Chapter 9: Statistical physicsfrom Maxwell's molecules to Fermi and Bose gases

9.1Historical overview

The mechanistic view developed by Newton, Lagrange, Hamilton, and Laplace rests on deterministic laws: with perfect knowledge of the universe, Laplace imagined, one could predict the future perfectly. Heisenberg's uncertainty principle created serious problems for that position. Statistical physics grew up alongside thermodynamics, which was driven by the Industrial Revolution (Sadi Carnot's Reflections on the Motive Power of Fire appeared in 1824). Around 1800 heat was commonly viewed as a fluid, "caloric"; in 1798 Benjamin Thompson (Count Rumford) proposed that heat is merely the motion of individual particles in a substance. Joseph Fourier published a quantitative theory of heat in 1822, and around 1843 James Prescott Joule measured the mechanical equivalent of heat: in his paddle-wheel experiment, the energy of falling weights was transferred to internal energy in the water. James Clerk Maxwell brought probability and statistics to bear on thermodynamics, deriving the distributions of velocities and speeds of molecules in an ideal gas and using them to derive observed macroscopic phenomena. Statistical mechanics won the day only in the early twentieth century: in 1905 Einstein's theory of Brownian (random) motion supported the view that atoms are real, and Perrin's experiments confirmed it a few years later.

9.2Maxwell velocity distribution

Laplace's dream: in principle we could know everything about an ideal gas by knowing the position \((x,y,z)\) and instantaneous velocity \((v_x,v_y,v_z)\) of every molecule, a point in a six-dimensional phase space. For the thermal properties of an ideal gas, the energy depends only on the velocities, not the positions. Maxwell asked: what is the distribution of velocities for an ideal gas at a given temperature? Define a velocity distribution function \(f(\vec v)\) so that \(f(\vec v)\,d^3\vec v\) is the probability of finding a particle with velocity between \(\vec v\) and \(\vec v + d^3\vec v\), where \(d^3\vec v = dv_x\,dv_y\,dv_z\) (a role similar to \(\Psi^*\Psi\) in quantum theory). Maxwell proved that

with \(m\) the molecular mass, \(v^2 = v_x^2+v_y^2+v_z^2\), and \(k_B\) Boltzmann's constant. It factors as \(g(v_x)g(v_y)g(v_z)\), with \(g(v_x)\,dv_x = C'\exp(-\tfrac12\beta mv_x^2)\,dv_x\) and \(C' = C^{1/3}\) by symmetry. Normalizing with \(\int_{-\infty}^{\infty}e^{-ax^2}dx = \sqrt{\pi/a}\),

a Gaussian. The mean \(\bar v_x = C'\int v_x\exp(-\tfrac12\beta mv_x^2)\,dv_x = 0\) (odd integrand, and by symmetry): the velocities are randomly distributed. Using \(\int x^2e^{-ax^2}dx = \tfrac{\sqrt\pi}{2}a^{-3/2}\) (from differentiating with respect to \(a\)), \(\overline{v_x^2} = 1/\beta m = k_BT/m\), so

one of the principal results of kinetic theory, derived from purely statistical considerations. Example 9.1: at room temperature (293 K) a molecule's mean translational kinetic energy is \(6.07\times10^{-21}\ \text{J} = 0.038\ \text{eV}\) (about \(\tfrac1{25}\) eV, a good number to remember), and a mole has \(3650\) J.

9.3Equipartition theorem (등분배 정리)

The result \(\tfrac12m\overline{v_x^2} = \tfrac12k_BT\) holds equally for \(v_y\) and \(v_z\), giving a net average translational kinetic energy of \(\tfrac32k_BT\) per molecule; for a monatomic gas such as He or Ar, virtually all the gas's energy is in this form. It generalizes into a statement relating a system's internal energy to its temperature:

