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I.2.3 First Examples of Linear Transport Phenomena

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I.2.3 First Examples of Linear Transport Phenomena 14 Thermodynamics of irreversible processes I.2.3 First examples of linear transport phenomena Following the example set by Onsager in his original articles [4, 5], we now enumerate several phenomenological linear transport laws formulated in the 19th century and re-express them in terms of relations between fluxes and affinities as formalised in section I.2.1. We shall begin with a few “direct” transport phenomena—for heat, particle number, or electric charges. Next, we turn to a case in which indirect transport plays a role, namely that of thermoelec- tric effects. These first examples will be studied in the respective rest frames of the systems under eq: study, so that the equilibrium fluxes Jj will vanish. Eventually, we describe the various transport phenomena in a simple fluid, which will allow us to derive the classical laws of hydrodynamics. In most of this section, we shall for the sake of brevity drop the (t;~r)-dependence of the various physical quantities under consideration. ::::::I.2.3 a :::::::::::::::Heat transport In an insulating solid with a temperature gradient, heat is transported through the vibrations of the underlying crystalline structure—whose quantum mechanical description relies on phonons— rather than through particle transport. Traditionally, this transport of energy is expressed in the form of Fourier’s(g) law (1822) J~E = −κ r~ T; (I.36a) with κ the heat conductivity of the insulator. Using the general formalism of linear irreversible thermodynamic processes, the relationship between the energy flux density and the conjugate affinity, in the case when there is no gradient of the ratio µ/T ,(8) reads in the linear regime (8)Phonons are massless and carry no conserved quantum number, so that their chemical potential vanishes every- where. (f)H. Casimir, 1909–2000 (g)J. Fourier, 1768–1830 I.2 Linear irreversible thermodynamic processes 15 1 J~ = L · r~ ; (I.36b) E EE T (9) with LEE a tensor of rank 2 of (first-order) kinetic coefficients. If the medium is isotropic, this tensor is for symmetry reasons proportional to the identity LEE = LEE13: The comparison between Eqs. (I.36a) and (I.36b) then gives the identification 1 κ = L : (I.36c) T 2 EE Since LEE ≥ 0 to ensure the positivity of the entropy production rate, κ is also nonnegative. The flux (I.36a) thus transports energy from the regions of higher temperatures to the colder ones. Combining Fourier’s law (I.36a) with the continuity equation (I.18a) applied to the energy density e yields @e = −r~ · J~ = r~ · κ r~ T @t E Assuming that the heat conductivity is uniform in the medium under study, κ can be taken out of the divergence, so that the right-hand side becomes κ4T , with 4 the Laplacian. According to a well known thermodynamic relation, at fixed volume the change in the internal energy equals the heat capacity at constant volume multiplied by the change in the temperature, which results in de = cV dT with cV the heat capacity per unit volume. If the latter is independent of temperature, one readily obtains the evolution equation @T κ = 4T: (I.37) @t cV This is the generic form of a diffusion equation [see Eq. (I.39) below], with diffusion coefficient κ/cV . ::::::I.2.3 b::::::::::::::::::Particle diffusion Consider now “particles” immersed in a motionless and homogeneous medium, in which they can move around—microscopically, through scatterings on the medium constituents—without affecting the medium characteristics. Examples are the motion of dust in the air, of micrometer-scale bodies in liquids, but also of impurities in a solid or of neutrons in the core of a nuclear reactor. Let n denote the number density of the particles. The transport of particles can be described by Fick’s(h) law (1855) [7] ~ ~ JN = −D rn; (I.38a) with J~N the flux density of particle number and D the diffusion coefficient. Remark: Relation (I.38a) is sometimes referred to as Fick’s first law, the second one being actually the diffusion equation (I.39). In the absence of temperature gradient and of collective motion of the medium, the general relation (I.31) yields for the particle number flux density µ J~ = L r~ − : (I.38b) N NN T (9). which is strictly speaking never the case at the microscopic level in a crystal, since the lattice structure is incompatible with local invariance under the whole set of three-dimensional rotations. Nevertheless, for lattices with a cubic elementary mesh, isotropy holds, yet at the mesoscopic level. (h)A. Fick, 1829–1901 16 Thermodynamics of irreversible processes Relating the differential of chemical potential to that of number density with @µ dµ = dn; @n T the identification of Fick’s law (I.38a) with