Dissipative Driving Term in Nonequilibrium Systems

• Dissipative driving term is a mechanism in nonequilibrium systems that simultaneously inputs energy and induces irreversible losses, altering steady-state dynamics.
• It appears across fields like nonlinear oscillators, quantum optics, and cosmology, where its formulation governs transport, phase transitions, and entropy production.
• The interplay between drive and dissipation informs regime classification and practical modeling of irreversible processes in both classical and quantum systems.

A dissipative driving term refers to any source or control parameter in a nonequilibrium dynamical system that simultaneously injects energy or resources (“drives”) and induces irreversible losses (“dissipates”), altering the steady-state structure, transport, or fluctuations relative to conservative or equilibrium analogues. Such terms appear in a diverse array of fields: nonlinear oscillator chains, quantum optics, open quantum systems, statistical mechanics of non-equilibrium steady states, hydrodynamics with relaxation, granular flows, and cosmology. The form and interpretation of the dissipative driving term are system-specific, but universally they encode both the pathway by which nonequilibrium steady states are maintained and the extra entropy production or irreversibility beyond conservative relaxation.

1. Mathematical Origin and Physical Interpretation

A dissipative driving term generically appears in the evolution equations of open or non-equilibrium systems:

• In classical chains, e.g., the boundary-driven Klein–Gordon model, dissipation and drive emerge from coupling ends to external baths and coherent drivers. For site $j=1$:

$$\ddot q_1 = -q_1 - \nu q_1^3 - \varepsilon(q_1-q_2) - \eta \dot q_1 + F_d\cos(\omega_d t)$$

Here, $-\eta\dot q_1$ is the purely dissipative term, and $F_d\cos(\omega_d t)$ is the coherent boundary driving. Interplay leads to diverse transport regimes (Prem et al., 2022).

• In quantum or semiclassical open systems, a dissipative driving term results from engineered couplings in Lindblad-type master equations. For a Kerr resonator:

$$\frac{d\hat\rho}{dt} = -i[\hat H, \hat\rho] + \gamma D[a]\rho + \eta D[a^2]\rho$$

The Hamiltonian $\hat H$ includes both one- and two-photon drive amplitudes, while each dissipator models corresponding loss (Bartolo et al., 2016).

• In hydrodynamics, dissipative driving is realized as energy and momentum relaxation proportional to velocity (e.g., $-\rho_m \tau_m^{-1} v_i$), essential to model steady flows under external fields and finite entropy production (Amoretti et al., 26 Jul 2024).
• In cosmology, the dissipative driving term $h_B(t) = \beta(2H^2 + \dot{H})$ arises as an effective bulk-viscous or matter-creation pressure, proportional to scalar curvature, modifying standard $\Lambda$CDM expansion (Komatsu, 22 Dec 2025).

The universal feature is that such terms represent, at the equation-of-motion level, the mechanisms by which the system is both pumped away from equilibrium and relaxed toward a steady regime with persistent fluxes or structures.

2. Representative Forms Across Physical Systems

The specific algebraic structure of dissipative driving terms varies:

System	Term Example	Reference
Klein-Gordon chain	$-\eta\dot q_1 + F_d\cos(\omega_d t)$	(Prem et al., 2022)
Quantum oscillator	$\gamma \dot Q$ in $\ddot Q + \gamma \dot Q$	(Chaki et al., 2021)
Kerr resonator	Hamiltonian drive + Lindblad loss	(Bartolo et al., 2016)
Fluids with relaxation	$-\rho_m\tau_m^{-1}v_i$	(Amoretti et al., 26 Jul 2024)
Cosmological expansion	$h_B(t) = \beta(2H^2 + \dot{H})$	(Komatsu, 22 Dec 2025)
Granular media	Bulk kicks + restitutional collisions	(Kranz et al., 2010)

In each, the driven terms may inject energy or particles (coherent drive, bath coupling, external field, parametric squeezing), while the dissipative contribution irreversibly removes energy or relaxes momentum.

3. Dynamical and Statistical Roles

Dissipative driving terms fundamentally alter system dynamics:

• They break detailed balance and time-reversal symmetry, ensuring that steady states are fundamentally nonequilibrium and characterized by finite irreversible currents or fluxes.
• In oscillator and chain systems, dissipative driving balances input and output, setting steady-state energy densities and controlling transport regimes (from ballistic to diffusive) (Prem et al., 2022).
• In quantum systems under strong driving, the interplay of drive and dissipation can induce rich phenomena: nontrivial steady states, phase transitions, collapse–revival dynamics, or robust multimode Wigner functions (Saiko et al., 2016, Bartolo et al., 2016).