(Its proof requires more advanced statistical physics, the Gibbs–Boltzmann statistics.) The quadratic term may be in a coordinate, a velocity component, an angular velocity component, or anything else that, squared, is proportional to energy; each such independent phase-space coordinate is a degree of freedom. A monatomic ideal gas has \(K = \tfrac12m(v_x^2+v_y^2+v_z^2)\), three degrees of freedom, so \(3\times\tfrac12k_BT\) per molecule and internal energy \(U = N\bar E = \tfrac32Nk_BT\). The heat capacity at constant volume (등적열용량), from the first law \(\Delta U = Q + W = Q - p\Delta V\) with \(\Delta V = 0\), is

with \(R = 8.31\) J/K the ideal gas constant. That matches Ar and He, but not diatomic gases. A diatomic molecule such as O₂ can also rotate: in the rigid-rotator model, \(\tfrac12I_x\omega_x^2\) and \(\tfrac12I_y\omega_y^2\) add \(2\times\tfrac12k_BT\), for \(\big(\tfrac32+1\big)k_BT = \tfrac52k_BT\) per molecule. Why not rotations about \(z\), the axis through both atoms? The answer is quantum: the rotator's allowed energies are \(E = L^2/2I = \hbar^2\ell(\ell+1)/2I\). Almost all of each atom's mass is in its tiny nucleus, so \(I_z\) is orders of magnitude smaller than \(I_x\) or \(I_y\), and the small \(I_z\) in the denominator gives a high energy even for small quantum numbers: that mode cannot be excited. If the atoms are connected by a spring, the vibrational kinetic energy \(\tfrac12m(dr/dt)^2\) and potential energy \(\tfrac12\kappa(r-r_0)^2\) add two more degrees of freedom (on average the kinetic and potential energies of an oscillator are equal, Problem 53), for \(\tfrac72R\) in total. But high temperatures are normally needed to excite vibrations: for H₂, \(c_V\approx\tfrac32R\) just above its boiling point (about 20 K), rotation turns on below 100 K until \(c_V = \tfrac52R\) from about 250 K to 1000 K, and vibration then turns on gradually, approaching \(\tfrac72R\) before the molecule dissociates. Polyatomic molecules have one vibration frequency per normal mode, each turning on at its own temperature. In a solid such as copper (a face-centered cubic lattice), each atom is a three-dimensional harmonic oscillator with six degrees of freedom, so the molar heat capacity is \(6(\tfrac12R) = 3R\), almost exactly what is measured for copper at room temperature.

9.4Maxwell speed distribution

The velocity distribution, \(f(\vec v)\,d^3\vec v = (\beta m/2\pi)^{3/2}\exp(-\tfrac12\beta mv^2)\,d^3\vec v\), depends only on the speed \(v\), yet it is still a velocity distribution. We want a speed distribution \(F(v)\), with \(F(v)\,dv\) the probability of finding a particle with speed between \(v\) and \(v+dv\). It is not possible simply to set \(f(\vec v) = F(v)\): the speed distribution must count all the particles in a spherical shell in velocity space between \(v\) and \(v+dv\), of volume \(4\pi v^2dv\). Thus \(F(v)\,dv = f(\vec v)\,4\pi v^2dv\):

the Maxwell speed distribution, with a new \(v^2\) dependence: \(F\to0\) as \(v\to0\), and as \(v\to\infty\) because the exponential dominates. Its asymmetry makes three characteristic speeds slightly different. Setting \(dF/dv = 0\) gives the most probable speed, and integrating gives the mean and rms speeds:

A molecule moving at \(v^*\) has \(K^* = k_BT\), exactly two-thirds of the mean kinetic energy; \(v_{\rm rms}\) is consistent with \(\bar K = \tfrac12m\overline{v^2} = \tfrac32k_BT\). The standard deviation, \(\sigma_v = (\overline{v^2}-\bar v^2)^{1/2} = [3-8/\pi]^{1/2}(k_BT/m)^{1/2}\approx0.48\,v^*\), grows as \(\sqrt T\): the distribution widens with temperature. Example 9.3: at 293 K the mean speed of H₂ (2.02 u) is 1750 m/s and that of radon (222 u) 167 m/s; other gases lie between. Example 9.4: the fraction of molecules with speeds within ±1% of \(v^*\), approximated as \(F(v^*)(0.02v^*) = \frac{4}{\sqrt\pi}e^{-1}(0.02)\approx0.017\); the book invites students to check this numerically. Conceptual Example 9.5: \(v_{\rm rms}>\bar v>v^*\) because the distribution's long tail lies at high speeds, and squaring weights high speeds even more heavily.

