Thermodynamics · Upper Division

Irreversible Processes & Transport

Real processes are irreversible — they produce entropy. The phenomenology of entropy production, driven by gradients in temperature, concentration, and velocity, leads to heat conduction, diffusion, and viscosity — the transport coefficients.

PrerequisitesLawsofthermodynamics(Ch.12)⋅Statisticalmechanics(Ch.S)⋅PartialderivativesLaws of thermodynamics (Ch. 12) \cdot Statistical mechanics (Ch. S) \cdot Partial derivatives
Learning Goals
  • Express entropy production rate as a sum of flux-force products and state why it is non-negative.
  • Apply Fourier's, Fick's, and Ohm's laws as examples of the linear flux-force relationship.
  • State the Onsager reciprocal relations and connect them to thermoelectric effects.
  • Derive the diffusion equation from Fick's law and the continuity equation.
  • Apply the Einstein-Stokes relation to compute diffusion coefficients from fluid viscosity.

IR.1 Entropy Production

The second law states dS ≥ δQ/T, with equality for reversible processes. For an isolated system, entropy never decreases. The entropy production rate σ (per unit volume) is always non-negative:

σ=Jq⋅∇(1/T)+J⋅(−∇μ/T)+⋯≥0\sigma = J_{q} \cdot \nabla(1/T) + J \cdot (-\nabla\mu/T) + \cdots \ge 0(IR.1)

Here J_q is heat flux and J is particle flux. Each term is a product of a fluxand a thermodynamic force (gradient of an intensive variable). Entropy production reaches zero only when all gradients vanish — equilibrium.

IR.2 Phenomenological Laws

The linear response between fluxes and forces (valid near equilibrium):

Jq=−κ\nablaT(Fourier′slaw,heatconduction)J_{q} = -\kappa \nablaT \qquad (Fourier's law, heat conduction)(IR.2)
J=−D\nablan(Fick′slaw,diffusion)J = -D \nablan \qquad (Fick's law, diffusion)(IR.3)
Jelec=σE=−σ\nablaV(Ohm′slaw)J_{elec} = \sigma E = -\sigma \nablaV \qquad (Ohm's law)(IR.4)

These three laws have the same form: flux = −(coefficient) × (gradient of intensive variable). The transport coefficients κ (thermal conductivity, W/m·K), D (diffusion coefficient, m²/s), and σ (electrical conductivity, S/m) are not independent — they are related by microscopic physics.

IR.3 Onsager Reciprocal Relations

Onsager (1931) proved that the cross-coefficients in the flux-force matrix are equal:

Lij=Lji(Onsagerreciprocalrelations)L_{ij} = L_{ji} \qquad (Onsager reciprocal relations)(IR.5)

These relate thermoelectric effects: the Seebeck effect (heat flow drives electric current) and the Peltier effect (electric current drives heat flow) have coefficients related by L₁₂ = L₂₁. The proof uses time-reversal symmetry of microscopic dynamics. Onsager won the 1968 Nobel Prize for this fundamental result.

Thermoelectric effects (Seebeck and Peltier): A temperature gradient drives a current (Seebeck, basis of thermocouples); a current drives heat flow (Peltier, basis of solid-state cooling — no moving parts). The figure of merit ZT = S²σT/κ (S = Seebeck coefficient) must exceed ~3 for competitive cooling devices. Current best: ZT ≈ 2.5 in some nanostructured materials.

Example IR.1 — Heat Conduction Through a Wall

Awall(areaA,thicknessL,conductivityκ)separatesregionsattemperaturesT1andT2>A wall (area A, thickness L, conductivity \kappa) separates regions at temperatures T_{1} and T_{2} > T1.FindtheheatfluxandentropyproductionrateT_{1}. Find the heat flux and entropy production rate

Fourier's law:Jq=−κ\nablaT=κ(T2−T1)/L(magnitude,directedfromhottocoldJ_{q} = -\kappa \nablaT = \kappa(T_{2}-T_{1})/L (magnitude, directed from hot to cold
Heat flow rate:Q˙=AJq=κA(T2−T1)/L(thermalresistanceR=L/(\kappaAQ̇ = A J_{q} = \kappa A(T_{2}-T_{1})/L \qquad (thermal resistance R = L/(\kappaA
Entropy production:σ=Jq⋅∇(1/T).In1D:σ=−Jq(1/T)′=Jq×(T2−T1)/(LT2)≈Q˙(T2−T1)/(AT2).Integrat\sigma = J_{q} \cdot \nabla(1/T). In 1D: \sigma = -J_{q} (1/T)' = J_{q} \times (T_{2}-T_{1})/(LT^{2}) \approx Q̇(T_{2}-T_{1})/(AT^{2}). Integrated: \sigmȧ_total = Q̇(1/T_{1} - 1/T_{2}) > 0 ✓.
Physical meaning:ThecoldsidegainsentropyQ˙/T1;thehotsidelosesQ˙/T2.NetentropyproductionQ˙(1/TThe cold side gains entropy Q̇/T_{1}; the hot side loses Q̇/T_{2}. Net entropy production Q̇(1/T1−1/T2)>0asrequiredbythe2ndlaw_{1}-1/T_{2}) > 0 as required by the 2nd law.

IR.4 Diffusion Equation

Combining Fick's law J = −D∇n with the continuity equation ∂n/∂t + ∇·J = 0:

\partialn/\partialt=D∇2n(diffusionequation)\partialn/\partialt = D \nabla^{2}n \qquad (diffusion equation)(IR.6)

This is the heat equation with n → T, D → κ/(ρc_p) ≡ α (thermal diffusivity). It is a parabolic PDE: information propagates instantaneously (infinite speed) — a result of the approximation that ignores the finite time for microscopic collisions.