formula (I.38b) yields 1 @µ D = LNN ; (I.38c) T @n T where the precise form of the partial derivative depends on the system under study. Diffusion equation If the number of diffusing particles is conserved—which is for instance not the case for neutrons in a nuclear reactor(10)—and if the diffusion coefficient is independent of position, the associated continuity equation (I.18a) leads to the diffusion equation @n(t;~r) = D 4n(t;~r): (I.39) @t To tackle this partial differential equation (considered on R3), one can introduce the Fourier transform with respect to space coordinates Z ~ n~(t;~k) ≡ n(t;~r) e−ik·~r d3~r of the number density. This transform then satisfies for each ~k the ordinary differential equation @n~(t;~k) = −D ~k 2n~(t;~k); @t where we assumed that the number density and its spatial derivatives vanish at infinity at every instant. These assumptions respectively guarantee the finiteness of the overall particle number and the absence of particle flux at infinity. ~ 2 The solution to the differential equation reads n~(t;~k) = e−Dk t n~(0;~k), with n~(0;~k) the initial condition at t = 0 in Fourier space. An inverse Fourier transform then yields 3 Z ~ 2 ~ d ~k n(t;~r) = e−Dk t n~(0;~k) eik·~r : (2π)3 (3) If the initial condition is n(0;~r) = n0 δ (~r)—which physically amounts to introducing a particle ~ density n0 at the point ~r = ~0 at time t = 0—then the Fourier transform is trivially n~(0; k) = n0, so that the inverse Fourier transform above is simply that of a Gaussian, which gives n0 2 n(t;~r) = e−~r =4Dt: (4πDt)3=2 p The typical width of the particle number density increases with t. ::::::I.2.3 c ::::::::::::::::::::::Electrical conduction Another example of particle transport is that of the moving charges in an electrical conductor in the presence of an electric field E~ = −r~ Φ, with Φ the electrostatic potential. The latter is assumed to vary very slowly at the mesoscopic scale, so as not to spoil the local equilibrium assumption. If q (10)There, one should also include various source and loss terms, to account for the production of neutrons through fission reactions, or their “destruction” through reactions with the nuclear fuel, with the nuclear waste present in the reactor, or with the absorber bars that moderate the chain reaction, or their natural decay. I.2 Linear irreversible thermodynamic processes 17 denotes the electric charge of the carriers, then the electric charge flux density, traditionally referred to as current density, is simply related to the number flux density of the moving charges through J~el: = qJ~N : (I.40a) The relation between electric field and current density in a microscopically isotropic conductor at constant temperature in the absence of magnetic field is (the microscopic version of) Ohm’s(i) law ~ J~el: = σel:E (I.40b) with σel: the (isothermal) electrical conductivity. To relate the electrical conductivity to the kinetic coefficients of Sec. I.2.1, and more specifically to LNN since J~el: is proportional to J~N , one needs to determine the intensive variable conjugate to particle number—or equivalently, thanks to the local equilibrium assumption, conjugate to particle number density. Now, if e denotes the (internal) energy density in the absence of electrostatic potential, then the energy density in presence of Φ becomes e + nqΦ: meanwhile, the particle number density n remains unchanged. The entropy per unit volume then satisfies—as can most easily be checked within the grand-canonical ensemble of statistical mechanics—the identity s(e; n; Φ) = s(e − nqΦ; n; 0); which yields @s µ + qΦ µ = − ≡ − Φ ; (I.41) @n T T with µ the chemical potential at vanishing electric potential. µΦ is referred to as electrochemical potential. Assuming a uniform temperature in the conductor, the linear relation (I.31) for the flux of particle number then reads µ + qΦ J~ = L r~ − : (I.42) N NN T Using again the uniformity of temperature, this gives ~ 1 @µ ~ q ~ JN = − LNN rn − LNN rΦ; T @n T T where the first term is the same as in Sec. I.2.3 b, while the second can be rewritten with the help of the electric field. If the density is uniform, the first term vanishes, and the identification with Eqs. (I.40a) and (I.40b) yields the electrical conductivity q2 σ = L : (I.43) el: T NN Einstein relation Equations (I.38c) and (I.43) show that the diffusion coefficient D and the electrical conductivity σ are both related to the same kinetic coefficient LNN , so that they are related with each other: σel: @µ D = 2 : (I.44) q @n T Let µel: denote the electrical mobility of the charge carriers, which is the proportionality factor ~ between the mean velocity ~vav: they acquire in an electric field E and this field ~ ~vav: = µel:E : (I.45) Obviously, the determination of µel: requires a microscopic model for the motion of the charges.
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