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Physics LibreTexts

5.S: Summary

  • Page ID
    18731
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    \( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
    \( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
    \( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
    \( \newcommand\veF{\ve\ns_\RF}\)
    \( \newcommand\kF{k\ns_\RF}\)
    \( \newcommand\kFu{k\ns_{\RF\uar}}\)
    \( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
    \( \newcommand\muB{\mu\ns_\ssr{B}}\)
    \( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
    \( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
    \( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
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    \( \newcommand\msa{m\ns_\ssr{A}}\)
    \( \newcommand\msb{m\ns_\ssr{B}}\)
    \( \newcommand\mss{m\ns_\Rs}\)
    \( \newcommand\HBx{\hat\Bx}\)
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    \( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
    \( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
    \( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
    \( \newcommand\vbar{\bar v}\)
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    \( \newcommand\TB{T\ns_\ssr{B}}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \newcommand\NS{N\ns_{\textsf S}}\)

    References

    • F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1987) This has been perhaps the most popular undergraduate text since it first appeared in 1967, and with good reason.
    • A. H. Carter, Classical and Statistical Thermodynamics (Benjamin Cummings, 2000) A very relaxed treatment appropriate for undergraduate physics majors.
    • D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 2000) This is the best undergraduate thermodynamics book I’ve come across, but only 40% of the book treats statistical mechanics.
    • C. Kittel, Elementary Statistical Physics (Dover, 2004) Remarkably crisp, though dated, this text is organized as a series of brief discussions of key concepts and examples. Published by Dover, so you can’t beat the price.
    • R. K. Pathria, Statistical Mechanics (\(2^{nd}\) edition, Butterworth-Heinemann, 1996) This popular graduate level text contains many detailed derivations which are helpful for the student.
    • M. Plischke and B. Bergersen, Equilibrium Statistical Physics (\(3^{rd}\) edition, World Scientific, 2006) An excellent graduate level text. Less insightful than Kardar but still a good modern treatment of the subject. Good discussion of mean field theory.
    • E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics (part I, \(3^{rd}\) edition, Pergamon, 1980) This is volume 5 in the famous Landau and Lifshitz Course of Theoretical Physics. Though dated, it still contains a wealth of information and physical insight.

