6.S: Summary

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References

• M. Kardar, Statistical Physics of Particles (Cambridge, 2007) A superb modern text, with many insightful presentations of key concepts.
• L. E. Reichl, A Modern Course in Statistical Physics (2nd edition, Wiley, 1998) A comprehensive graduate level text with an emphasis on nonequilibrium phenomena.
• 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.
• J.-P Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, 1990) An advanced, detailed discussion of liquid state physics.

Summary

$\bullet$ Lattice-based models: Amongst the many lattice-based models of physical interest are

$$\begin{aligned} &&\HH\ns_{ Ising} &=-J\sum_{\langle ij\rangle} \sigma\ns_i\,\sigma\ns_j - H\sum_i\sigma\ns_i &&;&& \sigma\ns_i \in\{-1,+1\}&& \\ &&\HH\ns_{ Potts} &=-J\sum_{\langle ij\rangle} \delta\ns_{\sigma\ns_i,\sigma\ns_j} - H \sum_i \delta\ns_{\sigma,1} &&;&& \sigma\ns_i\in \{1,\ldots,q\} &&\\ &&\HH\ns_{\SO(n)} &=-J\sum_{\langle ij\rangle} \nhat\ns_i\ncdot\nhat\ns_j - \BH\cdot\sum_i\nhat\ns_i &&;&& \nhat\ns_i\in S^{n-1} \ .&&\end{aligned}$$

Here $J$ is the coupling between neighboring sites and $H$ (or $\BH$) is a polarizing field which breaks a global symmetry (groups $\MZ\ns_2$ , $\CS\ns_q$ , and $\SO(n)$, respectively). $J>0$ describes a ferromagnet and $J<0$ an antiferromagnet. One can generalize to include further neighbor interactions, described by a matrix of couplings $J\ns_{ij}$. When $J=0$, the degrees of freedom at each site are independent, and $Z(T,N,J=0,H)=\zeta^N$, where $\zeta(T,H)$ is the single site partition function. When $J\ne 0$ it is in general impossible to compute the partition function analytically, except in certain special cases.

$\bullet$ Transfer matrix solution in $d=1$: One such special case is that of one-dimensional systems. In that case, one can write $Z=\Tra\!(R^N)$, where $R$ is the transfer matrix. Consider a general one-dimensional model with nearest-neighbor interactions and Hamiltonian

$$\HH=-\sum_n U(\alpha\ns_n,\alpha\ns_{n+1}) - \sum_n W(\alpha\ns_n)\ ,$$

where $\alpha\ns_n$ describes the local degree of freedom, which could be discrete or continuous, single component or multi-component. Then

$$R\ns_{\alpha\alpha'}=e^{U(\alpha,\alpha')/\kT}\,e^{W(\alpha')/\kT}\ .$$

The form of the transfer matrix is not unique, although its eigenvalues are. We could have taken $R\ns_{\alpha\alpha'}=e^{W(\alpha)/2\kT}\,e^{U(\alpha,\alpha')/\kT}\,e^{W(\alpha')/2\kT}$, for example. The interaction matrix $U(\alpha,\alpha')$ may or may not be symmetric itself. On a ring of $N$ sites, one has $Z=\sum_{i=1}^K\lambda_i^N$, where $\{\lambda_i\}$ are the eigenvalues and $K$ the rank of $R$. In the thermodynamic limit, the partition function is dominated by the eigenvalue with the largest magnitude.

$\bullet$ Higher dimensions: For one-dimensional classical systems with finite range interactions, the thermodynamic properties vary smoothly with temperature for all $T>0$. The lower critical dimension $d\ns_\ell$ of a model is the dimension at or below which there is no finite temperature phase transition. For models with discrete global symmetry groups, $d\ns_\ell=1$, while for continuous global symmetries $d\ns_\ell=2$. In zero external field the ($d=2$) square lattice Ising model has a critical temperature $T\ns_\Rc=2.269\,J$. On the honeycomb lattice, $T\ns_\Rc=1.519\,J$. For the $\SO(3)$ model on the cubic lattice, $T\ns_\Rc=4.515\,J$. In general, for unfrustrated systems, one expects for $d>d\ns_\ell$ that $T\ns_\Rc\propto z$, where $z$ is the lattice coordination number ( number of nearest neighbors).

