Nonequilibrium Thermodynamics and Transport

Equilibrium statistical mechanics describes states in which nothing happens. This chapter is about everything else: a bar with a temperature difference across it, a wire carrying current, a colloidal particle jostled by a fluid, a molecule pulled apart by an optical trap. Its central claim is that near equilibrium the whole of this behaviour is fixed by equilibrium fluctuations — Johnson–Nyquist noise [Johnson:1928] [Nyquist:1928b], the Einstein relation between diffusion and mobility, the Green–Kubo formulae [Green:1954] [Kubo:1957] and the fluctuation–dissipation theorem [Callen:1951] are four statements of one fact — and that far from equilibrium the second law survives in a sharpened, quantitative form: the fluctuation theorems [Evans:1993] [Gallavotti:1995] and the work relations [Jarzynski:1997] [Crooks:1999], which have been tested directly by pulling single RNA molecules [Liphardt:2002] [Collin:2005]. Onsager's reciprocal relations [Onsager:1931a] [Onsager:1931b], derived from microscopic time-reversal invariance, tie the cross coefficients together and are the reason a chapter on heat flow must also treat thermoelectricity.

The chapter builds on the kinetic theory of Kinetic Theory of Gases in Part VI — Classical Thermodynamics, where the Boltzmann equation and the H-theorem are set up, and on the ensembles of Statistical Mechanics, whose correlation functions supply every transport coefficient computed here. It supplies in turn the scattering-limited conductivities used in Electrons in Solids: Band Theory and Semiconductors, the critical dynamics that accompanies the static criticality of Phase Transitions and Critical Phenomena, and the classical limit of the open-system dynamics of Open Quantum Systems and Decoherence. Standard monographs are [deGroot:1962] [Prigogine:1967].

The kinetic route

The Boltzmann equation and the H-theorem

[Reserved: a summary, not a re-derivation, of the material of Kinetic Theory of Gases: the integro-differential equation for the one-particle distribution with its collision integral, and the H-theorem showing \(\dd H/\dd t\leq0\) [Boltzmann:1872]; the assumption of molecular chaos as the exact point at which time-reversal symmetry is given up, and the Loschmidt and Zermelo objections; why the chapter takes the H-theorem as its starting point and asks instead what survives when the system is driven.]

Chapman–Enskog and the transport coefficients

[Reserved: the systematic expansion of the Boltzmann equation in gradients about local equilibrium [Chapman:1916], which yields the Navier–Stokes equations of Fluid Dynamics together with first-principles expressions for viscosity and thermal conductivity; the predicted independence of the viscosity of a dilute gas from its density, and the measurements confirming it; the Prandtl number as a parameter-free test; the standard monograph [Chapman:1970].]

The phenomenological transport laws

[Reserved: the four linear constitutive laws that the microscopic theory must reproduce — Fourier's law of heat conduction [Fourier:1822], Fick's law of diffusion [Fick:1855], Ohm's law [Ohm:1827] and Newton's law of viscous shear [Newton:1687]; their common structure, flux proportional to gradient; measured coefficients in SI for representative solids, liquids and gases; the range of validity, and the ballistic and hydrodynamic regimes in which each fails.]

Linear irreversible thermodynamics

Entropy production and thermodynamic forces

[Reserved: local equilibrium as the working hypothesis, and the entropy balance \(\pp_{t}s+\nabla\cdot\vect{J}_{s}=\sigma\) with \(\sigma=\sum_{i}J_{i}X_{i}\geq0\) [deGroot:1962]; the canonical pairing of fluxes with forces — heat flux with \(\nabla(1/T)\), electric current with \(\nabla(-\mu/T)\) — and why that pairing, not another, makes the coefficient matrix symmetric; measured entropy production rates in simple conduction experiments.]

Onsager reciprocal relations

[Reserved: the theorem \(L_{ik}=L_{ki}\) for the linear coefficients relating fluxes to forces, derived from microscopic reversibility and the regression hypothesis for equilibrium fluctuations [Onsager:1931a] [Onsager:1931b]; the status of the regression hypothesis as the one non-trivial assumption; the independent experimental evidence, principally the Thomson relations of Section 124.3.2 and thermal diffusion in mixtures; why the relations count as a genuine prediction and not a definition.]

