Dissipative Distillation in Nonequilibrium Systems

• Dissipative distillation is a framework of nonequilibrium protocols that converts fluctuating, contaminated resources into stable, ordered outputs.
• It encompasses diverse applications such as autonomous entanglement distillation, thermodynamic free energy conversion, and stability-aware model compression.
• The approach redefines traditional purification by using irreversible dynamics as a design tool to optimize system performance and resource efficiency.

Searching arXiv for the cited works and related uses of the term. Dissipative distillation denotes a family of non-equilibrium procedures in which dissipation is not merely tolerated but engineered, exploited, or quantitatively controlled as the mechanism that converts a noisy, fluctuating, distributed, or contaminated resource into a more ordered, usable, or stable target. Current arXiv usage is heterogeneous rather than singular: the phrase names autonomous entanglement distillation in open quantum systems, thermodynamic conversion into fluctuation-free energetic resources, loss-assisted formation of supercritical Bose condensates, stability-aware compression of diffusion-model transports, and, by extension, distillation problems in which interface stability or vapor-duty minimization governs sustained operation (Vollbrecht et al., 2010, Biswas et al., 2021, Munoz et al., 2020, Gao et al., 2 Jun 2026, Ma et al., 2023, Gooty et al., 2020). This suggests that “dissipative distillation” functions as an umbrella label for several distinct research programs linked by irreversible dynamics, steady-state structure, and nonequilibrium control.

1. Scope and terminological range

The term has no single standardized definition across fields. In some papers it denotes a concrete protocol class, as in dissipative entanglement distillation or thermodynamic distillation. In others it is a broader interpretive lens applied to nonequilibrium transport, stability, or model reduction. A recurring structural pattern is nevertheless visible: a source resource with fluctuations, admixtures, or contaminant-induced instability is subjected to an irreversible process whose fixed point, effective output, or coarse-grained representation is more ordered than the source.

Usage	Distilled object	Representative paper
Open quantum systems	High-fidelity entangled steady state	(Vollbrecht et al., 2010)
Thermodynamic resource theory	Energy eigenstate or fluctuation-free free energy	(Biswas et al., 2021)
Ultracold quantum gases	Supercritical, metastable condensate	(Munoz et al., 2020)
Driven-dissipative RBM	Variance-ranked reduced operator basis	(Christiansen et al., 8 May 2025)
Diffusion-model flow distillation	Few-step approximation of long-horizon transport	(Gao et al., 2 Jun 2026)
Membrane/process distillation	Stable evaporation interface or minimum-vapor-duty configuration	(Ma et al., 2023, Gooty et al., 2020)

A common misconception is that distillation must mean direct selective removal of an unwanted component. Several of the cited works explicitly reject that interpretation. In the ultracold-gas setting, the hot bath induces approximately uniform one-body loss rather than preferential depletion of thermal atoms. In the hydrogel-modified membrane-distillation setting, surfactant molecules were shown to diffuse through the hydrogel, so anti-wetting does not arise from simple exclusion. In the reduced-basis method for driven-dissipative systems, “distillation” is not state purification or entanglement distillation at all, but variance-based compression of a reduced representation (Munoz et al., 2020, Ma et al., 2023, Christiansen et al., 8 May 2025).

2. Autonomous entanglement distillation in open quantum systems

In open-system quantum information, dissipative distillation means engineering a Liouvillian whose attractive steady state is not merely entangled, but distilled relative to noisy source pairs. The dynamical framework is Lindblad evolution,

$$\dot\rho=\mathcal L(\rho)=\sum_i \gamma_i\left(L_i\rho L_i^\dagger-\frac12\{L_i^\dagger L_i,\rho\}\right),$$

with dissipation itself used as the resource-conversion mechanism rather than as an adversary to be suppressed. The central construction embeds ordinary LOCC distillation maps into continuous-time generators of the form $\delta\bigl(T(\rho)-\rho\bigr)$, so that source pairs are continuously replenished while a target pair is continuously pumped toward the distilled state (Vollbrecht et al., 2010).

Two schemes are developed. Scheme I addresses source states close to pure entangled states. Two source pairs are driven by nonlocal entangling jump operators toward the steady state

$$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$

while local cooling, heating, and dephasing perturb that source. Distillation then transfers entanglement from a two-copy source block to a target pair either autonomously, through source-target couplings exploiting the symmetry of the maximally entangled sector, or through an LOCC-assisted dissipative map. Scheme II is more general and cleaner: it treats Werner states

$$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$

as source steady states and defines a continuous $n\to1$ distillation process

$$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$

The noisy single-pair source steady state is

$$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$

and the target converges to the image of an effective source ensemble under the underlying distillation map.