9.5Classical and quantum statistics

The Maxwell speed distribution is valid only in the classical (nonrelativistic) limit, so it fails at very high temperatures; quantum theory imposes a more severe restriction. In liquids and solids, with densities several orders of magnitude higher than gases, particles are packed closely and the Pauli exclusion principle prevents two particles in identical quantum states from sharing the same space, limiting the allowed energy states. In quantum systems, moreover, only certain energies are allowed. Rewriting the classical speed distribution in terms of energy, with \(E = \tfrac12mv^2\), \(dE = mv\,dv\), \(dv = dE/\sqrt{2mE}\),

where \(\sqrt E\) comes from the phase-space volume and \(e^{-\beta E}\), characteristic of any classical system, is the Maxwell–Boltzmann factor (or Boltzmann factor), \(F_{\rm MB} = Ae^{-\beta E}\) with a normalization constant \(A\). The number of particles with energy between \(E\) and \(E+dE\) is \(n(E)\,dE\), with

where \(g(E)\) is the density of states, the number of states per unit energy. Example 9.6: in atomic hydrogen, the ground state has \(g(E_1) = 2\) and the first excited state \(g(E_2) = 8\), so \(n(E_2)/n(E_1) = 4\exp[-\beta(10.2\ \text{eV})]\): about \(10^{-175}\) at 293 K, \(2\times10^{-10}\) at 5000 K (the Sun's surface), and 3.55 at \(10^6\) K (a star's interior, where hydrogen is actually ionized into a plasma).

Quantum distributions. Particles are described by wave functions, and identical particles cannot be distinguished from one another if their wave functions overlap significantly. Take two particles, each equally likely to be in either of two states. If they are distinguishable (A and B), there are four configurations (AB in state 1; A in 1 and B in 2; B in 1 and A in 2; AB in 2), each with probability \(\tfrac14\). If they are indistinguishable, there are only three, each with probability \(\tfrac13\). Whether particles obey the Pauli exclusion principle is determined by their spin: fermions with half-integer spin (\(\tfrac12,\tfrac32,\dots\)), such as protons, neutrons, and electrons, obey it; bosons with zero or integer spin, such as photons and pions, do not. Atoms and molecules made of an even number of fermions are bosons (their total spin is an integer), and those with an odd number are fermions.

The two quantum distributions are, for fermions and bosons respectively,

where the normalization factors \(B_i\) are related to a thermodynamic quantity called the chemical potential \(\mu\), the energy needed to add a particle, through \(Be^{\beta E} = e^{\beta(E-\mu)}\), i.e. \(B = e^{-\beta\mu}\). The Fermi–Dirac and Bose–Einstein distributions differ only by the normalization constant and the sign of the 1 in the denominator, but this sign difference has major consequences. Both reduce to the classical Maxwell–Boltzmann distribution when \(B_ie^{\beta E}\gg1\) (then \(A = 1/B_i\)): the probability that a particular state is occupied is much less than 1, as for a dilute gas. A good rule of thumb: if the spacing between particles is much greater than their de Broglie wavelength, Maxwell–Boltzmann statistics are accurate; otherwise use quantum statistics. With all normalization constants set to 1, \(F_{\rm BE}\) lies above \(F_{\rm FD}\) at every energy, because bosons can crowd into the lower states, and all three coincide at high energies, the classical limit.

9.6Fermi–Dirac statistics

Write \(F_{\rm FD} = 1/[B_{\rm FD}e^{\beta E}+1] = 1/[e^{\beta(E-E_F)}+1]\), with \(B_{\rm FD} = e^{-\beta E_F}\) and \(E_F\) the Fermi energy. Strictly speaking \(E_F\) is defined at \(T = 0\); for \(T>0\) it is replaced by the temperature-dependent chemical potential \(\mu\). At \(E = E_F\), \(F_{\rm FD} = \tfrac12\) exactly. As \(T\to0\) (\(\beta\to\infty\)), \(F_{\rm FD} = 1\) for \(E\lt E_F\) and 0 for \(E>E_F\): the fermions fill the lowest energy levels available. As the temperature rises, fermions jump to higher levels and the sharp step is smeared into a smoother curve. It is useful to define the Fermi temperature \(T_F\equiv E_F/k\): for \(T\ll T_F\) the step function is reasonably accurate, and for \(T\gg T_F\), \(F_{\rm FD}\) approaches a decaying exponential, the Maxwell–Boltzmann limit.

