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Dissipative system
Dissipative system
Dissipative system
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A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dissipative systems stand in contrast to conservative systems.

A dissipative structure is a dissipative system that has a dynamical regime that is in some sense in a reproducible steady state. This reproducible steady state may be reached by natural evolution of the system, by artifice, or by a combination of these two.

Overview

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A dissipative structure is characterized by the spontaneous appearance of symmetry breaking (anisotropy) and the formation of complex, sometimes chaotic, structures where interacting particles exhibit long range correlations. Examples in everyday life include convection, turbulent flow, cyclones, hurricanes and living organisms. Less common examples include lasers, Bénard cells, droplet cluster, and the Belousov–Zhabotinsky reaction.[1]

One way of mathematically modeling a dissipative system is given in the article on wandering sets: it involves the action of a group on a measurable set.

Dissipative systems can also be used as a tool to study economic systems and complex systems.[2] For example, a dissipative system involving self-assembly of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems.[3]

The Hopf decomposition states that dynamical systems can be decomposed into a conservative and a dissipative part; more precisely, it states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant dissipative set.

Dissipative structures in thermodynamics

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Russian-Belgian physical chemist Ilya Prigogine, who coined the term dissipative structure, received the Nobel Prize in Chemistry in 1977 for his pioneering work on these structures, which have dynamical regimes that can be regarded as thermodynamic steady states, and sometimes at least can be described by suitable extremal principles in non-equilibrium thermodynamics.

In his Nobel lecture,[4] Prigogine explains how thermodynamic systems far from equilibrium can have drastically different behavior from systems close to equilibrium. Near equilibrium, the local equilibrium hypothesis applies and typical thermodynamic quantities such as free energy and entropy can be defined locally. One can assume linear relations between the (generalized) flux and forces of the system. Two celebrated results from linear thermodynamics are the Onsager reciprocal relations and the principle of minimum entropy production.[5] After efforts to extend such results to systems far from equilibrium, it was found that they do not hold in this regime and opposite results were obtained.

One way to rigorously analyze such systems is by studying the stability of the system far from equilibrium. Close to equilibrium, one can show the existence of a Lyapunov function which ensures that the entropy tends to a stable maximum. Fluctuations are damped in the neighborhood of the fixed point and a macroscopic description suffices. However, far from equilibrium stability is no longer a universal property and can be broken. In chemical systems, this occurs with the presence of autocatalytic reactions, such as in the example of the Brusselator. If the system is driven beyond a certain threshold, oscillations are no longer damped out, but may be amplified. Mathematically, this corresponds to a Hopf bifurcation where increasing one of the parameters beyond a certain value leads to limit cycle behavior. If spatial effects are taken into account through a reaction–diffusion equation, long-range correlations and spatially ordered patterns arise,[6] such as in the case of the Belousov–Zhabotinsky reaction. Systems with such dynamic states of matter that arise as the result of irreversible processes are dissipative structures.

Recent research has seen reconsideration of Prigogine's ideas of dissipative structures in relation to biological systems.[7]

Dissipative systems in control theory

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Willems first introduced the concept of dissipativity in systems theory[8] to describe dynamical systems by input-output properties. Considering a dynamical system described by its state , its input and its output , the input-output correlation is given a supply rate . A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function such that , and

.[9]

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate .

The physical interpretation is that is the energy stored in the system, whereas is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by V.M. Popov, J.C. Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems[clarification needed], this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems[clarification needed].[10] Dissipative systems are still an active field of research in systems and control, due to their important applications.

Quantum dissipative systems

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Main article: Quantum dissipation

As quantum mechanics, and any classical dynamical system, relies heavily on Hamiltonian mechanics for which time is reversible, these approximations are not intrinsically able to describe dissipative systems. It has been proposed that in principle, one can couple weakly the system – say, an oscillator – to a bath, i.e., an assembly of many oscillators in thermal equilibrium with a broad band spectrum, and trace (average) over the bath. This yields a master equation which is a special case of a more general setting called the Lindblad equation that is the quantum equivalent of the classical Liouville equation. The well-known form of this equation and its quantum counterpart takes time as a reversible variable over which to integrate, but the very foundations of dissipative structures imposes an irreversible and constructive role for time.

Recent research has seen the quantum extension[11] of Jeremy England's theory of dissipative adaptation[7] (which generalizes Prigogine's ideas of dissipative structures to far-from-equilibrium statistical mechanics, as stated above).