The steady-state interpretation is the decisive departure from gate-based recurrence protocols. The target pair remains available continuously after convergence, and the fixed point is attractive independently of the initial condition. The paper also shows that arbitrary LOCC maps can be implemented approximately in continuous time using local dissipation plus fast classical communication, and extends the architecture to continuous entanglement swapping and continuous quantum repeaters. With booster blocks that amplify the desired target pumping rate against target noise, the repeater resources scale polynomially with distance,

$\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$

The work is notable for several counterintuitive consequences. Adding local cooling noise can improve the distilled steady-state entanglement in the LOCC-based version of Scheme I, although convergence becomes slower. The paper also identifies regimes in which none of the individual source pairs is entangled in steady state, yet entanglement can still be distilled because the two-copy source block retains distillable correlations. Dissipation is therefore not only compatible with purification-like tasks; it can be the mechanism by which those tasks are autonomously maintained.

3. Thermodynamic distillation and fluctuation–dissipation

In thermodynamic resource theory, thermodynamic distillation is a state-conversion problem under thermal operations. The free states are Gibbs states at inverse temperature $\beta$, and the allowed channels are thermal operations

$$\mathcal{E}(\rho)=\mathrm{Tr}_{E'}\!\left[U\left(\rho\otimes\gamma_E\right)U^\dagger\right],$$

with $\delta\bigl(T(\rho)-\rho\bigr)$0 commuting with the total Hamiltonian. A thermodynamic distillation process takes $\delta\bigl(T(\rho)-\rho\bigr)$1 to $\delta\bigl(T(\rho)-\rho\bigr)$2 where the target is an energy eigenstate,

$\delta\bigl(T(\rho)-\rho\bigr)$3

and approximation is measured by infidelity. The paper studies asymptotic families of $\delta\bigl(T(\rho)-\rho\bigr)$4 noninteracting subsystems, allowing both arbitrary independent energy-incoherent states and identical pure states, and explicitly allows the target Hamiltonian to differ from the initial one (Biswas et al., 2021).

The central quantitative objects are the nonequilibrium free energy based on relative entropy and its fluctuations. For the $\delta\bigl(T(\rho)-\rho\bigr)$5-partite input, the mean free-energy scale is

$\delta\bigl(T(\rho)-\rho\bigr)$6

and the fluctuation scale is

$\delta\bigl(T(\rho)-\rho\bigr)$7

The main second-order theorem states that the optimal error obeys a Gaussian law,

$\delta\bigl(T(\rho)-\rho\bigr)$8

with a Berry–Esseen correction for incoherent states. In the critical regime where $\delta\bigl(T(\rho)-\rho\bigr)$9, the unavoidable dissipated free energy in optimal distillation from identical incoherent states is

$$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$0

where

$$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$1

For identical pure states, the same Gaussian error law holds, and the same expression is proved as a lower bound. A key operational point is that dissipation is defined relative to the actual output state, not merely as a mismatch to the nominal target.

The conceptual content is a resource-theoretic fluctuation–dissipation theorem: converting fluctuating free energy into a fluctuation-free resource incurs an unavoidable finite-size penalty controlled not by the mean free energy alone but by its fluctuations. The paper makes this especially transparent in the i.i.d. pure-state case, where the relative-entropy variance reduces to ordinary energy variance. It also specializes the framework to work extraction, information erasure, and thermodynamically-free communication, thereby placing second-order thermodynamic irreversibility in direct correspondence with finite-blocklength information-theoretic structure. On the technical side, the incoherent-state proof relies on embedding Gibbs-rescaled distributions into a uniform space and reducing thermal convertibility to approximate majorization, drawing on results of Brandão et al. and Chubb et al.; the coherent pure-state analysis proceeds via dephasing and a multivariate central limit theorem. The paper is explicit, however, that mixed coherent initial states, more general targets, and deviation regimes beyond the $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$2 window remain open.

