ \sinh x(1+1/2s) }{ \sinh x/ 2s } $$
 \begin{eqnarray*}
  M(H,T) &=& \left(-\parcial[F']{H}\right)_T =M_0 B_s(x) \\
  & = & \left( N g_L \mu_B s \right) \left[ \left(1+\frac1{2s}\right) \coth\left(1+\frac1{2s}\right)x - \frac1{2s} \coth\frac x{2s}\right]
 \end{eqnarray*}
 $$ U' = - H M_0 B_s(x) \quad , \quad C_H = \left( \parcial[U']{T} \right)_H = N k_B x^2 B'_s(x) $$

 \sep
 $$ H = - \mu H \cos\theta \ , \quad Z_1 = \int \dd\varphi \dd\theta\, \sin\theta\, \expon{\beta \mu \cos\theta} = 4\pi \frac{\sinh x}{x} $$
 $$ M = M_0 L(x) = N \mu \left[ \coth x - \frac1x \right] $$ $ s\to\infty $ , $g_L \to 0$ , $g_L s \mu_B \to \mu$ , $B_s(x) \to L(x) $

\hrule\hrule\sep
 $$ \begin{array}{llllll}
   U(S,V) &           & \quad & \dd U = &   & T \dd S - p \dd V + \mu \dd N \\
   H(S,p) & = U+pV    & \quad & \dd H = &   & T \dd S + V \dd p + \mu \dd N \\
   F(T,V) & = U-TS    & \quad & \dd F = & - & S \dd T - p \dd V + \mu \dd N \\
   G(p,T) & = U+pV-TS & \quad & \dd G = &   & V \dd p - S \dd T + \mu \dd N \
 \end{array} $$


\hrule\hrule\sep
Integrales
 $$ \int_0^\infty x^m e^{-ax^2} \dd x = \frac {\Gamma \left( \frac{m+1}{2} \right) }{2 a^ \frac{m+1}{2}} \quad , \quad \int_0^\infty x^n e^{-ax} \dd x = \frac {\Gamma \left( n+1 \right) }{2 a^{n+1}} $$
 $$ \int_0^\infty \frac{x^{n-1}}{e^x-1} \dd x = \Gamma (n)  \xi (n) \quad , \quad \Gamma (1/2) = \sqrt{\pi} $$
 $$ \text{geométrica}\ \sum_{k=0}^{N-1} a r^k = \frac{a(1-r^N)}{1-r} = \frac{a}{1-r} $$ 
 $$ \text{Stirling}\ \ln n! \sim n \ln n -n + \ln n + \frac{1}{2} \ln(2\pi) $$
\hrule\sep
Combinatória
 $$ VR(N,m) = N^m \quad , \quad VSR (N,m) = \frac{N!}{(N-m)!} $$ 
 $$ CR (N,m) = \frac{(N+m-1)!}{m! (N-1)!} \quad , \quad CSR (N,m) = \frac{N!}{m!(N-m)!}$$

\vspace{1cm}
Gracias a Sergio Oller por las correcciones.
\end{document}