Applications on dissipative systems of dissipative structure concept

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The framework of dissipative structures as a mechanism to understand the behavior of systems in constant interexchange of energy has been successfully applied on different science fields and applications, as in optics,[12][13] population dynamics and growth[14][15][16] and chemomechanical structures.[17][18][19]

See also

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Notes

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References

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Dissipative system

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A dissipative system, also known as a dissipative structure, is a thermodynamically open system that operates far from equilibrium, exchanging energy and matter with its environment to sustain organized, coherent states through irreversible processes that produce entropy. These systems emerge via fluctuations and instabilities, leading to self-organization where non-equilibrium conditions act as a source of order rather than disorder, contrasting with the tendency toward equilibrium in closed systems. Introduced by Ilya Prigogine in his work on non-equilibrium thermodynamics, dissipative structures require continuous energy input to maintain stability, often exhibiting sensitivity to boundary conditions, system size, and nonlinear interactions such as autocatalysis. Key characteristics include , where small fluctuations amplify into macroscopic patterns, and an increase in that drives the system toward more complex configurations. For instance, in physical systems like Bénard convection cells, layers heated from below form hexagonal patterns when the exceeds a critical threshold ( > 1708), maximizing the rate of . In chemical contexts, reactions such as the model demonstrate oscillations and spatial patterns like Turing structures, illustrating how feedback loops sustain temporal and spatial order. Dissipative systems extend to biological applications, where living organisms function as prime examples, maintaining and through metabolic cycles that dissipate energy from nutrient flows while exporting to the surroundings. This framework links self-organization to , with phenomena like genetic mutations amplifying fluctuations akin to thermodynamic instabilities, enabling adaptive responses to environmental changes. In broader contexts, such as or , dissipative principles inform models of behaviors in particle systems or feedback control in dynamical systems, emphasizing the role of in achieving stability and . Overall, these systems unify aspects of physics, chemistry, and by revealing how irreversibility fosters emergent order in far-from-equilibrium environments.

Introduction

Definition and Basic Principles

A dissipative system is a thermodynamically open system that operates far from equilibrium, exchanging and/or with its environment, which leads to the dissipation of free energy and an overall increase in . Unlike closed or isolated systems, open systems allow continuous flows that sustain internal dynamics, preventing them from reaching where minimizes. The basic principles of dissipative systems revolve around irreversibility, where processes driven by the second law of convert ordered energy into disordered forms like , fostering complex behaviors rather than stasis. This irreversibility enables , such as temporal oscillations in chemical reactions or spatial patterns in fluid , and the emergence of ordered structures through amplified fluctuations that organize matter despite increasing global . In contrast to conservative systems, where forces like preserve without loss, dissipative systems inherently lose usable energy to irreversible processes, leading to and eventual stabilization or novel steady states. Simple examples illustrate these principles in physical contexts: a mechanical oscillator with , such as a damped , dissipates as through air resistance and material , causing oscillations to decay over time. Similarly, an electrical circuit with resistance converts electrical energy into thermal via , reducing current flow and preventing indefinite energy storage. These cases highlight how openness to environmental exchange drives , distinguishing dissipative dynamics from idealized reversible models.

Historical Development

The concept of dissipative systems traces its roots to 19th-century developments in , where the foundations of energy dissipation and irreversibility were laid. introduced the notion of in 1865 as a measure of the unavailability of energy for work in thermodynamic processes, emphasizing the dissipative nature of heat transfer and the second law of . Building on this, Lord Rayleigh analyzed instabilities in fluid motions during the 1880s, demonstrating how dissipative processes could lead to the breakdown of equilibrium states and the emergence of ordered patterns in viscous fluids under gravitational forces. In the mid-20th century, the framework for advanced significantly with Lars Onsager's formulation of reciprocal relations in 1931, which linked phenomenological coefficients in irreversible processes and provided a mathematical basis for understanding coupled dissipative fluxes near equilibrium. These ideas set the stage for exploring systems far from equilibrium, highlighting how dissipation could drive symmetry-breaking and . Ilya Prigogine played a pivotal role in the 1960s and 1970s by developing the theory of dissipative structures, which describe self-organizing systems maintained by continuous energy and matter exchange with their environment. For this work on and the role of fluctuations in , Prigogine received the in 1977. His influential book From Being to Becoming (1980) further synthesized these concepts, arguing that irreversibility and time's arrow are intrinsic to dissipative processes, bridging classical with complexity in physical and biological systems. Following Prigogine's contributions, the concept expanded into systems and in the early 1970s, with Jan C. Willems introducing dissipativity as a property of dynamical systems characterized by dissipation relative to a storage function. In the 1980s and 1990s, extensions to emerged through models of open , incorporating dissipation via environment interactions to describe decoherence and quantum . More recently, post-2020 research has applied dissipative principles to , such as in studies of dissipative adaptation enabling in driven , and further to quantum many-body correlations and gravitational effective field theories for open systems as of 2025.