$\bullet$ Nonideal classical gases: For $\HH=\sum_{i=1}^N {\Bp_i^2\over 2m} + \sum_{i<j}u\big(|\Bx\ns_i-\Bx\ns_j|\big)$, one has $Z(T,V,N)=\lambda_T^{-Nd}\,Q\ns_N(T,V)$, where

$$Q\ns_N(T,V)={1\over N!}\int\!\!d^d\!x\nd_1\cdots\int\!\! d^d\!x\nd_N\,\prod_{i<j} e^{-u(r\ns_{ij})/\kT}$$

is the configuration integral. For the one-dimensional Tonks gas of $N$ hard rods of length $a$ confined to the region $x\in[0,L]$, one finds $Q\ns_N(T,L)=(L-Na)^N$, whence the equation of state $p=n\kT/(1-na)$. For more complicated interactions, or in higher dimensions, the configuration integral is analytically intractable.

$\bullet$ Mayer cluster expansion: Writing the Mayer function $f\ns_{ij}\equiv e^{-u\ns_{ij}/\kT}-1$, and assuming $\int\!d^d\!r\,f(r)$ is finite, one can expand the pressure $p(T,z)$ and $n(T,z)$ as power series in the fugacity $z=\exp(\mu/\kT)$, viz.

$$\begin{aligned} p/\kT&=\sum_\gamma \big(z\lambda_T^{-d}\big)^{n\ns_\gamma}\,b\ns_\gamma\\ n&=\sum_\gamma n\ns_\gamma\, \big(z\lambda_T^{-d}\big)^{n\ns_\gamma}\,b\ns_\gamma\ .\end{aligned}$$

The sum is over unlabeled connected clusters $\gamma$, and $n\ns_\gamma$ is the number of vertices in $\gamma$. The cluster integral $b\ns_\gamma(T)$ is obtained by assigning labels $\{1,\ldots\,n\ns_\gamma\}$ to all the vertices, and computing

$$b\ns_\gamma(T)\equiv {1\over s\ns_\gamma}\cdot{1\over V} \!\int\!\!d^d\!x\nd_1\cdots d^d\!x\nd_{n\ns_\gamma}\prod_{i<j}^\gamma f\nd_{ij}\ ,$$

where $f\ns_{ij}$ appears in the product if there is a link between vertices $i$ and $j$. $s\ns_\gamma$ is the symmetry factor of the cluster, defined to be the number of elements from the symmetric group $\CS_{n\ns_\gamma}$ which, acting on the labels, would leave the product $\prod_{i<j}^\gamma f\nd_{ij}$ invariant. By definition, a cluster consisting of a single site has $b\ns_\bullet=1$. Translational invariance implies $b\ns_\gamma(T)\propto V^0$. One then inverts $n(T,z)$ to obtain $z(T,n)$, and inserting the result into the equation for $p(T,z)$ one obtains the virial expansion of the equation of state,

$$p=n\kT\,\Big\{1+B\ns_2(T)\,n + B\ns_3(T)\,n^2 + \ldots\Big\}\ .$$

where

$$B\ns_k(T)=-{1\over k (k-2)!}\sum_{\gamma\in \Gamma\ns_k} \int\!\!d^d\!x\ns_1\cdots\!\!\int\!\!d^d\!x\ns_{k-1}\prod_{\langle ij\rangle}^\gamma f\ns_{ij}$$

with $\Gamma\ns_k$ the set of all one-particle irreducible $j$-site clusters. An irreducible cluster is a connected cluster which does not break apart into more than one piece if any of its sites and all of that site’s connecting links are removed from the graph. Any site whose removal, along with all its connecting links, would result in a disconnected graph is called an articulation point. Irreducible clusters have no articulation points.

