4.S: Summary

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

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1. The generalization to the GCE is straightforward.↩

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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.↩