Thermodynamic Foundations

Non-Equilibrium Thermodynamics

In , dissipative systems operate as open systems exchanging and energy with their environment, allowing the second to permit local entropy decreases compensated by greater increases in the surroundings. The balance is expressed as dS=deS+diSdS = d_e S + d_i S, where deSd_e S represents entropy exchange and diS≥0d_i S \geq 0 the internal production, ensuring overall entropy growth in the . The rate σ=∑JiXi>0\sigma = \sum J_i X_i > 0, with JiJ_i as thermodynamic fluxes (e.g., or flow) and XiX_i as conjugate affinities (e.g., temperature or gradients), quantifies as the driving force for irreversible processes. Near equilibrium, the linear regime prevails, characterized by Onsager's reciprocal relations, where fluxes linearly depend on affinities: Ji=∑jLijXjJ_i = \sum_j L_{ij} X_j, with the phenomenological coefficients satisfying Lij=LjiL_{ij} = L_{ji} due to . In this domain, Prigogine's principle of minimum applies, positing that steady states minimize σ\sigma subject to fixed constraints, providing a variational criterion for stability akin to a . Far from equilibrium, however, nonlinear interactions emerge, transitioning to regimes where can increase through instabilities, enabling beyond linear approximations. Bifurcations mark critical transitions in these far-from-equilibrium conditions; for instance, a occurs when a stable equilibrium loses stability, giving rise to sustained oscillatory states through the emergence of a , reflecting the onset of temporal organization. Prigogine's theory elucidates how such order arises from fluctuations: near critical points, random perturbations are amplified by nonlinearities, breaking the law of large numbers and fostering coherent structures sustained by continuous dissipation. In chemical reactions, autocatalytic sets illustrate this principle, where self-amplifying cycles (e.g., product catalyzing reactant conversion) enhance local order via concentration patterns, while elevating global to comply with the second law.

Dissipative Structures

Dissipative structures refer to spatially or temporally organized patterns that emerge and persist in far-from-equilibrium systems through the continuous of and matter, often involving broken in steady states. These structures arise from irreversible processes that amplify fluctuations, leading to where order is maintained by ongoing exchanges with the environment, countering the tendency toward disorder predicted by equilibrium . Coined by in the context of , the concept highlights how such patterns function as "islands of decreasing " locally, while globally increasing . These structures are characterized by excess , where the system maximizes to maintain order, as opposed to minimizing it near equilibrium. Key characteristics of dissipative structures include driven by autocatalytic feedback loops, high reproducibility under consistent conditions, and acute sensitivity to external parameters such as temperature gradients or chemical concentrations. For instance, small perturbations near critical thresholds can trigger bifurcations, transitioning the system from uniform states to coherent patterns, with the scale and form dictated by boundary conditions and system size. This sensitivity underscores their role in understanding and order formation in open systems. Classic examples illustrate these principles vividly. Bénard convection cells, observed in the early 1900s, form when a thin fluid layer heated from below develops hexagonal patterns of upward and downward flows beyond a critical , dissipating heat more efficiently than conduction alone. The Belousov-Zhabotinsky reaction, discovered in the 1950s, produces temporal oscillations in color and chemical concentrations through autocatalytic cycles involving and , exemplifying spatiotemporal patterns in chemical systems. Lasers represent another paradigm, where in an excited medium, pumped by external energy, generates coherent light beams as a dissipative structure, with gain and loss balancing to sustain the ordered emission. Mathematically, the formation of dissipative structures is often described by reaction-diffusion equations, which couple local reaction kinetics with spatial . A prototypical form for a two-component is given by: ∂u∂t=Du∇2u+f(u,v),∂v∂t=Dv∇2v+g(u,v),\begin{align} \frac{\partial u}{\partial t} &= D_u \nabla^2 u + f(u, v), \\ \frac{\partial v}{\partial t} &= D_v \nabla^2 v + g(u, v), \end{align} where uu and vv are concentrations, DuD_u and DvD_v are diffusion coefficients, and ff and gg represent nonlinear reaction terms, such as in the model. These equations predict instabilities like Turing patterns when diffusion rates differ, leading to stationary spatial structures sustained by energy throughput. On a planetary scale, hurricanes and tornadoes exemplify large-scale dissipative structures, where solar energy input drives atmospheric convection and rotation, forming organized vortices that dissipate heat and moisture into the environment, thereby enhancing global entropy export.