4. Non-equilibrium many-body state preparation

In ultracold-gas experiments, dissipative distillation denotes loss-assisted production of a condensate with a condensed fraction larger than equilibrium would permit for the same measured atom number, temperature, and trap frequencies. The experiment immerses an evaporatively cooled $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$3 Bose gas in a much hotter $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$4 magneto-optical trap, so that the potassium cloud acts as a controllable dissipative bath. The bath causes approximately uniform one-body loss and introduces little or no heating, while evaporative cooling continues to lower the temperature. The condensed fraction is

$$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$5

and equilibrium is benchmarked against the interacting-harmonic-trap formula based on $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$6. Under dissipation, the measured $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$7 rises well above the equilibrium curve; $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$8 is obtained already at $$|\psi\rangle=\frac{|00\rangle-\lambda |11\rangle}{\sqrt{1+\lambda^2}},\qquad \lambda=\tanh(r),$$9, and with $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$0 stronger dissipation the system enters a regime where $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$1 increases even while $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$2 increases beyond $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$3, yielding condensates above the equilibrium critical temperature. The excess condensed fraction relaxes on a timescale of $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$4, much longer than the trap and elastic-collision timescales, so the distilled condensates are metastable and quasi-static rather than fleeting transients (Munoz et al., 2020).

The mechanism is not direct removal of only thermal atoms. The authors instead interpret the effect through a nonequilibrium imbalance between condensate growth and depletion, maintained by simultaneous evaporation and dissipation. In their quantum-kinetic picture, the rates $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$5 and $$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$6 for atoms entering and leaving the condensate do not balance, and condensate growth can be favored by an interaction-generated energy scale

$$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$7

The resulting state is explicitly out of phase equilibrium: the thermal chemical potential can decrease even while the condensed fraction continues increasing. Dissipative distillation in this setting therefore names a controlled route to supercritical, long-lived, nonequilibrium matter.

5. Distillation of reduced representations in driven-dissipative systems

A different usage appears in the reduced-basis method for driven-dissipative Markovian quantum systems. Here the underlying dynamics is again Lindbladian,

$$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$8

but “distillation” refers neither to entanglement purification nor to condensate extraction. Exact steady states at selected parameter points are first used to construct a reduced basis,

$$\rho_W(f)=f\,\Omega+(1-f)\frac{\mathbb I-\Omega}{3}$$9

with coefficients obtained by minimizing the Liouvillian residual

$n\to1$0

A greedy procedure chooses new snapshot points by maximizing that residual over parameter space, so expensive exact solutions are required only at selected locations (Christiansen et al., 8 May 2025).

The distillation step is a subsequent PCA-like compression of the reduced basis itself. Because the reduced model is cheap to evaluate on a dense parameter grid, one forms the covariance matrix of the parameter-dependent coefficients,

$n\to1$1

diagonalizes it, and rewrites the state as

$n\to1$2

with explained variances $n\to1$3 and ratios

$n\to1$4

The distilled basis vectors are therefore principal operator directions in Liouville space, ranked by how much of the total parameter-induced state variation they explain.

This usage is explicitly observable-independent and “unbiased” in the sense that it does not require choosing an order parameter in advance. In the boundary-driven Fermi-Hubbard or driven XZZ chain, the dominant principal component has $n\to1$5 and shows a clear precursor of the conducting-to-insulating transition. In the dissipative XYZ chain, the leading principal component accounts for $n\to1$6 of the total state variance and closely tracks $n\to1$7. Symmetry analysis then turns the principal components into interpretable response channels with selection rules for which observables can overlap with which distilled modes. Dissipative distillation here is thus best understood as variance-ranked compression of a steady-state manifold rather than purification of a quantum resource.

6. Stability-aware flow distillation and classical distillation interfaces

In diffusion models, flow or trajectory distillation compresses a long reverse-time probability-flow transport into a small composition of learned maps. The teacher dynamics is the probability-flow ODE

$n\to1$8

and the student approximates the flow map $n\to1$9 by a segmentwise composition $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$0. The distinctive contribution of the quantitative framework is to show that few-step distillation is not only a static approximation problem but a stability problem: local errors are amplified according to the integrated Lipschitz/Jacobian bound

$$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$1

where, in the Gaussian-mixture Ornstein–Uhlenbeck setting,

$$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$2

The resulting global error bound takes the form

$$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$3

In low-noise multimodal regimes, the teacher transport becomes stiff, a Lipschitz-mismatch regime can arise, and one-step distillation is shown to be structurally unfavorable. The proposed remedy is a non-uniform stability-balanced grid defined by the cumulative stability coordinate $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$4, with $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$5. Experiments support the theory and reduce end-to-end relative MSE by up to $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$6 with $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$7 segments compared with uniform grids (Gao et al., 2 Jun 2026).