$\bullet$ Liquids: In the ordinary canonical ensemble,

$$P(\Bx\ns_1,\ldots,\Bx\ns_N)=Q_N^{-1}\cdot{1\over N!}\>e^{-\beta W(\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N)}\ ,$$

where $W$ is the total potential energy, and $Q\ns_N$ is the configuration integral,

$$Q\ns_N(T,V)={1\over N!}\int\!\!d^d\!x\ns_1\cdots\int\!\! d^d\!x\ns_N\>e^{-\beta W(\xoN)}\ .$$

We can use $P$, or its grand canonical generalization, to compute thermal averages, such as the average local density

$$\begin{aligned} n\ns_1(\Br)&= \blangle \sum_i\delta(\Br-\Bx\ns_i) \brangle \\ &=N\!\int\!\!d^d\!x\ns_2\cdots\!\int\!\!d^d\!x\ns_N\> P(\Br,\Bx\ns_2,\ldots,\Bx\ns_N)\end{aligned}$$

and the two particle density matrix, two-particle density matrix $n\ns_2(\Br\ns_1,\Br\ns_2)$ is defined by

$$\begin{aligned} n\ns_2(\Br\ns_1,\Br\ns_2)&=\blangle\sum_{i\ne j} \delta(\Br\nd_1-\Bx\ns_i)\,\delta(\Br\ns_2-\Bx\ns_j) \brangle \\ &=N(N-1)\!\int\!\!d^d\!x\ns_3\cdots\!\int\!\!d^d\!x\ns_N\> P(\Br\ns_1,\Br\ns_2,\Bx\ns_3,\ldots,\Bx\ns_N)\ .\end{aligned}$$

$\bullet$ Pair distribution function: For translationally invariant simple fluids consisting of identical point particles interacting by a two-body central potential $u(r)$, the thermodynamic properties follow from the behavior of the pair distribution function (pdf),

$$g(r)={1\over Vn^2}\>\blangle \sum_{i\ne j}\delta(\Br-\Bx\ns_i+\Bx\ns_j)\brangle\ ,$$

where $V$ is the total volume and $n=N/V$ the average density. The average energy per particle is then

$$\ve(n,T)={\langle E\rangle\over N}= \frac{3}{2}\, \kT + 2\pi n\!\!\int\limits_0^\infty \!\!dr\>r^2\,g(r)\,u(r)\ .$$

Here $g(r)$ is implicitly dependent on $n$ and $T$ as well In the grand canonical ensemble, the pdf satisfies the compressibility sum rule, $\int\! d^3\!r\,\big[g(r)-1\big]=\kT\,\kappa\ns_T - n^{-1}$, where $\kappa\ns_T$ is the isothermal compressibility. Note $g(\infty)=1$. The pdf also implies the virial equation of state,

$$p=n\kT -\frac{2}{3}\pi n^2\!\!\int\limits_0^\infty\!\!dr\,r^3\,g(r)\,u'(r)\ .$$

$\bullet$ Scattering: Scattering experiments are sensitive to momentum transfer $\hbar\Bq$ and energy transfer $\hbar\omega$, and allow determination of the dynamic structure factor

$$\begin{aligned} S(\Bq,\omega)&={1\over N}\!\!\int\limits_{-\infty}^\infty\!\!\!dt\>e^{i\omega t}\>\blangle \sum_{l,l'} e^{i\Bq\cdot\Bx\ns_l(0)}\,e^{-i\Bq\cdot\Bx\ns_{l'}(t)}\brangle\ns_T\\ &={2\pi\hbar\over N}\sum_i P\ns_i\ \sum_j \big|\expect{j}{\sum_{l=1}^N e^{-i\Bq\cdot\Bx\ns_l}}{i}\big|^2\,\delta(E\ns_j-E\ns_i+\hbar\omega)\ ,\end{aligned}$$