Systems and Control Theory

Concept of Dissipativity

In systems and control theory, dissipativity refers to a property of dynamical systems where an abstract notion of "energy" or storage does not increase beyond what is supplied externally along system trajectories. Specifically, a system is dissipative with respect to a supply rate function w(u,y)w(u, y), which quantifies the rate of energy supply from inputs uu to outputs yy, if the accumulated supply is non-positive or bounded in a way that prevents indefinite energy growth. If the supply rate satisfies w(u,y)≤0w(u, y) \leq 0 for all admissible uu and yy, the system is strictly dissipative, implying that internal storage decreases over time without external input. The foundational framework for dissipativity was established by Jan C. Willems in 1972, who defined it for state-space models through the existence of a non-negative storage function V(x):X→R≥0V(x): X \to \mathbb{R}_{\geq 0}, where xx is the state. For a evolving from initial state x(0)x(0) to x(t)x(t), dissipativity holds if V(x(t))≤V(x(0))+∫0tw(u(s),y(s)) dsV(x(t)) \leq V(x(0)) + \int_0^t w(u(s), y(s)) \, ds for all t≥0t \geq 0 and all input trajectories u(⋅)u(\cdot). This inequality ensures that any increase in storage is accounted for by the integrated supply, preventing the system from generating spontaneously. Willems' approach unifies various stability concepts by treating V(x)V(x) as an abstract measure, applicable beyond physical interpretations. From an input-output perspective, dissipativity generalizes classical notions like passivity, where the supply rate is w(u,y)=uTyw(u, y) = u^T y, representing power flow into the . More broadly, arbitrary supply rates allow analysis of properties such as finite gain (w(u,y)=γ2∥u∥2−∥y∥2w(u, y) = \gamma^2 \|u\|^2 - \|y\|^2) or sector boundedness, providing a flexible tool for characterizing without requiring detailed internal dynamics. This framework applies directly to nonlinear dynamical systems of the form x˙=f(x,u)\dot{x} = f(x, u), y=h(x,u)y = h(x, u), assuming well-posedness such as of solutions for given . Unlike thermodynamic dissipative structures, which involve physical and openness to maintain far-from-equilibrium states, the systems-theoretic concept of dissipativity serves primarily as a tool for stability and control design. Here, the storage function V(x)V(x) is mathematical and not tied to thermodynamic , focusing instead on bounding energy-like quantities to ensure robust behavior under feedback.

Stability and Passivity

In , the dissipativity property of a system establishes a direct link to when the supply rate satisfies specific conditions. For a dynamical system x˙=f(x,u)\dot{x} = f(x, u), y=h(x,u)y = h(x, u), dissipativity with respect to a supply rate s(u,y)s(u, y) implies the existence of a nonnegative storage function V(x)V(x) such that V˙(x)≤s(u,y)\dot{V}(x) \leq s(u, y) along system trajectories. If s(u,y)s(u, y) is negative semi-definite (i.e., s(u,y)≤0s(u, y) \leq 0 for all u,yu, y) and the system is zero-state observable, then V(x)V(x) serves as a Lyapunov function, ensuring asymptotic stability of the equilibrium. This connection, originally derived for nonlinear systems, unifies energy-based arguments with stability analysis, where the dissipation inequality bounds the increase in stored "energy." Passivity represents a of dissipativity, where the supply rate is s(u,y)=uTys(u, y) = u^T y, corresponding to power flow in physical systems. Passive systems are dissipative with respect to this rate, and the passivity theorem guarantees that the interconnection of two strictly passive systems is asymptotically stable, assuming detectability of the outputs. Conversely, the converse Lyapunov theorem provides a method to verify passivity by constructing a quadratic storage function that satisfies the inequality, particularly for linear time-invariant systems via the Kalman-Yakubovich-Popov lemma. These results enable feedback design that preserves or enforces passivity, facilitating stabilization through energy-dissipating interconnections. Dissipativity further supports applications, such as the framework of integral quadratic constraints (IQCs), which extend dissipation inequalities to frequency-domain bounds for uncertain systems. IQCs model uncertainties (e.g., nonlinearities or delays) as constraints on input-output signals, allowing linear matrix inequality (LMI)-based tests for robust stability of feedback loops. This approach is widely used in , where passivity ensures persistent excitation and parameter convergence without destabilizing the system. A representative example is the analysis of RLC electrical networks, which are passive dissipative systems with storage function given by the total magnetic and electric 12Li2+12Cv2\frac{1}{2} L i^2 + \frac{1}{2} C v^2, and supply rate s(u,y)=vis(u, y) = v i, where vv and ii are voltage and current; this underpins passivity-based stabilization of power systems. Extensions of dissipativity theory post-2000 have addressed hybrid systems, incorporating discrete switching events while maintaining stability guarantees. In the 2010s, research developed notions of quadratic supply rate (QSR) dissipativity for hybrid interconnections, ensuring that mode transitions preserve overall dissipation and Lyapunov-like stability through multiple storage functions. These advancements enable analysis of cyber-physical systems, such as switched control networks, where dissipativity certifies robustness to abrupt changes.