    Summary

    \def\tpar{t\ns_\parallel} \def\mhat{\hat\Bm} \parindent=0pt \renewcommand*\rmdefault{ppl}\normalfont\upshape \physgreek \font\seventeenbf=cmbx10 scaled \magstep3 \setcounter{section}{4} \section{Quantum Statistics : Summary} $\bullet$ {\it Second-quantized Hamiltonians\/}: A noninteracting quantum system is described by a Hamiltonian $\HH=\sum_\alpha\ve\ns_\alpha\,\Hn\ns_\alpha$, where $\ve\ns_\alpha$ is the energy eigenvalue for the single particle state $\psi\ns_\alpha$ (possibly degenerate), and $\Hn\ns_\alpha$ is the number operator. Many-body eigenstates $\tket{\Vn}$ are labeled by the set of occupancies $\Vn=\{n\ns_\alpha\}$, with $\Hn\ns_\alpha\,\tket{\Vn}=n\ns_\alpha\tket{\Vn}$. Thus, $\HH\,\tket{\Vn}=E\ns_\Vn\>\tket{\Vn}$, where $E\ns_\Vn=\sum_\alpha n\ns_\alpha\,\ve\ns_\alpha$. $\bullet$ {\it Bosons and fermions\/}: The allowed values for $n\ns_\alpha$ are $n\ns_\alpha\in\{0,1,2,\ldots,\infty\}$ for bosons and $n\ns_\alpha\in\{0,1\}$ for fermions. $\bullet$ {\it Grand canonical ensemble\/}: Because of the constraint $\sum_\alpha n\ns_\alpha=N$, the ordinary canonical ensemble is inconvenient. Rather, we use the grand canonical ensemble, in which case \begin{equation*} \Omega(T,V,\mu)=\pm\kT\,\sum_\alpha\ln\!\Big(1\mp e^{-(\ve\ns_\alpha-\mu)/\kT}\Big)\ , \end{equation*} where the upper sign corresponds to bosons and the lower sign to fermions. The average number of particles occupying the single particle state $\psi\ns_\alpha$ is then \begin{equation*} \langle \Hn\ns_\alpha\rangle={\pz\Omega\over\pz\ve\ns_\alpha}={1\over e^{(\ve\ns_\alpha-\mu)/\kT}\mp 1}\ . \end{equation*} In the Maxwell-Boltzmann limit, $\mu\ll -\kT$ and $\langle n\ns_\alpha\rangle = z\,e^{-\ve\ns_\alpha/\kT}$, where $z=e^{\mu/\kT}$ is the fugacity. Note that this low-density limit is common to both bosons and fermions. $\bullet$ {\it Single particle density of states\/}: The single particle density of states per unit volume is defined to be \begin{equation*} g(\ve)={1\over V}\Tra\,\delta(\ve-\Hh) = {1\over V}\sum_\alpha \delta(\ve-\ve\ns_\alpha)\ , \end{equation*} where $\Hh$ is the one-body Hamiltonian. If $\Hh$ is isotropic, then $\ve=\ve(k)$, where $k=|\Bk|$ is the magnitude of the wavevector, and \begin{equation*} g(\ve)={\Sg\,\Omega\ns_d\over (2\pi)^d}\,{k^{d-1}\over {d\ve/dk}}\ , \end{equation*} where $\Sg$ is the degeneracy of each single particle energy state (due to spin, for example). $\bullet$ {\it Quantum virial expansion\/}: From $\Omega=-pV$, we have \begin{align*} n(T,z) &=\int\limits_{-\infty}^\infty\!\!\!d\ve\>{g(\ve)\over z^{-1}\,e^{\ve/\kT}\mp 1} = \sum_{j=1}^\infty (\pm 1)^{j-1}\,z^j\,C\ns_j(T)\\ {p(T,z)\over\kT}&=\mp\!\!\int\limits_{-\infty}^\infty\!\!\!d\ve\>g(\ve)\,\ln\!\big(1\mp z\,e^{-\ve/\kT}\big)= \sum_{j=1}^\infty (\pm 1)^{j-1}\,{z^j\over j}\, C\ns_j(T)\ , \end{align*} where \begin{equation*} C\ns_j(T)=\int\limits_{-\infty}^\infty\!\!