Non-Quadratic Dual Dissipation Pair

• Non-Quadratic Dual Dissipation Pair is defined by a pair of Legendre dual convex functionals that govern irreversible, jump-type, and bounded dissipative fluxes in non-equilibrium dynamics.
• It appears within the GENERIC framework to couple reversible Hamiltonian dynamics with nonlinear irreversible processes via non-quadratic Legendre duality.
• Its applications span quantum impurity problems, boundary phenomena, and kinetic theory, with convergence to quadratic limits under singular scaling.

A non-quadratic dual dissipation pair refers to a pair of Legendre dual convex functionals (often denoted as Ψ, Ψ*) governing the irreversible, dissipative part of non-equilibrium dynamics where at least one of the dissipation functionals is genuinely non-quadratic. Such pairs appear throughout dissipative field theories, particularly in the GENERIC (General Equations for Non-Equilibrium Reversible–Irreversible Coupling) framework, and in boundary critical phenomena, quantum impurity problems, and kinetic theory with non-Gaussian noise or jump-type processes. Unlike the classic quadratic dissipation generated by linear Onsager operators (as in the heat equation or Landau theory), non-quadratic pairs encode nonlinear, typically bounded or jump-driven dissipative fluxes and thus capture rich non-equilibrium behaviors and singular limits.

1. General Structure and Definition

In the GENERIC formalism, the dynamics on a state manifold Z are governed by the superposition of:

• A reversible (Hamiltonian) part encoded by a Poisson operator L(z) applied to the gradient of an energy functional E(z), and
• An irreversible (dissipative) part, generated by a dissipation potential Ψ*(z, ξ) that is the Legendre dual (with respect to the tangent-cotangent pairing) of a convex potential Ψ(z, v).

The dual potentials Ψ, Ψ* are linked by

$$\Psi(z,v) = \sup_{\xi} \langle\xi, v\rangle - \Psi^*(z,\xi)$$

and, in applications to large deviations and stochastic processes, arise from a Hamiltonian H(z, ξ) on the cotangent bundle with a generalized fluctuation symmetry (Kraaij et al., 2017):

$H_2(z,\xi) = H_2(z, dS(z) - \xi)$

where S is an entropy functional, and Ψ*, Ψ encode the irreversible (entropic) part of the dynamics. When H_2 is non-quadratic in ξ, the resulting dissipative terms are non-quadratic.

The corresponding evolution equation is

$$\dot{z} = L(z) dE(z) + \partial_\xi \Psi^*(z, -\frac{1}{2} dS(z))$$

with Ψ* typically non-quadratic for jump-type or bounded dissipative processes.

2. Characteristic Properties and Fluctuation Symmetry

Non-quadratic dual dissipation pairs are characterized by convexity, symmetry, and their relationship to entropy production. For instance, in the GENERIC context, the dissipation potential Ψ* is constructed to be

$$\Psi^*(z, \xi) = H_2(z, \xi + \frac{1}{2} dS(z)) - H_2(z, \frac{1}{2} dS(z))$$

and Ψ via Legendre transform. Ψ* satisfies Ψ*(z, 0) = 0 and is symmetric under ξ ↔ -ξ, reflecting detailed balance.

The quadratic case, with Ψ(z, ξ) = ½⟨ξ, M(z)ξ⟩, yields linear evolution laws, whereas for genuinely non-quadratic Ψ, the resulting dissipation law is a nonlinear function of the thermodynamic force, possibly reflecting finite dissipation "capacity," bounded generators, or jump processes (Kraaij et al., 2017).

Table: Properties of Dissipation Pairs

Property	Quadratic Dissipation	Non-Quadratic Dissipation
Functional form	Ψ*(ξ) = ½⟨ξ, Mξ⟩	e.g., Ψ*(ξ) ~ cosh ξ – 1
Irreversible law	Linear (Onsager)	Nonlinear (bounded/jump)
Legendre dual	Both quadratic	Duals not both quadratic

A canonical example is the Andersen thermostat, where

$$\Psi^*(\rho,\xi) = \gamma \iint \left[ \cosh(\xi(q,p')-\xi(q,p)) - 1 \right] M(dp')\, d\rho(q,p)$$

encoding jump-type momentum redistributions rather than Gaussian diffusion (Kraaij et al., 2017).

3. Exemplary Occurrences: Quantum Dissipative Systems and Duality

In the dissipative Hofstadter model (DHM) on a triangular lattice, exact T-duality transformations (an O(2,2;ℝ) subgroup) relate commutative and non-commutative bases in the space of bosonic fields. The zero-temperature phase diagram is structured by equivalence classes in parameter space labeled by a complex variable $z = \beta + i\alpha$, where α encodes Caldeira–Leggett dissipation and β is the dimensionless magnetic flux (Lee, 2016). The exact duality mapping

$$\frac{1}{\hat{z}} = \frac{1}{z} + i \frac{2n}{\sqrt{3}}, \quad n \in \mathbb{Z}$$

identifies points yielding equivalent boundary dynamics.

