Non Equilibrium Thermodynamics

Concept

Non-Equilibrium Thermodynamics — Transport, Coupling, and Entropy Production

Scope: detailed derivation of irreversible thermodynamics from first principles. Includes entropy balance, flux–force relations, Onsager reciprocity, and coupled transport of heat, mass, charge, and momentum. Connects macroscopic continuum laws to microscopic statistical mechanics and extends to nonlinear transport and finite-rate relaxation.

1. Local Equilibrium Hypothesis

Non-equilibrium thermodynamics assumes that each infinitesimal element of a system can be described by local intensive properties T(r,t),P(r,t),μi(r,t)T(\mathbf{r},t), P(\mathbf{r},t), \mu_i(\mathbf{r},t) obeying the equilibrium equations of state.

Although gradients exist, each local element satisfies: du=T ds−P dv+∑iμi dni.du = T\,ds - P\,dv + \sum_i \mu_i\,dn_i.

This allows thermodynamic quantities (e.g., entropy density s) to be defined locally even in non-uniform systems.

2. Balance Equations for Conserved Quantities

For a continuum: ∂(ρψ)∂t+∇⋅(ρψv+Jψ)=σψ,\frac{\partial (ρψ)}{\partial t} + ∇·(ρψ\mathbf{v} + \mathbf{J}_ψ) = σ_ψ, where ψ\psi is any specific property, Jψ\mathbf{J}_ψ its diffusive flux, and σψσ_ψ its source term.

2.1 Mass Conservation

∂ρ∂t+∇⋅(ρv)=0.\frac{∂ρ}{∂t} + ∇·(ρ\mathbf{v}) = 0.

2.2 Energy Conservation

∂(ρe)∂t+∇⋅(ρev+Jq+∑ihiJi)=ρr,\frac{∂(ρe)}{∂t} + ∇·(ρe\mathbf{v} + \mathbf{J}_q + \sum_i h_i\mathbf{J}_i) = ρr, where Jq\mathbf{J}_q is the heat flux and rr the volumetric heat source.

2.3 Entropy Balance

∂(ρs)∂t+∇⋅(ρsv+Js)=σs.\frac{∂(ρs)}{∂t} + ∇·(ρs\mathbf{v} + \mathbf{J}_s) = σ_s. The term σs≥0σ_s ≥ 0 is the entropy production rate density.

3. Derivation of Entropy Production Density

From local energy and mass balances and Gibbs relation: Tds=de+Pd(1/ρ)−∑iμid(ni/m).T ds = de + P d(1/ρ) - \sum_i μ_i d(n_i/m).

Substitute into the entropy balance and perform algebraic elimination to get: σs=Jq⋅∇(1T)−∑iJi⋅∇(μiT)+1Tτ:∇v.σ_s = \mathbf{J}_q·∇\left(\frac{1}{T}\right) - \sum_i \mathbf{J}_i·∇\left(\frac{μ_i}{T}\right) + \frac{1}{T} \boldsymbol{τ} : ∇\mathbf{v}.

Here:

  • Jq\mathbf{J}_q: heat flux (relative to mass-averaged motion)
  • Ji\mathbf{J}_i: diffusion fluxes
  • τ\boldsymbol{τ}: viscous stress tensor

This is the general expression for local entropy production.

4. Thermodynamic Fluxes and Forces

Identify conjugate pairs:

FluxThermodynamic ForcePhysical Phenomenon
Jq\mathbf{J}_q∇(1/T)∇(1/T)Heat conduction
Ji\mathbf{J}_i−∇(μi/T)-∇(μ_i/T)Diffusion
τ\boldsymbol{τ}∇v/T∇\mathbf{v}/TViscous dissipation
Electric current Je\mathbf{J}_eE/T\mathbf{E}/TElectrical conduction

Then: σs=∑kJk⋅Xk≥0.σ_s = \sum_k \mathbf{J}_k · \mathbf{X}_k ≥ 0.

This scalar must be nonnegative for all processes, ensuring the second law locally.

5. Linear Irreversible Thermodynamics and Onsager Reciprocity

5.1 Linear Flux–Force Relations

For small deviations from equilibrium: Ji=∑jLijXj.\mathbf{J}_i = \sum_j L_{ij} \mathbf{X}_j.

Here LijL_{ij} are phenomenological coefficients satisfying: σs=∑i,jLijXi⋅Xj≥0.σ_s = \sum_{i,j} L_{ij} \mathbf{X}_i·\mathbf{X}_j ≥ 0. The matrix L must be symmetric and positive semi-definite.