Reciprocity with a magnetic field

[Reserved: the extension \(L_{ik}(\vect{B})=L_{ki}(-\vect{B})\) required when the microscopic dynamics is reversible only under simultaneous reversal of the field [Casimir:1945]; the consequence for the Hall and Nernst–Ettingshausen coefficients, and the fact that reciprocity does not forbid an antisymmetric part; the same structure in rotating frames (Rigid Bodies and Rotating Frames); the quantized case in Experiment: The Quantum Hall Effect.]

Steady states and minimum entropy production

[Reserved: the theorem that in the linear regime a stationary state with fixed boundary constraints minimizes the entropy production [Prigogine:1967], with an explicit statement of its narrow conditions of validity; the honest caveat that no such variational principle is known beyond the linear regime, and that claims to the contrary are not supported by evidence; dissipative structures and pattern formation as the far-from-equilibrium alternative, whose dynamics belongs to Nonlinear Dynamics and Chaos.]

Thermoelectricity

The Seebeck, Peltier and Thomson effects

[Reserved: Seebeck's discovery that a circuit of two metals with junctions at different temperatures deflects a compass needle, and its correct reading as a thermally driven electromotive force [Seebeck:1826]; Peltier's converse effect, heat absorbed or released at a junction carrying current [Peltier:1834]; Thomson's prediction and observation of the third, bulk effect in a single conductor with a temperature gradient [Thomson:1857]; measured Seebeck coefficients in \(\mu\mathrm{V}/\mathrm{K}\).]

The Thomson relations as a test of reciprocity

[Reserved: the two relations \(\Pi=ST\) and \(\mu_{T}=T\,\dd S/\dd T\) connecting the three effects, obtained by Thomson from a thermodynamic argument he knew to be incomplete [Thomson:1857] and derived properly only from reciprocity [Onsager:1931b]; the measured agreement, quoted with uncertainties, as the classic experimental confirmation of the Onsager relations; the thermocouple as a practical thermometer (Measurement, SI Units, and the Theory of Errors).]

Conduction in metals and the figure of merit

[Reserved: the Wiedemann–Franz law, the near-constancy of \(\kappa/\sigma T\) across metals [Wiedemann:1853]; the Drude model's accidental success and its failure on the specific heat [Drude:1900a], resolved only by Fermi statistics in Electrons in Solids: Band Theory; the Lorenz number \(L_{0}=\pi^{2}k_{B}^{2}/3e^{2}=2.44\times 10^{-8}\,\mathrm{W}\,\mathrm{\Omega}/\mathrm{K}^{2}\) and the conditions under which it is violated; the thermoelectric figure of merit \(ZT\) and the materials strategy of decoupling \(\kappa\) from \(\sigma\) [Snyder:2008].]

Brownian motion and stochastic dynamics

The Einstein–Smoluchowski relation

[Reserved: the derivation of the diffusion equation from a random-walk continuity argument and the identification \(D=k_{B}T/6\pi\eta a\) [Einstein:1905d], obtained independently by Sutherland [Sutherland:1905] and by a different route by Smoluchowski [Smoluchowski:1906]; the general Einstein relation \(D=\mu k_{B}T\) between diffusion constant and mobility as the first fluctuation–dissipation statement in physics; Perrin's measurement of Avogadro's number from displacement statistics [Perrin:1909], treated in full in Experiment: Brownian Motion and Avogadro's Number.]

The Langevin equation

[Reserved: the stochastic equation \(m\ddot{x}=-\gamma\dot{x}+\xi(t)\) with white noise satisfying \(\langle\xi(t)\xi(t')\rangle=2\gamma k_{B}T\delta(t-t')\) [Langevin:1908]; the second fluctuation–dissipation relation built into that variance, without which the particle would not thermalize; the exact velocity autocorrelation and mean square displacement, crossing from ballistic to diffusive at the momentum relaxation time [Uhlenbeck:1930]; the observation of the ballistic regime in an optically trapped bead [Li:2010].]

The Fokker–Planck equation

[Reserved: the evolution equation for the probability density in phase space, obtained for a rotating dipole in a radiation field [Fokker:1914] and given its general statistical-dynamical form by Planck [Planck:1917]; drift and diffusion coefficients as the first two Kramers–Moyal moments; the stationary solution reproducing the Boltzmann distribution as the consistency condition; Kramers' escape rate over a potential barrier, with its low- and high-friction limits [Kramers:1940], and the Arrhenius factor it explains.]