Classical theory of electrical conduction. In 1900 Paul Drude modeled a metal as a lattice of positive ions with a gas of free electrons flowing through it. Electrons have thermal speeds of about \(10^5\) m/s at room temperature, directed randomly; under an applied field they drift, restricted by collisions with the lattice ions, which gives Ohm's law. But the conductivity \(\sigma = ne^2\tau/m\), with \(n\) the electron density and \(\tau = \ell/\bar v\) the mean time between collisions (\(\ell\) the mean free path), came out an order of magnitude too small. Since \(\bar v\propto\sqrt T\), Drude's model also predicts \(\sigma\propto T^{-1/2}\), while for most conductors \(\sigma\propto T^{-1}\). And equipartition would add \(\tfrac32R\) for the electron gas to the \(3R\) of the lattice, giving \(\tfrac92R\); the measured electronic contribution is only about \(0.02R\) per mole at room temperature.

Quantum theory of electrical conduction. Electrons are fermions, so we use Fermi–Dirac statistics, and the main problem is to find \(g(E)\). Recall the three-dimensional infinite well, physically a cubic box of ions of side \(L\): \(E = \frac{h^2}{8mL^2}(n_1^2+n_2^2+n_3^2) = r^2E_1\), with \(E_1 = h^2/8mL^2\) and \(r^2 = n_1^2+n_2^2+n_3^2\) the squared "radius" in number space. At room temperature the distribution is close to the \(T = 0\) step (\(T_F\approx80{,}000\) K for copper). The number of allowed states up to radius \(r\) is one octant of a sphere (the \(n_i\) are positive), doubled for the spin degeneracy:

At \(T = 0\) the Fermi energy is the energy of the highest occupied level, so for \(N\) electrons

where \(N/L^3\) is the number density of conduction electrons, a quantity measured with the Hall effect. Example 9.7: for copper, \(N/L^3 = 8.47\times10^{28}\ \text{m}^{-3}\) gives \(E_F = 7.03\) eV and \(T_F = 8.16\times10^4\) K.

The density of states is \(g(E) = dN_r/dE = \frac\pi2E_1^{-3/2}E^{1/2} = \frac{3N}{2}E_F^{-3/2}E^{1/2}\propto\sqrt E\), and at \(T = 0\), \(n(E) = g(E)\) for \(E\lt E_F\) and 0 above. The mean electronic energy is

a reasonable result, so \(U = \tfrac35NE_F\) at \(T = 0\). For \(T>0\), only electrons within about \(k_BT\) of \(E_F\) can absorb thermal energy and jump to higher states: a fraction of order \(k_BT/E_F\), each gaining of order \(k_BT\). So \(U = \tfrac35NE_F + \alpha N\frac{k_BT}{E_F}(k_BT)\) and \(C_V = \partial U/\partial T = 2\alpha Nk_B\frac{T}{T_F}\), or per mole \(c_V = 2\alpha R\,T/T_F\), with \(\alpha = \pi^2/4\) from Arnold Sommerfeld's more advanced calculation. Example 9.8: at 293 K the electronic contribution is \(c_V = \frac{\pi^2T}{2T_F}R = 0.0177R\) for copper and \(0.0227R\) for silver (\(T_F = 6.38\times10^4\) K): of the order of the measured 0.02R, and small compared with the lattice's \(3R\).

For the conductivity, replace \(\bar v\) with the Fermi speed \(u_F = \sqrt{2E_F/m}\approx1.6\times10^6\) m/s for copper, since the conduction electrons are those at the highest levels. That is even faster than \(\bar v\), but the classical mean free path was also wrong: the ions are not hard spheres, and electrons behave as waves. Einstein calculated \(\ell\) to be about 40 nm in copper at room temperature, giving \(\sigma = ne^2\ell/mu_F\approx6\times10^7\ \Omega^{-1}\text{m}^{-1}\), just right. As temperature increases, ionic vibrations grow, the mean free path shrinks, and the conductivity falls. Treating the ions as harmonic oscillators whose effective cross section is proportional to the square of their amplitude, and hence to their energy and to \(T\), gives \(\sigma\propto\ell\propto r^{-2}\propto U^{-1}\propto T^{-1}\), as observed for most pure metals.