Solution for a point source at origin at t=0:

n(r,t)=N/(4\piDt)3/2×exp(−r2/(4Dt))(Gaussianspreading)n(r, t) = N/(4\piDt)^{3/2} \times exp(-r^{2}/(4Dt)) \qquad (Gaussian spreading)(IR.7)

The concentration spreads as a Gaussian with width σ = √(2Dt). The mean-square displacement ⟨r²⟩ = 6Dt — Einstein's 1905 result for Brownian motion. Einstein's relation connects D to the mobility μ = D/(k_BT) — both arise from the same thermal fluctuations.

Theorem IR.1 — Fluctuation-Dissipation Theorem
The dissipation (irreversibility) and the fluctuations of a system in equilibrium are related:D=kBT/(6π\etaR)=kBTμ(Einstein−StokesrelationD = k_{BT} / (6\pi\etaR) = k_{BT} \mu \qquad (Einstein-Stokes relationThesamefriction(η)thatdissipatesenergyalsodrivestherandomkicks(BrownianmotionThe same friction (\eta) that dissipates energy also drives the random kicks (Brownian motion) thatdiffuseparticles.Thisfundamentalrelationextendsto:Johnson−Nyquistnoise⟨V2⟩=that diffuse particles. This fundamental relation extends to: Johnson-Nyquist noise ⟨V^{2}⟩ =kBTR(thermalnoiseinresistorR),Wiener−Khinchin(powerspectrumoffluctuations=disk_{BTR} (thermal noise in resistor R), Wiener-Khinchin (power spectrum of fluctuations = dissipativepartofresponse),andquantumversion(zero−pointfluctuationsatT=0sipative part of response), and quantum version (zero-point fluctuations at T=0.
Definition IR.2 — Common Traps
  • Entropy production is local and nonnegative near equilibrium: individual flux terms can be coupled, but the total production cannot be negative.
  • Flux points down the gradient: the minus sign in Fourier's and Fick's laws encodes hot-to-cold and high-to-low flow.
  • Onsager symmetry needs microscopic reversibility: magnetic fields and active driving can modify reciprocal relations.
  • Diffusionwidthgrowsas\sqrttDiffusion width grows as \sqrtt: mean-square displacement is linear in time, not the displacement itself.
Exercises — IR.1–IR.4 Irreversible Processes
0.
UsetheEinstein−StokesrelationD=kBT/(6π\etaR)tofindthediffusioncoefficient(in\mum2Use the Einstein-Stokes relation D = k_{BT}/(6\pi\etaR) to find the diffusion coefficient (in \mum^{2}/s)ofaproteinwithStokesradiusR=5nminwater(η=10−3Pa⋯ )atT=300Ks) of a protein with Stokes radius R = 5 nm in water (\eta = 10^{-3} Pa\cdots) at T = 300 K
μm²/s
Straightforward
1.Apply Fick's law to find theoxygenfluxacrossacellmembrane10nmthick,givenaconcentrationdifferenceof101he oxygen flux across a cell membrane 10 nm thick, given a concentration difference of 10^{1}n adequate for cell respiration?
Straightforward
2.Analyze a Peltier cooler: find the optimal current for maximum cooling power and the maximum achievable temperature difference in terms of ZT.
Intermediate
3.Calculate the diffusion coefficient of glucose (radius 0.5 nm)inwaterat37°CusingtheEinstein−Stokesrelation.Howlongtodiffuseacrossa1\mumnm) in water at 37°C using the Einstein-Stokes relation. How long to diffuse across a 1 \mum
Intermediate
4.DerivetheJohnson−Nyquistnoisevoltage⟨V2⟩=4kBTR\Deltaffromthefluctuation−dissipationDerive the Johnson-Nyquist noise voltage ⟨V^{2}⟩ = 4k_BTR\Deltaf from the fluctuation-dissipation theorem.Whatisthermsnoisevoltagefora1kΩresistoratroomtemperaturein1MHzbatheorem. What is the rms noise voltage for a 1 k\Omega resistor at room temperature in 1 MHz badwidth?
Challenging
Key Takeaways
  • Entropyproductionσ=∑JiXi≥0(flux×force).ZeroonlyatequilibriumEntropy production \sigma = \sum J_{i} X_{i} \ge 0 (flux \times force). Zero only at equilibrium.
  • Fourier:Jq=−κ\nablaT.Fick:J=−D\nablan.Ohm:J=\sigmaE.Samestructure:flux=coefficient×graFourier: J_{q} = -\kappa\nablaT. Fick: J = -D\nablan. Ohm: J = \sigmaE. Same structure: flux = coefficient \times gradient.
  • Onsager:Lij=Lji—cross−effects(Seebeck/Peltier,Soret/Dufour)arerelatedbytime−rOnsager: L_{ij} = L_{ji} — cross-effects (Seebeck/Peltier, Soret/Dufour) are related by time-reversal.
  • Diffusionequation:\partialn/\partialt=D∇2n.Gaussianspreading:⟨r2⟩=6DtDiffusion equation: \partialn/\partialt = D\nabla^{2}n. Gaussian spreading: ⟨r^{2}⟩ = 6Dt.
  • Einstein−Stokes:D=kBT/(6π\etaR).MobilityanddiffusivityrelatedbykBTEinstein-Stokes: D = k_{BT}/(6\pi\etaR). Mobility and diffusivity related by k_{BT}.
  • Fluctuation−dissipation:Johnson−Nyquistnoise⟨V2⟩=4kBTR\Deltaf.SamephysicsasfrictionFluctuation-dissipation: Johnson-Nyquist noise ⟨V^{2}⟩ = 4k_BTR\Deltaf. Same physics as friction.