where $\sket{i}$ and $\sket{j}$ are (quantum) states of the system being studied, and $P\ns_i$ is the equilibrium probability for state $i$.¹ Integrating over all frequency, one obtains the static structure factor,

$$\begin{aligned} S(\Bq)&=\!\!\int\limits_{-\infty}^\infty\!\!{d\omega\over 2\pi}\>S(\Bq,\omega) ={1\over N}\sum_{l,l'} \blangle e^{i\Bq\cdot(\Bx\ns_l-\Bx\ns_{l'})}\brangle\\ &=N\,\delta\ns_{\Bq,0}+1+n\!\!\int\!\!d^d\!r\,e^{-i\Bq\cdot\Br}\,\big[g(r)-1\big]\ .\end{aligned}$$

$\bullet$ Theories of fluid structure – The BBGKY hierarchy is set of coupled integrodifferential equations relating $k$- and $(k+1)$-particle distribution functions. In order to make progress, a truncation must be performed, expressing higher order distributions in terms of lower order ones. This results in various theories of fluids, known by their defining equations for the pdf $g(r)$. Examples include the Born-Green-Yvon equation, the Percus-Yevick equation, the hypernetted chains equation, the Ornstein-Zernike approximation, Except in the simplest cases (such as the OZ approximation), these equations must be solved numerically. OZ approximation deserves special mention. There we write $S(\Bq)\approx {1\over (R/\xi)^2 + R^2 \Bq^2}$ for small $q$, where $\xi(T)$ is the correlation length and $R(T)$ is related to the range of interactions.

$\bullet$ Debye-Hückel theory – Due to the long-ranged nature of the Coulomb interaction, the Mayer function decays so slowly as $r\to\infty$ that it is not integrable, so the virial expansion is problematic. Progress can be made by a self-consistent mean field approach. For a system consisting of charges $\pm e$, one assumes a local electrostatic potential $\phi(\Br)$. Boltzmann statistics then gives a charge density

$$\rho(\Br)=e\lambda_+^{-d} z\ns_+\,e^{-e\,\phi(\Br)/\kT} - e\lambda_-^{-d} z\ns_-\,e^{e\,\phi(\Br)/\kT}\ ,$$

where $\lambda\ns_\pm$ and $z\ns_\pm$ are the thermal de Broglie wavelengths and fugacities for the $+$ and $-$ species. Assuming overall charge neutrality at infinity, one has $\lambda_+^{-d} z\ns_+ = \lambda_-^{-d} z\ns_- = n\ns_\infty$ , and we have $\rho(\Br)=-2e n\ns_\infty\,\sinh\!\big(e\,\phi(\Br)/\kT\big)$. The local potential is then determined self-consistently, using Poisson’s equation:

$$\nabla^2\phi = 8\pi e n\ns_\infty\,\sinh(e\phi/\kT) - 4\pi\rho\ns_{ ext}\ .$$

If $e\phi\ll\kT$, we can expand the sinh function to obtain $\nabla^2\phi =\kappa_{\ssr{D}}^2\,\phi -4\pi\rho\ns_{ ext}$ , where the Debye screening wavevector is $\kappa\ns_{\ssr{D}}=(8\pi n\ns_\infty e^2/\kT)^{1/2}$. The self-consistent potential arising from a point charge $\rho\ns_{ ext}(\Br)=Q\,\delta(\Br)$ is then of the Yukawa form $\phi(\Br)=Q\exp(-\kappa\ns_{\ssr{D}} r)/r$ in three space dimensions.

