Onsager Reciprocity and Entropy Production

Microscopic time-reversal symmetry constrains the transpose of the linear transport matrix, while the local second law constrains its symmetric part to be positive semidefinite. Magnetic fields and time-reversal-odd backgrounds require Onsager–Casimir reversal. Antisymmetric transport can be nondissipative, and none of these positivity statements establishes causal propagation.

Required background. Relativistic Dissipative Hydrodynamics fixes the flux and force conventions. KMS Relations and Fluctuation–Dissipation supplies the equilibrium real-time identity. Internal, Spacetime, Discrete, and Antiunitary Symmetries fixes antiunitary parity.

Helpful background. Detailed Balance and Fluctuation–Dissipation gives the stochastic counterpart. Contact Terms, Magnetization Currents, and Order of Limits treats transport-current subtraction.

Entropy production in Landau frame

For one charge, take

$$s^\mu = su^\mu-\frac{\mu}{T}j^\mu$$

through first order. Using conservation, the first law, and the Landau-frame constitutive relations yields

$$\nabla_\mu s^\mu = \frac{\eta}{2T}\sigma_{\mu\nu}\sigma^{\mu\nu} +\frac{\zeta}{T}\theta^2 +\frac{\sigma_Q}{T}\, P_{\mu\nu}\mathcal V^\mu\mathcal V^\nu +O(\partial^3).$$

Every displayed quadratic form is nonnegative in the local rest space if

$$\eta\ge0, \qquad \zeta\ge0, \qquad \sigma_Q\ge0.$$

The factor $1/2$ compensates for the shear definition used in this volume. Substituting a rest-frame velocity gradient reproduces the usual positive viscous heating.

The entropy current is not unique: it can be shifted by identically conserved terms and higher-derivative improvements. Consequently, the local second-law method constrains transport within a declared order and basis; it does not prove a unique entropy density far from equilibrium.

Coupled thermoelectric transport

Let $J_A^\mu$ denote transverse fluxes and $X_B^\mu$ their thermodynamic forces:

$$J_A^\mu=L_{AB}X_B^\mu.$$

For charge and heat, one convenient rest-frame choice is

$$\begin{pmatrix} \mathbf j\\ \mathbf q/T \end{pmatrix} = \begin{pmatrix} L_{11}&L_{12}\\ L_{21}&L_{22} \end{pmatrix} \begin{pmatrix} \mathbf E/T-\boldsymbol\nabla(\mu/T)\\ -\boldsymbol\nabla T/T \end{pmatrix}.$$

Other force normalizations move powers of $T$ among the matrix entries. Entropy production is

$$\dot s_{\mathrm{prod}} = X_A\!\cdot L_{AB}X_B = X_A\!\cdot L_{(AB)}X_B.$$

Only the symmetric part contributes. For a $2\times2$ matrix, positivity requires

$$L_{11}\ge0, \qquad L_{22}\ge0, \qquad L_{11}L_{22} \ge \left(\frac{L_{12}+L_{21}}{2}\right)^2.$$

The antisymmetric part describes reversible transverse conversion and is unconstrained by entropy production alone.

Onsager–Casimir reciprocity

Let $\varepsilon_A=\pm1$ be the time-reversal parity of the underlying variable whose flux is $J_A$. Microscopic reversibility gives

$$L_{AB}(\mathbf B,\boldsymbol\Omega,\ldots) = \varepsilon_A\varepsilon_B\, L_{BA}(-\mathbf B,-\boldsymbol\Omega,\ldots).$$

At zero time-reversal-odd background and for equal parity, $L_{AB}=L_{BA}$. At nonzero magnetic field, the coefficient at fixed $\mathbf B$ need not be symmetric; its transpose is related to the coefficient in the reversed field. Onsager’s two original papers give the fluctuation-regression argument and reciprocal relations Onsager 1931, part I, pp. 405–426 Onsager 1931, part II, pp. 2265–2279; Casimir gives the magnetic and parity refinement Casimir 1945, §§2–3, pp. 343–350.

Magnetization currents must be removed before identifying $L_{AB}$ with through-sample transport. Otherwise a hydrostatic circulation can appear to violate reciprocity or produce a spurious Hall-type coefficient.

KMS and the evidence chain

In quantum equilibrium, the KMS condition relates symmetric fluctuations to the dissipative part of retarded response. In the low-frequency limit this supplies the same transport matrix that enters entropy production. The logical chain is:

$$\text{antiunitary symmetry} \Longrightarrow \text{Onsager--Casimir relation},$$ $$\text{KMS plus spectral positivity} \Longrightarrow \text{nonnegative dissipative response},$$ $$\text{constitutive matching} \Longrightarrow \text{local entropy-production form}.$$

Each arrow has hypotheses. A driven state need not satisfy KMS; a system with broken time reversal need not have a symmetric transport matrix; an anomalous or Hall sector can be nondissipative; and an indefinite operator pairing can invalidate a naive positivity claim.

The relativistic hydrodynamic consistency reference records these constraints separately from causality and well-posedness.

Positivity is not causality

The diffusion equation with $D>0$ has positive entropy production and damped Fourier modes, yet its Green function has nonzero support at every distance for any $t>0$. Conversely, a hyperbolic system can have an unstable lower-order source term. Entropy production controls the sign of dissipative quadratic forms; the principal symbol controls characteristics; the full spectrum controls linear stability.

This separation is crucial in relativistic theory. Imposing $\eta,\zeta,\sigma_Q\ge0$ is necessary physical input, but no manipulation of $\nabla_\mu s^\mu$ replaces the characteristic analysis on Strong Hyperbolicity, Stability, and Causal Propagation.

Exercise

Assume time-reversal symmetry at zero magnetic field, so $L_{12}=L_{21}$. Determine the positivity conditions and the zero-production direction at saturation.

Solution

The symmetric matrix is positive semidefinite iff

$$L_{11}\ge0,\qquad L_{22}\ge0,\qquad L_{11}L_{22}-L_{12}^2\ge0.$$

At saturation, $L_{11}L_{22}=L_{12}^2$. If $L_{11}>0$, a null force obeys

$$X_1=-\frac{L_{12}}{L_{11}}X_2.$$

The associated coupled flux produces no entropy at this order. Whether that direction represents an exact nondissipative mode requires additional symmetry and higher-order analysis.

Where this leads

Frame-Invariant Dissipative Data identifies which transport combinations survive changes of hydrodynamic variables. Schwinger–Keldysh Effective Actions for Fluids derives response and noise constraints at action level.

References

• Casimir, H. B. G. 1945. “On Onsager’s Principle of Microscopic Reversibility.” Reviews of Modern Physics 17: 343–350. DOI.

• Onsager, Lars. 1931. “Reciprocal Relations in Irreversible Processes. I.” Physical Review 37: 405–426. DOI.

• Onsager, Lars. 1931. “Reciprocal Relations in Irreversible Processes. II.” Physical Review 38: 2265–2279. DOI.