A related, though physically different, interpretation appears in membrane distillation. The hydrogel-modified membrane work does not use the phrase “dissipative distillation” explicitly, but it treats membrane distillation as a fundamentally dissipative, non-equilibrium heat-and-mass-transfer process in which interface stability sets the operating envelope. A sodium polyacrylate hydrogel is grafted onto the polypropylene support side of a PTFE/PP membrane, yielding a hydrogel-modified PTFE membrane whose PTFE active side remains hydrophobic while the hydrogel side is superhydrophilic in air and underwater oleophobic. The central problem is evaporation-interface instability induced by low-surface-tension contaminants such as surfactants and oils. The hydrogel does not block SDS transport; rather, it redistributes contaminants and suppresses the damaging surface effect at the liquid–vapor interface and triple-phase line. For a $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$8 PSA layer, the membrane maintains $$\dot{\rho}= \sum_{i=1}^n\left(\gamma E_i(\rho)+\frac{\epsilon}{2}N_i(\rho)\right)+\delta_D\bigl(T_D(\rho)-\rho\bigr).$$9–$$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$0 bar breakthrough pressure over varying SDS concentration, and increasing the hydrogel thickness to $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$1 yields $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$2 bar. In direct-contact membrane distillation at $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$3, the membrane maintains constant mass flux and $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$4 salt rejection across all tested NaCl/SDS combinations, including $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$5 NaCl and $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$6 SDS, and runs stably for $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$7 on synthetic wastewater containing $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$8 NaCl, $$\rho_s=\frac{\gamma \rho_W(f)+\epsilon \,\mathbb I/4}{\gamma+\epsilon}, \qquad f_s=\frac{\gamma f+\epsilon/4}{\gamma+\epsilon},$$9 SDS, and $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$0 mineral oil, whereas bare PTFE fails in roughly $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$1 (Ma et al., 2023).

These two literatures share a stability-centric view of distillation. In diffusion modeling, the question is whether transport error grows under an expansive teacher flow. In membrane distillation, the question is whether an evaporation interface can sustain latent-heat-driven transport without surfactant-induced collapse. In both cases, the decisive variable is not only the nominal target map or membrane morphology, but the dynamical amplification or suppression of perturbations during ongoing nonequilibrium operation.

7. Process-level energy minimization and interpretive boundaries

In chemical-process synthesis, dissipative distillation is best understood as distillation whose thermal driving-force expenditure is minimized. The MINLP-based framework for multicomponent distillation sequence synthesis formulates the objective as minimization of total vapor duty,

$\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$2

over conventional and thermally coupled configurations. The difficult part is simultaneous treatment of combinatorial topology decisions and nonconvex fractional Underwood equations. The paper develops a lifted binary formulation using stream, section, split, and parent-link variables; combines partial fraction decomposition with Reformulation-Linearization Technique, yielding what it calls Reformulation-Division-Linearization Technique; proves simultaneous convex-hull results for several special structures; and introduces adaptive partitioning of Underwood-root domains. The formulation is stated to be strictly tighter than prior approaches of Caballero and Grossmann, Giridhar and Agrawal, and Tumbalam Gooty et al. On a benchmark of $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$3 five-component cases, the method solves all instances to $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$4 optimality, whereas the cited prior Caballero-based approach solves about $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$5 of cases and the prior 2019 MINLP approach solves $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$6. Within $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$7 minutes, the new method already outperforms the prior best approach run for $\left(\frac{L}{L_0}\right)^{\log_2(2m^2n)}.$8 hours (Gooty et al., 2020).

This process-systems usage sits at an interpretive boundary of the term. The paper is not about a new dissipative apparatus or a steady-state open-system protocol; it is about rigorous synthesis of the least energy-intensive configuration under shortcut thermodynamic assumptions. Likewise, the membrane-distillation paper is not framed by its authors as “dissipative distillation,” but it is directly informative to that theme because membrane distillation is a dissipative nonequilibrium process and because interfacial failure determines whether latent-heat-driven transport can be sustained. These cases suggest that the phrase can also denote a broader research orientation: engineering distillation so that dissipation is either stabilized, redistributed, or reduced, rather than merely endured.

Taken together, these literatures suggest that dissipative distillation is unified less by a single mathematical formalism than by a common inversion of perspective. Dissipation ceases to be a parasitic correction to an otherwise ideal protocol and instead becomes the operative object of design. What is distilled may be entanglement, free energy, condensate fraction, operator-basis variance, transport maps, evaporation interfaces, or vapor-duty expenditure. The differences among these usages are substantial, but the shared theme is that irreversible dynamics is treated as the mechanism, constraint, or optimization target through which ordered outputs emerge from nonequilibrium sources.