$\bullet$ Thomas-Fermi screening – In an electron gas with $\kT\ll\ve\ns_{\ssr{F}}$, we may take $T=0$. If the Fermi energy is constant, we write $\ve\ns_{\ssr{F}}={\hbar^2 k_{\ssr{F}}^2(\Br)\over 2m} - e\,\phi(\Br)$, and local electron number density is $n(\Br)=k_{\ssr{F}}^3(\Br)/3\pi^2$. Assuming a compensating smeared positive charge background $\rho\ns_+=e\,n\ns_\infty$, Poisson’s equation takes the form

$$\nabla^2\phi=4\pi e\,n\ns_\infty\cdot\left\{ \bigg(1+{e\phi(\Br)\over\veF}\bigg)^{\!\!3/2}-1\right\} -4\pi\rho\ns_{ ext}(\Br)\ .$$

If $e\phi\ll\ve\ns_{\ssr{F}}$, we expand in the presence of external sources to obtain $\nabla^2\phi =\kappa_{\ssr{TF}}^2\,\phi -4\pi\rho\ns_{ ext}$, where $\kappa_{\ssr{TF}}= (6\pi n\ns_\infty e^2/\ve\ns_{\ssr{F}})^{1/2}$ is the Thomas-Fermi screening wavevector. In metals, where the electron dispersion is a more general function of crystal momentum, the density response to a local potential $\phi(\Br)$ is $\delta n(\Br)=e\,\phi(\Br) \,g(\ve\ns_{\ssr{F}})$ to lowest order, where $g(\ve\ns_{\ssr{F}})$ is the density of states at the Fermi energy. One then finds $\kappa_{\ssr{TF}}=\sqrt{4\pi e^2 g(\ve\ns_{\ssr{F}})}$.

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1. In practice, what is measured is $S(q,\omega)$ convolved with spatial and energy resolution filters appropriate to the measuring apparatus.↩

1. Here we modify slightly the discussion in chapter 5 of the book by L. Peliti.
2. See. J. L. Lebowitz and A. E. Mazel, J. Stat. Phys. 90, 1051 (1998).
3. A corresponding mapping can be found between a cubic lattice and the linear chain as well.
4. Not that I personally think there’s anything wrong with that.
5. Disambiguation footnote: Take care not to confuse Philipp Lenard (Hungarian-German, cathode ray tubes, Nazi), Alfred-Marie Liénard (French, Liénard-Wiechert potentials, not a Nazi), John Lennard-Jones (British, molecular structure, definitely not a Nazi), and Lynyrd Skynyrd (American, "Free Bird”, possibly killed by Nazis in 1977 plane crash). I thank my colleague Oleg Shpyrko for setting me straight on this.
6. We assume that the long-ranged behavior of $f(r)\approx -\beta u(r)$ is integrable.
7. See C. N. Yang and R. D. Lee, Phys. Rev. 87, 404 (1952) and ibid, p. 410
8. See http://en.Wikipedia.org/wiki/Close-packing. For randomly close-packed hard spheres, one finds, from numerical simulations, $f\ns_{\ssr{RCP}}=0.644$.
9. To derive this expression, note that $\BF^{ (surf)}$ is directed inward and vanishes away from the surface. Each Cartesian direction $\alpha=(x,y,z)$ then contributes $-F^{ (surf)}_\alpha L_\alpha$, where $L_\alpha$ is the corresponding linear dimension. But $F^{ (surf)}_\alpha=pA_\alpha$, where $A_\alpha$ is the area of the corresponding face and $p$. is the pressure. Summing over the three possibilities for $\alpha$, one obtains Equation $\ref{surfvir}$.
10. We may write $\delta\ns_{\Bq,0}={1\over V}\,(2\pi)^d\,\delta(\Bq)$.
11. So named after Bogoliubov, Born, Green, Kirkwood, and Yvon.
12. I am grateful to Jonathan Lam and Olga Dudko for explaining this to me.
13. There are logarithmic corrections to the SAW result exactly at $d=4$, but for all $d>4$ one has $\nu=\half$.