9.7Bose–Einstein statistics

Two examples: the Planck formula for blackbody radiation, and the properties of liquid helium.

Blackbody radiation. Recall \(\mathcal I(\lambda,T) = \frac{2\pi c^2h}{\lambda^5}\frac{1}{e^{hc/\lambda k_BT}-1}\). Assume the radiation is a collection of photons of energy \(hc/\lambda\), bosons with spin 1. As for electrons, the key is the density of states. The "photon gas" is a collection of free particles in a three-dimensional infinite well, but photons are massless: \(E = pc = c\sqrt{p_x^2+p_y^2+p_z^2} = \frac{hc}{2L}\sqrt{n_x^2+n_y^2+n_z^2}\), since \(p_x = h/\lambda_x = hn_x/2L\). Counting again, with two photon polarizations in place of two spin states, \(N_r = 2\times\tfrac18\times\tfrac43\pi r^3\) with \(r = 2LE/hc\), so \(N_r = \frac{8\pi L^3}{3h^3c^3}E^3\) and \(g(E) = \frac{8\pi L^3}{h^3c^3}E^2\). With \(B_{\rm BE} = 1\) (\(\mu = 0\)), because the number of photons is not constant (they are absorbed and emitted by the cavity walls),

the energy per unit volume. With \(E = hc/\lambda\) and \(|dE| = (hc/\lambda^2)d\lambda\),

the same as Chapter 3. (The factor \(c/4\) converts energy per unit volume into power per unit area: of the radiation crossing a small area, only the part heading outward escapes, and averaging over directions gives \(\tfrac14\); integrating over wavelength yields the Stefan–Boltzmann law with \(\sigma = 2\pi^5k_B^4/15h^3c^2\).) Planck did not use the Bose–Einstein distribution; the problem was first solved this way in 1924 by Satyendra Nath Bose, for whom the boson is named. Einstein helped him publish and extended the distribution, all before quantum mechanics and spin were fully developed.

Liquid helium. Helium has the lowest boiling point of any element (4.2 K at 1 atm) and no solid phase at normal pressures; Heike Kamerlingh Onnes liquefied it in 1908. Measurements of its density (Kamerlingh Onnes and Boks, 1924) and specific heat (Keesom and Clausius, 1932) show something extraordinary near 2.17 K, the critical temperature, or lambda point after the shape of the specific-heat curve. Cooled toward 2.17 K the liquid boils vigorously, then suddenly becomes calm: a transition from the normal phase to the superfluid phase. The flow rate through a capillary rises dramatically below the lambda point (the superfluid can even creep up and over the walls of its container as a film), yet a plate dragged through the liquid feels a viscosity like a normal fluid's. Fritz London (1938) resolved the paradox with a two-fluid model: below the lambda point the liquid is part superfluid and part normal, with superfluid fraction

With two protons, two neutrons, and two electrons, the \(^4\)He atom is a boson, so the superfluid is a Bose–Einstein condensation into a single state: all the particles in a condensate are in the same quantum state, impossible for fermions. The fermion isotope \(^3\)He becomes superfluid only at 2.7 mK, by a mechanism like electron pairing in superconductors (Section 10.5). To estimate \(T_c\), take the fermion density of states and divide by 2 (no spin degeneracy for spin-zero bosons): \(g_{\rm BE}(E) = \frac{2\pi V}{h^3}(2m)^{3/2}E^{1/2}\). Normalizing, with \(u = E/k_BT\),

using the minimum allowed \(B_{\rm BE} = 1\) (the integral is then \(2.315\)). Rearranging,

for liquid helium's number density \(N/V = 2.11\times10^{28}\ \text{m}^{-3}\): the minimum temperature for a normal Bose–Einstein gas. Below it, atoms must condense into the ground state. The estimate is within 1 K of the observed 2.17 K, even though it uses a density of states for a noninteracting gas rather than a liquid. Example 9.9: for nitrogen gas at room conditions (\(N/V = 2.50\times10^{25}\ \text{m}^{-3}\)), normalizing \(n(E) = g(E)Ae^{-E/kT}\) gives \(A = \frac{h^3N}{V}(2\pi mkT)^{-3/2} = 1.8\times10^{-7}\ll1\), so Maxwell–Boltzmann statistics are justified.

