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Physics LibreTexts

4.S: Summary

  • Page ID
    18730
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    \)
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    ParseError: invalid DekiScript (click for details)
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        at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1
        at template()
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        at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1
        at template()
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        at template()
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    ParseError: invalid DekiScript (click for details)
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        at template()
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        at template()
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    \)
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        at template()
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        at template()
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    \)
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        at template()
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    \)
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        at template()
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        at template()
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    \)
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        at template()
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        at template()
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    ParseError: invalid DekiScript (click for details)
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        at template()
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    ParseError: invalid DekiScript (click for details)
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        at template()
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    ParseError: invalid DekiScript (click for details)
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    ParseError: invalid DekiScript (click for details)
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    ParseError: invalid DekiScript (click for details)
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    \( \newcommand\veF{\ve\ns_\RF}\)
    \( \newcommand\kF{k\ns_\RF}\)
    \( \newcommand\kFu{k\ns_{\RF\uar}}\)
    \( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
    \( \newcommand\muB{\mu\ns_\ssr{B}}\)
    \( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
    \( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
    \( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
    \( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
    \( \newcommand\xhihOZ
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    \( \newcommand\jhz{\HJ(0)}\)
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    \)
    \( \newcommand\msa{m\ns_\ssr{A}}\)
    \( \newcommand\msb{m\ns_\ssr{B}}\)
    \( \newcommand\mss{m\ns_\Rs}\)
    \( \newcommand\HBx{\hat\Bx}\)
    \( \newcommand\HBy{\hat\By}\)
    \( \newcommand\HBz{\hat\Bz}\)
    \( \newcommand\thm{\theta\ns_m}\)
    \( \newcommand\thp{\theta\ns_\phi}\)
    \( \newcommand\mtil{\widetilde m}\)
    \( \newcommand\phitil{\widetilde\phi}\)
    \( \newcommand\delf{\delta\! f}\)
    \( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
    \( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
    \( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
    \( \newcommand\vbar{\bar v}\)
    \( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
    \( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
    \( \newcommand\gla{g\nd_{\RLambda\nd}}\)
    \( \newcommand\TA{T\ns_\ssr{A}}\)
    \( \newcommand\TB{T\ns_\ssr{B}}\)
    \( \newcommand\ncdot{\!\cdot\!}\)
    \( \newcommand\NS{N\ns_{\textsf S}}\)

    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.
    • M. Kardar, Statistical Physics of Particles (Cambridge, 2007) A superb modern text, with many insightful presentations of key concepts.
    • 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

    \(\bullet\) Distributions: Let \(\vrh(\Bvphi)\) be a normalized distribution on phase space. Then

    \[\blangle f(\Bvphi)\brangle =\Tra\!\big[\vrh(\Bvphi)\,f(\Bvphi)\big]=\int\!d\mu\>\vrh(\Bvphi)\,f(\Bvphi)\ ,\]

    where \(d\mu=W(\Bvphi)\prod_i d\vphi_i\) is the phase space measure. For a Hamiltonian system of \(N\) identical indistinguishable point particles in \(d\) space dimensions, we have

    \[d\mu={1\over N!}\prod_{i=1}^N {d^d\!p\ns_i\,d^d\!q\ns_i\over (2\pi\hbar)^d}\ .\]

    The \({1\over N!}\) prefactor accounts for indistinguishability. Normalization means \(\Tra\!\vrh = 1\).

    \(\bullet\) Microcanonical ensemble (\(\mu\)CE): \(\vrh(\Bvphi)=\delta\big(E-\HH(\Bvphi)\big)/D(E)\), where \(D(E)=\Tra\delta\big(E-\HH(\Bvphi)\big)\) is the density of states and \(\HH(\Bvphi)=\HH(\Bq,\Bp)\) is the Hamiltonian. The energy \(E\), volume \(V\), and particle number \(N\) are held fixed. Thus, the density of states \(D(E,V,N)\) is a function of all three variables. The statistical entropy is \(S(E,V,N)=\kB\ln D(E,V,N)\), where \(\kB\) is Boltzmann’s constant. Since \(D\) has dimensions of \(E^{-1}\), an arbitrary energy scale is necessary to convert \(D\) to a dimensionless quantity before taking the log. In the thermodynamic limit, one has

    \[S(E,V,N)=N\kB\,\phi\bigg({E\over N},{V\over N}\bigg)\ .\]

    The differential of \(E\) is defined to be \(dE=T\,dS - p\,dV + \mu\,dN\), thus \(T=\pabc{E}{S}{\,V,N}\) is the temperature, \(p=-\pabc{E}{V}{\,S,N}\) is the pressure, and \(\mu=\pabc{E}{N}{\,S,V}\) is the chemical potential. Note that \(E\), \(S\), \(V\), and \(N\) are all extensive quantities, they are halved when the system itself is halved.

