Stability–Dissipation Relation

• Stability–dissipation relation is a principle linking dissipative mechanisms to the asymptotic stability of systems in classical, quantum, and continuum frameworks.
• It employs mathematical formulations from operator semigroup theory, thermodynamics, and IQC control frameworks to quantify energy decay and spectral gaps.
• The concept underpins applications ranging from feedback system stabilization and open quantum dynamics to numerical schemes ensuring discrete stability.

A stability–dissipation relation is any mathematical principle, structure, or theorem that explicitly links the stability properties of a dynamical system—linear or nonlinear, deterministic or stochastic, classical or quantum—to the presence, strength, or structure of dissipative mechanisms. This conceptual thread unites results across operator semigroup theory, continuum mechanics, non-equilibrium thermodynamics, system control, statistical physics, and numerical analysis. At its core, the stability–dissipation relation formalizes the intuition that dissipation—whether microscopic or macroscopic, local or global—can enforce the asymptotic stability (e.g., exponential decay, Lyapunov attraction, spectral gap) of systems whose conservative limits are (at best) marginally stable.

1. Mathematical Formulations in Operator Semigroup Theory

The abstract semigroup framework captures stability–dissipation relations in infinite-dimensional settings, particularly for evolution equations and coupled PDEs. A prototypical result is provided by closure-relation techniques for differential operators (Glück et al., 2022). Consider Hilbert spaces $H_1,H_2$ and an "extended" operator $A_{\mathrm{ext}}$ on $H_1\oplus H_2$, together with a coercive, bounded "closure operator" $S$. Impose the closure relation $h_2 = Sg_2$ to obtain the "closed" operator $A_S$ on $H_1$. If $A_{\mathrm{ext}}$ generates a contraction semigroup (i.e., $\|e^{tA_{\mathrm{ext}}}\|\leq 1$), is strictly dissipative (numerical range in $\{\mathrm{Re}\,\lambda\leq-\delta\}$), and certain compactness or norm-continuity holds, then $A_S$ generates an exponentially stable semigroup: $$\|e^{tA_S}\|_{\mathcal{L}(H_1)}\leq M e^{-\omega t}.$$ This fully quantifies how structural dissipativity in the extended system enforces exponential stability in the closed system, with the decay rate $\omega$ set by the dissipation.

2. Thermodynamic and Continuum Mechanics Perspectives

In generalized dissipative mechanics, stability–dissipation relations arise at the intersection of the second law and the Routh–Hurwitz criterion for the spectrum of linearized operators (Ván, 2012). For a weakly nonlocal, non-equilibrium continuum, the dissipation function $D$ governs the entropy production rate. If $D\geq 0$ (equivalently, Onsager matrix positive semi-definite), then all coefficients of the characteristic polynomial for plane-wave perturbations are non-negative, and additional Hurwitz positivity conditions yield

$\mathrm{Re}\,\Gamma(k) < 0 \quad \forall k,$

implying asymptotic decay of all perturbations. Thus, thermodynamically admissible dissipation is both necessary and essentially sufficient for linearized stability—actual dissipation constants directly control spectral gaps and decay rates. Extensions to non-convex dissipation potentials, as in non-equilibrium phase transitions of complex fluids (Janečka et al., 2017), show that loss of convexity in the dissipation potential leads to instabilities, multivalued constitutive relations, and dissipative phase transitions.

3. Feedback System Theory, Dissipativity, and IQC Frameworks

In nonlinear and feedback system control, stability–dissipation relations are formalized via dissipativity inequalities, both with static and dynamic supply rates (Khong et al., 2022). A system is said to be dissipative if there exists a storage function $S$ such that, along trajectories,

$$S(x(T),z(T)) \leq S(x(0),z(0)) + \int_0^T \xi(t)\,dt$$

for a supply rate $\xi(t)$. These inequalities—potentially involving dynamic auxiliary systems—anchor Lyapunov and asymptotic stability results for nonlinear interconnections. When the supply rate is quadratic in filtered inputs and outputs, classical small-gain and passivity theorems as well as integral quadratic constraint (IQC) frameworks (Scherer et al., 2018) are recovered, linking frequency-domain or time-domain energy dissipation directly to internal or input–output stability. Strict dissipativity (with coercivity on certain variables) yields exponential convergence, while mere dissipativity guarantees Lyapunov (non-increasing energy) stability.

4. Quantum Systems, Open Dynamics, and Dissipative State Stabilization

Stability–dissipation relations manifest in open quantum systems via the structure of the Lindblad (or GKSL) generator (Pan et al., 2015). In the Heisenberg picture, if a Lyapunov observable $V$ satisfies

$\mathcal{G}(V)\leq -\gamma(V-d\mathbb{I}),$

then the expectation $\mathrm{Tr}[\rho_t V]$ decays exponentially to its ground value $d$. Weaker dissipative conditions, involving the dissipation superoperator $\mathfrak{D}(V)\geq c(V-d)$, still yield asymptotic (possibly non-exponential) ground-state stabilization. The explicit algebraic relations between the dissipative structure of the Lindblad operators and Lyapunov stability form a direct stability–dissipation correspondence.

5. Stability–Dissipation Relations in Relativistic and Transport Theories

In relativistic hydrodynamics and kinetic theory, the invariance of stability under Lorentz transformations is tied to causality and dissipation (Gavassino, 2021). Dissipative decay in one frame can correspond to growth in another unless the dissipative mechanisms themselves are strictly sub-luminal; only causal dissipative theories guarantee Lorentz-invariant stability. The mathematical condition is that all characteristics (signals, dissipation) are confined to or within the light cone. When acausal dissipative modes exist (e.g., in naive extensions of classical diffusion laws to relativistic settings), stability is frame-dependent and can fail globally.

6. Numerical Methods: Artificial and Structural Dissipation for Discrete Stability

Stability–dissipation relations are foundational in the design and analysis of numerical algorithms for evolution equations. In explicit time-integration of semidiscretely stable schemes, discrete energy growth due to nonphysical terms must be compensated by artificial dissipation or filtering (Öffner et al., 2016). The introduction of dissipative operators or modal filters can be parametrized to exactly preserve or match the decay of implicit (energy-dissipating) schemes, and in SBP finite-difference or spectral-element methods, structural volume dissipation preserves (discrete) energy or entropy decay in analogy to continuous energy estimates (Bercik et al., 16 Mar 2025). The discrete energy or entropy is non-increasing due to judiciously constructed compatible dissipation, making stability and dissipation inextricable.

7. Dynamical Systems and Phase-Space Measures

In general nonlinear dynamical systems, the stability–dissipation relation is reflected in empirical and theoretical connections between phase-space contraction, Lyapunov spectra, and entropy production (Williams et al., 2020). Phase-space contraction rates (sum of Lyapunov exponents, average divergence $\nabla\cdot F$) are negative in dissipative systems and correlate with asymptotic stability. Temporal asymmetry in local expansion/contraction rates serves as a quantitative empirical proxy for dissipation and stability.

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In summary, the stability–dissipation relation is a mathematically precise, structurally robust principle that is realized in various operator-theoretic, thermodynamic, control-theoretic, quantum, relativistic, numerical, and statistical frameworks. In each instance, dissipative mechanisms—appropriately quantified—yield operator inequalities, spectral gaps, or energy/entropy decay laws that guarantee stability of the underlying system, while loss or misuse of dissipation (insufficient, ill-posed, non-convex, or acausal) leads to instability, phase transitions, or breakdown of control. This cross-disciplinary architecture provides algebraic, analytic, and computational tools to design, certify, and understand dissipative stabilization in both finite and infinite dimensions.