At "magic circles," where the non-commutative parameter θ can be set to zero modulo 2π, the theory maps to a boundary sine-Gordon model with explicit non-quadratic cosine interactions. This provides a genuine non-quadratic dual dissipation pair: under duality, both the direct and dual models have non-quadratic dissipative boundary terms, associated with tunneling in a three-wire quantum junction (Luttinger liquid wires joined at a point in a magnetic field) (Lee, 2016).

4. Non-Quadratic Dissipation in Kinetic Theory

The kinetic theory of the fuzzy Boltzmann equation with delocalized collisions provides a canonical framework for non-quadratic dual dissipation pairs. Proposition 2.11 in (Duong et al., 4 Dec 2025) details the construction of a “cosh-pair”:

$$\Psi^*(r) = 4 \left( \cosh\frac{r}{2} - 1 \right),\quad \Psi(u) = 2u \log\left( \frac{u+\sqrt{u^2+4}}{2} \right) - \sqrt{u^2+4} + 4$$

with Θ(s,t) = √(st) as the mean.

The GENERIC structure is implemented via variational principles:

• Dissipation potential at f:

$$D_{\Psi^*}(f) = \frac{1}{4}\int\Psi^*( -\overline{\nabla}\log f )\,\Theta(f'f'^*, ff^*)\, B(\cdot)\kappa(\cdot)\, d\eta d\sigma$$

• Dual potential R(f,U) via Legendre duality.

In the grazing-collision limit, as the nonlocal collisional kernel becomes sharply peaked, Ψ*, Ψ converge to quadratic forms and the limiting equation becomes Landau-type with standard quadratic dissipation (Duong et al., 4 Dec 2025). The Γ–convergence of the non-quadratic (cosh-type) dissipation to the quadratic limit is established via compactness/semi-continuity arguments, ensuring that non-quadratic dissipation controls limits of entropy production and action.

5. Mathematical Properties and Variational Formulations

A non-quadratic dual dissipation pair consists of convex, lower-semicontinuous, and Legendre-dual functionals Ψ, Ψ* with compatibility enforced by entropy/energy structures or fluctuation symmetries. In kinetic theory, compatibility is achieved through the identity

$$(\Psi^*)'( \log s - \log t )\, \Theta(s, t) = s - t$$

for all s, t > 0, ensuring that dissipation vanishes only at equilibrium.

The variational GENERIC formulations embed these pairs into generalizations of De Giorgi's action functionals; in weak solution theory for kinetic equations, the pair appears as the convex duals controlling entropy dissipation and irreversible action. For example, for solutions (f, U) of the fuzzy Boltzmann equation, the action functional

$$J^\epsilon(f,U) = S(f_T) - S(f_0) + \int_0^T R^\epsilon(f_t, U_t) dt + \int_0^T \langle U_t, \partial_t f_t - L(f_t) S'(f_t) \rangle dt$$

is minimized precisely when (f, U) solves the equation in weak GENERIC form, with R and D derived from (Ψ, Ψ*) (Duong et al., 4 Dec 2025).

6. Physical Relevance and Applications

Non-quadratic dual dissipation pairs are physically realized in models with:

• Strongly nonlinear friction (quantum impurity, boundary sine-Gordon, quantum wire junctions) (Lee, 2016)
• Jump-process thermostats (Andersen thermostat) with bounded flux/dissipation (Kraaij et al., 2017)
• Kinetic equations with nonlocal, non-Gaussian collision or interaction terms (Boltzmann equations, fuzzy Landau equations) (Duong et al., 4 Dec 2025)
• Strong-to-weak coupling dualities and self-dual points, as in the three-wire Hubbard/Luttinger junctions

Their analysis is crucial for understanding:

• Phase diagram topology (exact dualities, equivalence classes, and magic points (Lee, 2016))
• Nonlinear entropy production and convergence to equilibrium
• Non-Gaussian dissipation, hypocoercivity, and singular limit behaviors (e.g., grazing-collision limits (Duong et al., 4 Dec 2025))
• Generalized fluctuation–dissipation theorems in non-quadratic settings

A plausible implication is that in models with non-quadratic dissipation, the resulting macroscopic dynamics can exhibit strong nonlinearity, finite propagation, or bounded entropy fluxes, fundamentally differing from their quadratic (linear) analogues.

7. Limiting Cases and Convergence to Quadratic Dissipation

A central result is that, under appropriate scaling limits (such as the grazing-collision limit in kinetic theory), non-quadratic dual dissipation pairs can Γ-converge to quadratic pairs, and the associated weak solutions converge to those of the quadratic limit equation. For the fuzzy Boltzmann to fuzzy Landau limit, the cosh-type dissipation potentials yield to quadratic Landau dissipation through careful control of entropy and action in the limit ε → 0 (Duong et al., 4 Dec 2025). This establishes a rigorous connection between genuinely nonlinear dissipative systems and their linearized (classical) limits, providing a comprehensive framework for studying singular limits and non-equilibrium variational structures.