    \(\bullet\) Ordinary canonical ensemble (OCE): In the OCE, energy fluctuates, while \(V\), \(N\), and the temperature \(T\) are fixed. The distribution is \(\vrh=Z^{-1}\,e^{-\beta\HH}\), where \(\beta=1/\kT\) and \(Z=\Tra e^{-\beta\HH}\) is the partition function. Note that \(Z\) is the Laplace transform of the density of states: \(Z=\int\!dE\>D(E)\,e^{-\beta E}\). The Boltzmann entropy is \(S=-\kB\Tra(\vrh\ln\vrh)\). This entails \(F=E-TS\), where \(F=-\kT\ln Z\) is the Helmholtz free energy, a Legendre transform of the energy \(E\). From this we derive \(dF=-S\,dT - p\,dV + \mu\,dN\).

    \(\bullet\) Grand canonical ensemble (GCE): In the GCE, both \(E\) and \(N\) fluctuate, while \(T\), \(V\), and chemical potential \(\mu\) remain fixed. Then \(\vrh=\RXi^{-1}\,e^{-\beta(\HH-\mu\HN)}\), where \(\RXi=\Tra e^{-\beta(\HH-\mu\HN)}\) is the grand partition function and \(\Omega=-\kT\ln\RXi\) is the grand potential. Assuming \([\HH,\HN]=0\), we can label states \(\sket{n}\) by both energy and particle number. Then \(P\ns_n=\RXi^{-1}\,e^{-\beta (E\ns_n-\mu N\ns_n)}\). We also have \(\Omega=E-TS-\mu N\), hence \(d\Omega=-S\,dT - p\,dV\,-N\,d\mu\).

    \(\bullet\) Thermodynamics: From \(E=\Tra(\vrh\>\HH)\), we have \(dE=\Tra(\HH\,d\vrh) + \Tra(\vrh\,d\HH)= \dbar Q - \dbar W\), where \(\dbar Q=T \, dS\) and

    \[\dbar W = -\Tra(\vrh\>d\HH)=-\sum_n P\ns_n\sum_i {\pz E\ns_n\over\pz X\ns_i} \> dX\ns_i = \sum_i F\ns_i \> dX\ns_i\ ,\]

    with \(P\ns_n=Z^{-1}e^{-E\ns_n/\kT}\). Here \(F\ns_i=-\big\langle{\pz\HH\over\pz X\ns_i}\big\rangle\) is the generalized force conjugate to the generalized displacement \(X\ns_i\).

    \(\bullet\) Thermal contact: In equilibrium, two systems which can exchange energy satisfy \(T\ns_1=T\ns_2\). Two systems which can exchange volume satisfy \(p\ns_1/T\ns_1=p\ns_2/T\ns_2\). Two systems which can exchange particle number satisfy \(\mu\ns_1/T\ns_1=\mu\ns_2/T\ns_2\).

    \(\bullet\) Gibbs-Duhem relation: Since \(E(S,V,N)\) is extensive, Euler’s theorem for homogeneous functions guarantees that \(E=TS-pV+\mu N\). Taking the differential, we obtain the equation \(S\,dT - V dp + N d\mu=0\), so there must be a relation among any two of the intensive quantities \(T\), \(p\), and \(\mu\).

    \(\bullet\) Generalized susceptibilities: Within the OCE1, let \(\HH(\Blambda)=\HH\ns_0-\sum_i\lambda\ns_i\,\HQ\ns_i\), where \(\HQ\ns_i\) are observables with \([\HQ\ns_i,\HQ\ns_j]=0\). Then

    \[Q\ns_k(T,V,N;\Blambda)=\langle \HQ\ns_k\rangle = -{\pz F\over\pz\lambda\ns_k} \qquad,\qquad \xhi\ns_{kl}(T,V,N;\Blambda)={1\over V}\,{\pz Q\ns_k\over\pz\lambda\ns_l} = -{1\over V}\,{\pz^2\!F\over\pz\lambda\ns_k\, \pz\lambda\ns_l}\ .\]

    The quantities \(\xhi\ns_{kl}\) are the generalized susceptibilities.