\!d\ve\,g(\ve)\,e^{-j\ve/\kT}\ . \end{equation*} One now inverts $n=n(T,z)$ to obtain $z=z(T,n)$, then substitutes this into $p=p(T,z)$ to obtain a series expansion for the equation of state, \begin{equation*} p(T,n)=n\kT\Big( 1 + B\ns_2(T)\,n + B\ns_3(T)\,n^2 + \ldots\Big)\ . \end{equation*} The coefficients $B\ns_j(T)$ are the {\it virial coefficients\/}. One finds \begin{equation*} B\ns_2=\mp {C\ns_2\over 2 C_1^2} \qquad,\qquad B\ns_3={C_2^2\over C_1^4} - {2\,C_3\over 2\,C_1^3}\ . \end{equation*} $\bullet$ {\it Photon statistics\/}: Photons are bosonic excitations whose number is not conserved, hence $\mu=0$. The number distribution for photon statistics is then $n(\ve)=1/(e^{\beta\ve}-1)$. Examples of particles obeying photon statistics include phonons (lattice vibrations), magnons (spin waves), and of course photons themselves, for which $\ve(k)=\hbar c k$ with $\Sg=2$. The pressure and number density for the photon gas obey $p(T) = A\ns_d\,T^{d+1}$ and $n(T)=A'_d\,T^d$, where $d$ is the dimension of space and $A\ns_d$ and $A'_d$ are constants. $\bullet$ {\it Blackbody radiation\/}: The energy density per unit frequency of a three-dimensional blackbody is given{P by \begin{equation*} \ve(\nu,T)={8\pi h\over c^3}\cdot{\nu^3\over e^{h\nu/\kT}-1}\ . \end{equation*} The total power emitted per unit area of a blackbody is ${dP\over dA}=\sigma T^4$, where $\sigma=\pi^2 k_\ssr{B}^4/60\hbar^3 c^2 =5.67\times 10^{-8}\,\RW/\Rm^2\,\RK^4$ is Stefan's constant. $\bullet$ {\it Ideal Bose gas\/}: For Bose systems, we must have $\ve\ns_\alpha > \mu$ for all single particle states. The number density is \begin{equation*} n(T,\mu)=\impi d\ve\, {g(\ve)\over e^{\beta(\ve-\mu)} - 1}\ . \end{equation*} This is an increasing function of $\mu$ and an increasing function of $T$. For fixed $T$, the largest value $n(T,\mu)$ can attain is $n(T,\ve\ns_0)$, where $\ve\ns_0$ is the lowest possible single particle energy, for which $g(\ve)=0$ for $\ve < \ve\ns_0$. If $n\ns_\Rc(T)\equiv n(T,\ve\ns_0) < \infty$, this establishes a {\it critical density\/} above which there is {\it Bose condensation\/} into the energy $\ve\ns_0$ state. Conversely, for a given density $n$ there is a {\it critical temperature\/} $T\ns_\Rc(n)$ such that $n\ns_0$ is finite for $T<t\ns_\rc$\,.>T\ns_\Rc$, $n(T,\mu)$ is given by the integral formula above, with $n\ns_0=0$. For a ballistic dispersion $\ve(\Bk)=\hbar^2\Bk^2/2m$, one finds $n\lambda_{T\ns_\Rc}^d=\Sg\,\zeta(d/2)$, \ie\ $\kB T\ns_\Rc={2\pi\hbar^2\over m} \left(n\big/\Sg\,\zeta(d/2)\right)^{2/d}$. For $T<t\ns_\rc(n)$,>T\ns_\Rc(n)$, one has $n=\Sg\,{Li}\ns_{d\over 2}(z)\,\lambda_T^{-d}$ and $p=\Sg\,{Li}\ns_{{d\over 2}+1}(z)\,\kT\,\lambda_T^{-d}$, where \begin{equation*} {Li}\ns_q(z)\equiv\sum_{n=1}^\infty {z^n\over n\nsub^q}. \end{equation*} $\bullet$ {\it Ideal Fermi gas\/}: The Fermi distribution is $n(\ve)=f(\ve-\mu)=1\big/\!\left(e^{(\ve-\mu)/\kT}+1\right)$. At $T=0$, this is a step function: $n(\ve)=\RTheta(\mu-\ve)$, and $n=\int\limits_{-\infty}^\mu\!\! d\ve\>g(\ve)$. The chemical potential at $T=0$ is called the {\it Fermi energy\/}: $\mu(T=0,n)=\veF(n)$. If the dispersion is $\ve(\Bk)$, the locus of $\Bk$ values satisfying $\ve(\Bk)=\veF$ is called the {\it Fermi surface\/}. For an isotropic and monotonic dispersion $\ve(k)$, the Fermi surface is a sphere of radius $\kF$, the {\it Fermi wavevector\/}. For isotropic three-dimensional systems, $\kF=(6\pi^2 n/\Sg)^{1/3}$. $\bullet$ {\it Sommerfeld expansion\/}: Let $\phi(\ve)={d\Phi\over d\ve}$. Then \begin{align*} \impi d\ve\>f(\ve-\mu)\>\phi(\ve)&=\pi D\csc(\pi D)\,\Phi(\mu) \\ &=\Bigg\{1+{\pi^2\over 6}\,(\kT)^2\,{d^2\over d\mu^2} + {7\pi^4\over 360}\,(\kT)^4\,{d^4\over d\mu^4} + \ldots\Bigg\}\ \Phi(\mu)\ , \end{align*} where $D=\kT\,{d\over d\mu}$. One then finds, for example, $C\ns_V=\gamma V T$ with $\gamma=\third \pi^2 k_\ssr{B}^2\,g(\veF)$.