Symmetry of boson and fermion wave functions

Consider two identical particles, "1" and "2", described by \(\Psi(1,2)\). Interchanging them cannot change the probability density, \(|\Psi(1,2)|^2 = |\Psi(2,1)|^2\), so \(\Psi(1,2) = \pm\Psi(2,1)\): symmetric for the plus sign and antisymmetric for the minus sign. Bosons have symmetric wave functions and fermions antisymmetric ones. For noninteracting particles in states \(a\) and \(b\),

This justifies the Pauli exclusion principle: if two fermions are in the same state, \(a = b\) and \(\Psi_A = 0\). Bosons in the same state have \(\Psi_S = \frac{2}{\sqrt2}\Psi_a(1)\Psi_a(2)\), so \(\Psi_S^*\Psi_S = 2|\Psi_a(1)\Psi_a(2)|^2\): two bosons have a nonzero, enhanced probability of occupying the same state, and this generalizes to the Bose–Einstein condensates observed.

Bose–Einstein condensation in gases. Strong interactions long prevented the low temperatures and high densities needed to produce a condensate in a gas. In 1995 Eric Cornell and Carl Wieman at Boulder laser-cooled \(^{87}\)Rb atoms to about 1 mK, then evaporatively cooled them in a magnetic trap (driving away the faster atoms with radio waves) to about 20 nK. At about 170 nK the cloud's broad velocity distribution suddenly gave way to an extremely narrow one, with the condensed fraction growing as the temperature fell further, just as in a superfluid. Wolfgang Ketterle's group at MIT condensed sodium later that year, beginning at about 2 µK, and the three shared the 2001 Nobel Prize in Physics. Ketterle's group demonstrated the atom laser, a coherent beam of atoms drawn from a condensate, in 1996; atom lasers are used in atomic clocks and other precision measurements, thin-film deposition, and lithography.

This completes Chapter 9, and with it this set of Modern Physics notes. For where statistical physics goes from here, see Where to go next.

Where to go next

This chapter is a first look at statistical physics. The department's statistical physics courses take it up properly, following D. V. Schroeder, An Introduction to Thermal Physics, and each has its own set of interactive demos.

Statistical Physics 1 (통계물리1)

The foundations that this chapter takes for granted: energy, heat, and work; microstates, multiplicity, and entropy; what temperature really is; heat engines; thermodynamic potentials; and phase transformations.

Statistical Physics 2 (통계물리2)

The Boltzmann factor and the partition function, from which the results of this chapter follow. Several demos continue its sections directly:

• §9.2 and §9.4, the Maxwell distributions: Boltzmann statistics of an ideal gas.
• §9.3, heat capacities beyond equipartition: Debye theory of solids.
• §9.5, classical and quantum statistics: Boson vs Fermion, and the supplement Fermions and bosons on a ladder.
• §9.6, the Fermi electron gas: Degenerate Fermi gas.
• §9.7, blackbody radiation and Bose–Einstein condensation: Photon statistics and Bose–Einstein condensation.
• Interacting particles and phase transitions, which this chapter does not cover: Ising model.

Graduate courses

Beyond the undergraduate courses, Phase Transitions and Critical Phenomena (상전이와 임계현상) and Special Topics in Complex Systems (복잡계 특론) treat phase transitions, critical phenomena, percolation, and the renormalization group.

Further reading

• S. T. Thornton, A. Rex, and C. E. Hood, Modern Physics for Scientists and Engineers, 5th ed. (Cengage Learning), Chapter 9.
• 이상훈, "[사디 카르노 논문 200주년] 불꽃과 같은 짧은 삶 속 영원한 불꽃으로부터의 동력 이론," 지식의 지평 (대우재단): daewooacademia.com/horizon-of-knowledge/797/1840.
• C. A. Gearhart, "Specific heats and the equipartition law in introductory textbooks," Am. J. Phys. 64, 995 (1996).
• S. N. Bose, "Plancks Gesetz und Lichtquantenhypothese," Z. Phys. 26, 178 (1924).
• F. London, "The λ-phenomenon of liquid helium and the Bose–Einstein degeneracy," Nature 141, 643 (1938).
• M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, "Observation of Bose–Einstein condensation in a dilute atomic vapor," Science 269, 198 (1995).