    \(\bullet\) Ideal gases: For \(\HH=\sum_{i=1}^N{\Bp_i^2\over 2m}\), one finds \(Z(T,V,N)={1\over N!} \big({V\over\lambda_T^d}\big)^N\), where \(\lambda\ns_T=\sqrt

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    \) is the thermal wavelength. Thus \(F=N\kT\ln(N/V)-\half d N\kT\ln T + Na\), where \(a\) is a constant. From this one finds \(p=-\pabc{F}{V}{\,T,N}=n\kT\), which is the ideal gas law, with \(n={N\over V}\) the number density. The distribution of velocities in \(d=3\) dimensions is given by

    \[f(\Bv)=\Big\langle {1\over N}\sum_{i=1}^N \delta(\Bv-\Bv\ns_i\big)\Big\rangle = \bigg({m\over2\pi\kT}\bigg)^{3/2} e^{-m\Bv^2/2\kT}\ ,\]

    and this leads to a speed distribution \({\bar f}(v)=4\pi v^2 f(v)\).

    \(\bullet\) Example: For \(N\) noninteracting spins in an external magnetic field \(H\), the Hamiltonian is \(\HH=-\mu\ns_0 H\sum_{i=1}^N \sigma\ns_i\), where \(\sigma\ns_i=\pm 1\). The spins, if on a lattice, are regarded as distinguishable. Then \(Z=\zeta^N\), where \(\zeta=\sum_{\sigma=\pm 1} e^{\beta\mu\ns_0 H \sigma}= 2\cosh(\beta\mu\ns_0 H)\). The magnetization and magnetic susceptibility are then

    \[M=-\pabc{F}{H}{T,N}=N\mu\ns_0\tanh\!\bigg({\mu\ns_0 H\over\kT}\bigg) \qquad,\qquad \xhi={\pz M\over\pz H}={N\mu_0^2\over \kT}\>{sech}^2\bigg({\mu\ns_0 H\over\kT}\bigg)\ .\]

    \(\bullet\) Example: For noninteracting particles with kinetic energy \({\Bp^2\over 2m}\) and internal degrees of freedom, \(Z\ns_N={1\over N!}\big({V\over\lambda_T^d}\big)^N\xi^N(T)\), where \(\xi(T)=\Tra e^{-\beta \Hh\ns_{int}}\) is the partition function for the internal degrees of freedom, which include rotational, vibrational, and electronic excitations. One still has \(pV=N\kT\), but the heat capacities at constant \(V\) and \(p\) are

    \[C\ns_V=T\pabc{S}{T}{V,N}=\half d N\kB - N T\vphi''(T) \qquad,\qquad C\ns_p=T\pabc{S}{T}{p,N}=C\ns_V+N\kB\ ,\]

    where \(\vphi(T)=-\kT\ln\xi(T)\).


    1. The generalization to the GCE is straightforward.

    Endnotes

    1. We write the Hamiltonian as \(\HH\) (classical or quantum) in order to distinguish it from magnetic field (\(H\)) or enthalpy \((\CH)\).
    2. More on this in chapter 5.
    3. The factor of \(\half\) preceding \(\Omega\ns_M\) in Equation [nrdos] appears because \(\delta(u^2-1)=\half\,\delta(u-1)+\half\,\delta(u+1)\). Since \(u=|\Bu|\ge 0\), the second term can be dropped.
    4. Note that for integer argument, \(\RGamma(k)=(k-1)!\)
    5. See §2.7.4.
    6. See T.-C. Lu and T. Grover, arXiv 1709.08784.
    7. In applying Equation [EminusTS] to the denominator of Equation [PEOCE], we shift \(\CE'\) by \(E\) and integrate over the difference \(\delta\CE'\equiv\CE'-E\), retaining terms up to quadratic order in \(\delta\CE'\) in the argument of the exponent.
    8. In deriving Equation [thermforce], we have used the so-called Feynman-Hellman theorem of quantum mechanics: \(d\texpect{n}{\HH}{n}=\texpect{n}{\,d\HH\,}{n}\), if \(\tket{n}\) is an energy eigenstate.
    9. Nota bene we are concerned with classical spin configurations only – there is no superposition of states allowed in this model!
    10. Note that while we cannot simultaneously specify the eigenvalues of two components of \(\BL\) along axes fixed in space, we can simultaneously specify the components of \(\BL\) along one axis fixed in space and one axis rotating with a body. See Landau and Lifshitz, Quantum Mechanics, §103.
    11. See §72 of Landau and Lifshitz, Quantum Mechanics, which, in my humble estimation, is the greatest physics book ever written.
    12. See Landau and Lifshitz, Quantum Mechanics, §86.
    13. Note that there is no prime on the \(\Bk\) sum for \(F\), as we have divided the logarithm of \(Z\) by two and replaced the half sum by the whole sum.
    14. The hyperfine splitting in hydrogen is on the order of \((m\ns_\Re/m\ns_\Rp)\,\alpha^4\,m\ns_\Re c^2\sim 10^{-6}\,\)eV, which is on the order of \(0.01\,\)K. Here \(\alpha=e^2/\hbar c\) is the fine structure constant.