Master equations

[Reserved: the gain–loss balance equation for the occupation probabilities of discrete states, and detailed balance as the condition for an equilibrium stationary solution; Pauli's derivation of an H-theorem for the quantum transition rates [Pauli:1928]; the quantum generalization to completely positive dynamics [Lindblad:1976], whose physical content is developed in Open Quantum Systems and Decoherence; birth–death processes and chemical kinetics as classical instances.]

Linear response and fluctuation–dissipation

Johnson–Nyquist noise

[Reserved: Johnson's measurement of a temperature-dependent voltage fluctuation across any resistor, independent of its material and composition [Johnson:1928]; Nyquist's derivation of \(\langle V^{2}\rangle=4k_{B}TR\,\Delta f\) from equipartition among the modes of a transmission line, in the same issue [Nyquist:1928b]; its use as a primary thermometer and as the noise floor of every amplifier; shot noise as the distinct, non-thermal fluctuation, and the measured crossover between them.]

The fluctuation–dissipation theorem

[Reserved: the general theorem relating the imaginary part of a generalized susceptibility to the equilibrium spectral density of the conjugate variable, with the quantum form carrying a factor \(\operatorname{coth}(\hbar\omega/2k_{B}T)\) [Callen:1951]; Nyquist's formula and the Einstein relation as the two special cases already met; the review that established the modern statement [Kubo:1966]; violation of the theorem as an operational definition of “out of equilibrium”, and the effective temperatures inferred from it in glasses.]

Kubo formulae and Green–Kubo relations

[Reserved: transport coefficients as time integrals of equilibrium current autocorrelation functions — conductivity from the current–current correlator [Kubo:1957], viscosity and thermal conductivity from the corresponding fluxes [Green:1954]; the remarkable content of these formulae, that a dissipative quantity is computed entirely within the equilibrium ensemble; long-time tails and the divergence of the two-dimensional transport integrals; molecular-dynamics evaluation as the standard numerical route.]

Projection operators and memory kernels

[Reserved: the exact reduction of the full dynamics onto a chosen set of slow variables, producing a generalized Langevin equation with a memory kernel and a residual force [Zwanzig:1960] [Mori:1965]; the second fluctuation–dissipation theorem identifying the kernel with the residual force correlation; Markovian approximation as the limit of separated timescales, and what is lost when it fails; the formal link to coarse graining in The Renormalization Group.]

Fluctuation theorems

The steady-state fluctuation theorem

[Reserved: the observation in shearing molecular dynamics that trajectories with negative entropy production occur with a probability ratio \(\ee^{-\Sigma t/k_{B}}\) against their conjugates [Evans:1993]; the proof for chaotic dynamical systems obeying the chaotic hypothesis [Gallavotti:1995]; the reading of the second law as an inequality on an average that individual small systems may violate for short times; the smooth recovery of the thermodynamic statement as system size grows.]

Work relations far from equilibrium

[Reserved: Jarzynski's equality \(\langle\ee^{-W/k_{B}T}\rangle=\ee^{-\Delta F/k_{B}T}\), valid for arbitrarily fast driving and yielding an equilibrium free-energy difference from irreversible measurements [Jarzynski:1997]; Crooks's stronger relation between the forward and reverse work distributions, whose crossing point locates \(\Delta F\) directly [Crooks:1999]; recovery of \(\langle W\rangle\geq\Delta F\) by Jensen's inequality, so the second law is a corollary; the exponential average's practical dominance by rare trajectories.]

Experimental verification

[Reserved: the mechanically unfolded RNA hairpin, whose measured work distributions recovered the equilibrium folding free energy to within a few \(k_{B}T\) and so verified the Jarzynski equality [Liphardt:2002]; the Crooks relation tested on RNA with and without a bound magnesium ion, including the far-from-equilibrium regime where the Jarzynski estimator is biased [Collin:2005]; the colloidal bead in a moving optical trap that exhibited transient second-law violations [Wang:2002]; the SI energy scale of these experiments, set by \(k_{B}T\approx4.14\times 10^{-21}\,\mathrm{J}\) at room temperature.]

Stochastic thermodynamics and information

[Reserved: heat, work and entropy defined along a single stochastic trajectory, and the resulting integral and detailed fluctuation theorems [Seifert:2012]; the generalized second law including an information term, and its experimental demonstration by a feedback-controlled Brownian ratchet converting information into work [Toyabe:2010]; Maxwell's demon and Landauer's principle as the resolution — the cost lies in erasure, not in measurement; molecular motors as the biological instance.]