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 (\rho\psi)}{\partial t} + \nabla\cdot(\rho\psi\mathbf{v} + \mathbf{J}_\psi) = \sigma_\psi, where ψ\psiis any specific property, Jψ\mathbf{J}_\psiits diffusive flux, and σψ\sigma_\psiits source term.

2.1 Mass Conservation

∂ρ∂t+∇⋅(ρv)=0.\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) = 0.

2.2 Energy Conservation

∂(ρe)∂t+∇⋅(ρev+Jq+∑ihiJi)=ρr,\frac{\partial(\rho e)}{\partial t} + \nabla\cdot(\rho e\mathbf{v} + \mathbf{J}_q + \sum_i h_i\mathbf{J}_i) = \rho r, where Jq\mathbf{J}_qis the heat flux and rrthe volumetric heat source.

2.3 Entropy Balance

∂(ρs)∂t+∇⋅(ρsv+Js)=σs.\frac{\partial(\rho s)}{\partial t} + \nabla\cdot(\rho s\mathbf{v} + \mathbf{J}_s) = \sigma_s. The term σs≥0\sigma_s \geq 0is 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/\rho) - \sum_i \mu_i d(n_i/m).

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

Here:

  • Jq\mathbf{J}_q: heat flux (relative to mass-averaged motion)
  • Ji\mathbf{J}_i: diffusion fluxes
  • τ\boldsymbol{\tau}: 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)\nabla(1/T)Heat conduction
Ji\mathbf{J}_i−∇(μi/T)-\nabla(\mu_i/T)Diffusion
τ\boldsymbol{\tau}∇v/T\nabla\mathbf{v}/TViscous dissipation
Electric current Je\mathbf{J}_eE/T\mathbf{E}/TElectrical conduction

Then: σs=∑kJk⋅Xk≥0.\sigma_s = \sum_k \mathbf{J}_k \cdot \mathbf{X}_k \geq 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.\sigma_s = \sum_{i,j} L_{ij} \mathbf{X}_i\cdot\mathbf{X}_j \geq 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} \nabla(1/T) \\ \mathbf{E}/T \end{bmatrix}.

Cross-coefficients produce:

  • Seebeck effect: voltage induced by ∇T\nabla T(LqeL_{qe}).
  • Peltier effect: heat flow caused by electric current (LeqL_{eq}).
  • Thomson effect: continuous heating/cooling in ∇T\nabla 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} \nabla(1/T) \\ -\nabla(\mu_1/T) \end{bmatrix}.

Coupling produces:

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

Experimentally, Soret coefficient ST=(1/x(1−x))(∂x/∂T)JS_T = (1/x(1-x)) (\partial x/\partial T)_Jquantifies 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.-\nabla\mu_i = RT \sum_{j\neq 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.\sigma_s = \frac{1}{T^2} \mathbf{q}\cdot\nabla T + \frac{1}{T} \boldsymbol{\tau}:\nabla\mathbf{v} + \sum_i \frac{\mathbf{J}_i\cdot\nabla\mu_i}{T}.

Specific contributions:

  • Heat conduction: σq=q⋅∇(1/T)\sigma_q = \mathbf{q}\cdot\nabla(1/T)
  • Viscous dissipation: σv=(1/T)τ:∇v\sigma_v = (1/T) \boldsymbol{\tau}:\nabla\mathbf{v}
  • Diffusion: σd=−∑i(Ji⋅∇μi)/T\sigma_d = -\sum_i (\mathbf{J}_i\cdot\nabla\mu_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 \nabla TFourier’s law, k=LqqT2k = L_{qq}T^2
Viscous flowτij=2μeij\tau_{ij} = 2\mu e_{ij}Newtonian viscosity
Mass diffusionJi=−ρD∇Yi\mathbf{J}_i = -\rho D\nabla Y_iFick’s law
Electrical conductionJe=σE\mathbf{J}_e = \sigma \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^\infty \langle J_i(0) J_j(t)\rangle 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.\tau_q \frac{\partial\mathbf{q}}{\partial t} + \mathbf{q} = -k \nabla T.

Introduces finite heat propagation speed vq=k/(ρcpτq)v_q = \sqrt{k/(\rho c_p \tau_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.\sigma_s = \sum_r \frac{A_r \dot \xi_r}{T} - \sum_i \frac{\mathbf{J}_i\cdot\nabla\mu_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 \nabla c_i - \frac{z_i D_i F}{RT} c_i \nabla\varphi, \quad r = k_0 [c_{ox} e^{−\alpha F\eta/RT} - c_{red} e^{(1−\alpha)F\eta/RT}].

12. Entropy Generation and Exergy Dissipation Density

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

Total exergy destruction: ED=∫VT0σsdV.E_D = \int_V T_0 \sigma_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\nabla T + \Pi J_e, \quad E = \alpha\nabla T + \rho J_e. Hers Π=αT\Pi = \alpha T(Kelvin relation). Efficiency optimization uses balance between Joule heating and Peltier effects.

13.2 Electrolyte Transport

Entropy production: σs=∑iJi⋅(−∇μiT+ziFT∇φ).\sigma_s = \sum_i J_i\cdot\left(-\nabla\frac{\mu_i}{T} + \frac{z_i F}{T}\nabla\varphi\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:∇v\sigma_s = \mathbf{J}_q\cdot\nabla(1/T) - \sum_i \mathbf{J}_i\cdot\nabla(\mu_i/T) + (1/T) \boldsymbol{\tau}:\nabla\mathbf{v}| | Flux–force linear laws | Ji=∑jLijXj\mathbf{J}_i = \sum_j L_{ij} \mathbf{X}_j| | Onsager reciprocity | Lij=LjiL_{ij} = L_{ji}| | Cattaneo heat law | τq∂q/∂t+q=−k∇T\tau_q \partial\mathbf{q}/\partial t + \mathbf{q} = -k\nabla T| | Maxwell–Stefan diffusion | −∇μi=RT∑j≠i(xjJi−xiJj)/(cDij)-\nabla\mu_i = RT \sum_{j\neq i} (x_j J_i - x_i J_j)/(c D_{ij})| | Local exergy destruction | eD=T0σse_D = T_0 \sigma_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.