5.2 Onsager Reciprocal Relations

From microscopic reversibility (fluctuation–dissipation theorem): Lij=Lji.L_{ij} = L_{ji}.

This symmetry arises from time-reversal invariance of underlying molecular dynamics.

6. Examples of Coupled Transport Phenomena

6.1 Thermoelectric Coupling

In a conducting medium: [JqJe]=[LqqLqeLeqLee][∇(1/T)E/T].\begin{bmatrix} \mathbf{J}_q \\ \mathbf{J}_e \end{bmatrix} = \begin{bmatrix} L_{qq} & L_{qe} \\ L_{eq} & L_{ee} \end{bmatrix} \begin{bmatrix} ∇(1/T) \\ \mathbf{E}/T \end{bmatrix}.

Cross-coefficients produce:

  • Seebeck effect: voltage induced by ∇T∇T (LqeL_{qe}).
  • Peltier effect: heat flow caused by electric current (LeqL_{eq}).
  • Thomson effect: continuous heating/cooling in ∇T∇T–E overlap.

By reciprocity, Lqe=LeqL_{qe} = L_{eq}.

6.2 Thermal Diffusion and Dufour Effects

For binary mixture: [JqJ1]=[LqqLq1L1qL11][∇(1/T)−∇(μ1/T)].\begin{bmatrix} \mathbf{J}_q \\ \mathbf{J}_1 \end{bmatrix} = \begin{bmatrix} L_{qq} & L_{q1} \\ L_{1q} & L_{11} \end{bmatrix} \begin{bmatrix} ∇(1/T) \\ -∇(μ_1/T) \end{bmatrix}.

Coupling produces:

  • Soret effect (thermal diffusion): mass flux from ∇T∇T.
  • Dufour effect: heat flux from composition gradients.

Experimentally, Soret coefficient ST=(1/x(1−x))(∂x/∂T)JS_T = (1/x(1-x)) (∂x/∂T)_J quantifies this coupling.

6.3 Thermoosmosis and Electroosmosis

Fluid flow through porous medium due to gradients:

  • Temperature gradient → thermoosmosis.
  • Electric potential → electroosmosis.

Both arise from coupling between mechanical and thermal/electrical forces via boundary-layer interactions.

6.4 Cross Diffusion and Maxwell–Stefan Formalism

For multicomponent mixtures: −∇μi=RT∑j≠ixjJi−xiJjcDij.-∇μ_i = RT \sum_{j≠i} \frac{x_j \mathbf{J}_i - x_i \mathbf{J}_j}{c D_{ij}}.

These equations inherently include cross-coupling terms and reduce to Fick’s law for binary diffusion.

7. Entropy Production in Transport Processes

For a Newtonian fluid: σs=1T2q⋅∇T+1Tτ:∇v+∑iJi⋅∇μiT.σ_s = \frac{1}{T^2} \mathbf{q}·∇T + \frac{1}{T} \boldsymbol{τ}:∇\mathbf{v} + \sum_i \frac{\mathbf{J}_i·∇μ_i}{T}.

Specific contributions:

  • Heat conduction: σq=q⋅∇(1/T)σ_q = \mathbf{q}·∇(1/T)
  • Viscous dissipation: σv=(1/T)τ:∇vσ_v = (1/T) \boldsymbol{τ}:∇\mathbf{v}
  • Diffusion: σd=−∑i(Ji⋅∇μi)/Tσ_d = -\sum_i (\mathbf{J}_i·∇μ_i)/T

Each term ≥ 0 under linear laws.

8. Classical Transport Laws from Linear Theory

ProcessFlux–Force RelationCoefficients
Heat conductionq=−k∇T\mathbf{q} = -k ∇TFourier’s law, k=LqqT2k = L_{qq}T^2
Viscous flowτij=2μeijτ_{ij} = 2μ e_{ij}Newtonian viscosity
Mass diffusionJi=−ρD∇Yi\mathbf{J}_i = -ρD∇Y_iFick’s law
Electrical conductionJe=σE\mathbf{J}_e = σ \mathbf{E}Ohm’s law

These are linear approximations valid near equilibrium.

9. Microscopic Basis: Fluctuation–Dissipation Theorem

From statistical mechanics, transport coefficients relate to time correlations of microscopic fluxes: Lij=1kBV∫0∞⟨Ji(0)Jj(t)⟩dt.L_{ij} = \frac{1}{k_B V} \int_0^∞ ⟨J_i(0) J_j(t)⟩ dt.

This connects macroscopic irreversibility with microscopic fluctuations and underlies Onsager symmetry (Lij=LjiL_{ij} = L_{ji}).