In non-equilibrium statistical mechanics, such terms determine the nature of entropy production. The extra entropy per unit time beyond conservative relaxation quantifies the “housekeeping” cost needed to sustain current-carrying or structure-forming steady states (Hanel et al., 2018).

4. Steady-State and Thermodynamic Implications

The presence of dissipative driving has precise consequences for steady-state statistical and thermodynamic properties:

• In maximum configuration entropy formalisms, the dissipative driving term $S_D$ is defined as the non-equilibrium component of the cross-entropy functional, reflecting the additional combinatorial weight (or rarity) of observing driven transitions above equilibrium relaxation (Hanel et al., 2018).
• In hydrodynamic flows, the entropy production rate is directly proportional to the dissipative driving components; Onsager reciprocity can constrain or neutralize certain contributions (e.g., vanishing incoherent conductivity in time-reversible fluids) (Amoretti et al., 26 Jul 2024).
• In cosmology, a dissipative driving term in the expansion equation generates a positive dynamic creation pressure, ensuring compliance with the second law (entropy growth on the cosmic horizon) and enabling transitions from deceleration to acceleration (Komatsu, 22 Dec 2025).

5. Regime Classification and Physical Manifestations

Dissipative driving leads to distinct dynamical regimes, often associated with bifurcations, phase transitions, or qualitative changes in observables:

• In the Klein-Gordon chain, increasing dissipative driving leads to transitions from non-transmitting (localized/diffusive) to transmitting (ballistic/wave-propagating) regimes, with intermediate superdiffusive and resonant nonlinear-wave behaviors (Prem et al., 2022).
• For two-level and bosonic systems, tuning the ratio of drive to dissipation produces collapse–revival oscillations, population stabilization, or steady-state synchronization/desynchronization (Saiko et al., 2016, Li et al., 21 May 2024).
• In periodically driven, dissipative quantum oscillators, dissipation suppresses parametric instabilities (shrinks “Arnold tongues”) and localizes the Wigner function in phase space (Chaki et al., 2021).
• In driven granular fluids, the balance of stochastic driving and dissipative inelastic collisions (modeled via restitution coefficients) controls the emergence and location of the glass transition, shifting critical density as dissipation varies (Kranz et al., 2010).

6. Connections to Entropy, Information, and Irreversibility

The dissipative driving term has central importance for the thermodynamic characterization of nonequilibrium steady states:

• In the maximum configuration principle for SSR processes, $S_D$ quantifies the entropy produced solely by the presence of driving, i.e., transitions counter to the natural relaxation direction. $S_D$ appears in the modified Helmholtz free energy, interpretable as the dissipated “housekeeping heat” distinct from “useful work” in the driven system (Hanel et al., 2018).
• In quantum parameter estimation under dissipation, engineering two-photon (parametric) drives enables systems to maintain extremely high sensitivity (quantum Fisher information) even as single-photon losses would otherwise render estimation impossible. The drive–dissipation interplay defines the achievable precision bounds (Xie et al., 2019).
• In forced, dissipative hydrodynamics, the precise identification of entropy-producing (dissipative) terms enables the derivation of exact relationships for AC/DC conductivities and thermoelectric response, and constrains possible forms under microscopic symmetries (Amoretti et al., 26 Jul 2024).

7. Broader Implications and Universality

The emergence, structural form, and physical consequences of dissipative driving terms are universal features of open, non-equilibrium systems across classical, quantum, and statistical domains. They underpin the existence of nonequilibrium steady states, control transport and fluctuation phenomena, and mediate the balance between order (through driving) and disorder (via dissipation). Control and measurement of these terms is at the core of contemporary research in quantum technologies, condensed matter, statistical physics, and cosmological modeling. Their mathematical analysis connects Langevin equations, Lindblad master equations, non-equilibrium thermodynamics, large deviation formalisms, and hydrodynamic transport frameworks (Prem et al., 2022, Komatsu, 22 Dec 2025, Amoretti et al., 26 Jul 2024, Hanel et al., 2018, Bartolo et al., 2016, Kranz et al., 2010).