Quantum Mechanics

Open Quantum Systems

Open quantum systems describe quantum mechanical entities that interact non-negligibly with an external environment, or "bath," leading to phenomena such as decoherence, where quantum superpositions decay, and energy dissipation, where the system loses or gains energy to the surroundings. In the quantum domain, dissipative structures emerge through non-equilibrium dynamics, extending Prigogine's classical concepts to quantum fluctuations and coherence. In contrast, closed quantum systems evolve unitarily according to the , preserving coherence and energy without external influences. This coupling to the environment fundamentally alters the dynamics, making open systems central to understanding realistic quantum processes in fields like and . A common simplification in modeling open quantum systems is the Markovian approximation, which assumes the environment's memory effects are negligible, allowing the system's evolution to depend only on its current state. This approximation relies on the , valid for weak system-bath coupling where the system's state has minimal back-action on the bath, and the secular approximation, which neglects rapidly oscillating terms in the to focus on resonant energy exchanges. These assumptions enable tractable derivations of master equations but break down in strong-coupling regimes or structured environments. The dynamics of open quantum systems are rigorously described using the density operator formalism, where the system's state is represented by a ρ\rho, capturing mixed states arising from environmental entanglement. The time evolution of ρ\rho is governed by completely positive trace-preserving (CPTP) maps, ensuring that probabilities remain non-negative and normalized after any , including dissipative effects. This framework generalizes unitary evolution and accommodates irreversible processes without violating quantum axioms. Environments in open quantum systems are often modeled as thermal baths consisting of infinite collections of non-interacting oscillators, providing a bosonic reservoir at a given . The Caldeira-Leggett model, developed in the , exemplifies this approach by coupling a central quantum system—typically a —to such a bath via bilinear interactions, yielding Ohmic or sub-Ohmic spectra that capture realistic frictional and noisy effects. This model has become foundational for studying quantum and in condensed matter systems. Recent advancements since 2020 have emphasized non-Markovian effects in open quantum systems within , where memory correlations in the bath can lead to backflow and modified fluctuation relations. Reviews from 2022 onward highlight how these effects enable thermodynamic advantages, such as enhanced work extraction in quantum engines, challenging classical Markovian limits. For instance, non-Markovian dynamics have been shown to influence and heat statistics, with fluctuation theorems extended to account for temporal correlations in structured reservoirs. This focus underscores the interplay between and coherence in emerging quantum technologies.

Quantum Dissipative Models

Quantum dissipative models describe the dynamics of open quantum systems interacting with their environment, leading to irreversible processes such as decoherence and relaxation. The most general framework for Markovian dynamics in these systems is provided by the , which ensures complete positivity and trace preservation of the density operator ρ\rho. The equation takes the form dρdt=−i[H,ρ]+∑k(LkρLk†−12{Lk†Lk,ρ}),\frac{d\rho}{dt} = -i [H, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right), where HH is the system Hamiltonian and the LkL_k are Lindblad operators representing the dissipative channels, such as coupling to a bosonic bath. This form was derived independently by Gorini, Kossakowski, Sudarshan, and Lindblad in 1976 as the generator of quantum dynamical semigroups. A example is the , modeling phenomena like loss in optical cavities. For a Hamiltonian H=ωa†aH = \omega a^\dagger a with via the jump operator L=γaL = \sqrt{\gamma} a
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