10. Nonlinear and Extended Thermodynamics

For large gradients or fast processes, linear theory breaks down.

10.1 Cattaneo–Vernotte Heat Flux (Finite Propagation)

τq∂q∂t+q=−k∇T.τ_q \frac{∂\mathbf{q}}{∂t} + \mathbf{q} = -k ∇T.

Introduces finite heat propagation speed vq=k/(ρcpτq)v_q = \sqrt{k/(ρ c_p τ_q)}, avoiding infinite speed paradox of Fourier’s law.

10.2 Extended Thermodynamic Variables

Non-equilibrium variables (e.g., fluxes themselves) treated as independent state variables: dS=dSeq+∑kAkdJk.dS = dS_{eq} + \sum_k A_k dJ_k. Leads to hyperbolic transport equations and better modeling of relaxation phenomena.

11. Coupled Chemical–Diffusion Systems

For reactive mixtures: σs=∑rArξ˙rT−∑iJi⋅∇μiT.σ_s = \sum_r \frac{A_r \dot ξ_r}{T} - \sum_i \frac{\mathbf{J}_i·∇μ_i}{T}.

Cross terms Lr,iL_{r,i} represent coupling between reaction rates and diffusion fluxes (e.g., catalytic or electrochemical systems).

Example: Electrochemical reaction diffusion (Nernst–Planck + Butler–Volmer coupling): Ji=−Di∇ci−ziDiFRTci∇φ,r=k0[coxe−αFη/RT−crede(1−α)Fη/RT].J_i = -D_i ∇c_i - \frac{z_i D_i F}{RT} c_i ∇φ, \quad r = k_0 [c_{ox} e^{−αFη/RT} - c_{red} e^{(1−α)Fη/RT}].

12. Entropy Generation and Exergy Dissipation Density

Local exergy destruction per unit volume: e˙D=T0σs.\dot e_D = T_0 σ_s.

Total exergy destruction: ED=∫VT0σsdV.E_D = ∫_V T_0 σ_s dV. This connects microscopic irreversibility to macroscopic efficiency loss (Gouy–Stodola theorem in differential form).

13. Applications and Coupled Examples

13.1 Thermoelectric Generator

Combine Fourier and Ohm laws with Seebeck coupling: q=−k∇T+ΠJe,E=α∇T+ρJe.q = -k∇T + ΠJ_e, \quad E = α∇T + ρJ_e. Hers Π=αTΠ = αT (Kelvin relation). Efficiency optimization uses balance between Joule heating and Peltier effects.

13.2 Electrolyte Transport

Entropy production: σs=∑iJi⋅(−∇μiT+ziFT∇φ).σ_s = \sum_i J_i·\left(-∇\frac{μ_i}{T} + \frac{z_i F}{T}∇φ\right). Leads to coupled ionic conduction, electroosmosis, and diffusion.

13.3 Thermodiffusion in Gases

From Boltzmann equation expansions, cross-term L12L_{12} yields measurable mass transport in temperature gradient. Used in isotope separation and hydrocarbon reservoirs.

14. Summary Equations

| Concept | Relation | |----------|-----------|| | Entropy production density | σs=Jq⋅∇(1/T)−∑iJi⋅∇(μi/T)+(1/T)τ:∇vσ_s = \mathbf{J}_q·∇(1/T) - \sum_i \mathbf{J}_i·∇(μ_i/T) + (1/T) \boldsymbol{τ}:∇\mathbf{v} | | Flux–force linear laws | Ji=∑jLijXj\mathbf{J}_i = ∑_j L_{ij} \mathbf{X}_j | | Onsager reciprocity | Lij=LjiL_{ij} = L_{ji} | | Cattaneo heat law | τq∂q/∂t+q=−k∇Tτ_q ∂\mathbf{q}/∂t + \mathbf{q} = -k∇T | | Maxwell–Stefan diffusion | −∇μi=RT∑j≠i(xjJi−xiJj)/(cDij)-∇μ_i = RT ∑_{j≠i} (x_j J_i - x_i J_j)/(c D_{ij}) | | Local exergy destruction | eD=T0σse_D = T_0 σ_s |

  • 07_Exergy_and_Irreversibility.md — macroscopic connection between entropy production and exergy loss.
  • 09_Phase_Transitions_and_Critical_Phenomena.md — dynamic spinodal decomposition (Cahn–Hilliard form).
  • Fluid_Dynamics/03_Transport_Equations.md — Navier–Stokes and heat/mass transport PDE formulations.