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A '''dissipative system''' is a thermodynamically [[open system (systems theory)|open system]] which is operating out of, and often far from, [[thermodynamic equilibrium]] in an environment with which it exchanges [[energy]] and [[matter]]. | |||
A '''dissipative structure''' is a dissipative system that has a dynamical régime 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 == | |||
A [[Dissipation|dissipative]] structure is characterized by the spontaneous appearance of symmetry breaking ([[anisotropy]]) and the formation of complex, sometimes [[Chaos theory|chaotic]], structures where interacting particles exhibit long range correlations. The term ''dissipative structure'' was coined by Russian-Belgian physical chemist [[Ilya Prigogine]], who was awarded the [[Nobel Prize in Chemistry]] in 1977 for his pioneering work on these structures. The dissipative structures considered by Prigogine have dynamical régimes that can be regarded as thermodynamically steady states, and sometimes at least can be described by suitable [[extremal principles in non-equilibrium thermodynamics]]. | |||
Examples in every day life include [[convection]], [[cyclone]]s, [[Tropical cyclone|hurricane]]s and [[life|living organisms]]. Less common examples include [[laser]]s, [[Bénard cells]], and the [[Belousov–Zhabotinsky reaction]].{{citation needed|date=September 2012}} | |||
One way of mathematically modeling a dissipative system is given in the article on ''[[wandering set]]s'': it involves the action of a [[group (mathematics)|group]] on a [[measure (mathematics)|measurable set]]. | |||
== In control theory == | |||
In systems and [[control theory]], dissipative systems are dynamical systems with a state <math>x(t)</math>, inputs <math>u(t)</math> and outputs <math>y(t)</math>, which satisfy the so-called "dissipation inequality". | |||
Given a function <math>w</math> on <math>U \times Y</math>, with finite integral of its modulus for any input function <math>u</math> and initial state <math>x(0)</math> over any finite time <math>t</math>, called the "supply rate", a system is said to be dissipative if there exist a continuous nonnegative function <math>V(x)</math>, with <math>x(0) = 0</math>, called the storage function, such that for any input <math>u</math> and initial state <math>x(0)</math>, the following inequality, known as dissipation inequality, always holds: | |||
:<math>V(x(t)) - V(x(0)) \le \int_{0}^{t} u(\tau) \cdot y(\tau) d \tau</math>, | |||
Dissipative systems with supply rate | |||
:<math>w= u \cdot y</math> | |||
where <math> \cdot </math> denotes the scalar product, | |||
Dissipative systems satisfy the inequality: | |||
:<math>\frac{dV(x(t))}{dt} \le u(t) \cdot y(t)</math> | |||
The physical interpretation is that <math>V(x)</math> is the energy in the system, whereas <math>u \cdot y</math> 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, 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. Dissipative systems are still an active field of research in systems and control, due to their important applications. | |||
== Quantum dissipative systems == | |||
{{main|Quantum dissipation}} | |||
As [[quantum mechanics]], and any classical [[dynamical system]], relies heavily on [[Hamiltonian mechanics]] for which [[Time reversibility|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's theorem (Hamiltonian)|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 [[H-theorem|irreversible]] and constructive role for time. | |||
== See also == | |||
{{Div col}} | |||
* [[Non-equilibrium thermodynamics]] | |||
* [[Extremal principles in non-equilibrium thermodynamics]] | |||
* [[Autowave]] | |||
* [[Self-organization]] | |||
* [[Autocatalytic reactions and order creation]] | |||
* [[Dynamical system]] | |||
* [[Autopoiesis]] | |||
* [[Relational order theories]] | |||
* [[Loschmidt's paradox]] | |||
{{Div col end}} | |||
== References== | |||
* [http://sciphilos.info/doc%20PAGES%20/docDaviesSelfOrgStru.html Davies, Paul ''The Cosmic Blueprint'']{{dead link|date=September 2012}} Simon & Schuster, New York 1989 (abridged— 1500 words) (abstract— 170 words) — self-organized structures. | |||
* B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative Systems Analysis and Control. Theory and Applications. Springer Verlag, London, 2nd Ed., 2007. | |||
* J.C. Willems. Dissipative dynamical systems, part I: General theory; part II: Linear systems with quadratic supply rates. Archive for Rationale mechanics Analysis, vol.45, pp. 321–393, 1972. | |||
== External links == | |||
* [http://epress.anu.edu.au/info_systems/mobile_devices/ch11s05.html The dissipative systems model] The Australian National University | |||
[[Category:Thermodynamics]] | |||
[[Category:Systems theory]] | |||
[[Category:Non-equilibrium thermodynamics]] |
Latest revision as of 10:20, 7 March 2013
Template:No footnotes 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 dissipative structure is a dissipative system that has a dynamical régime 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
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. The term dissipative structure was coined by Russian-Belgian physical chemist Ilya Prigogine, who was awarded the Nobel Prize in Chemistry in 1977 for his pioneering work on these structures. The dissipative structures considered by Prigogine have dynamical régimes that can be regarded as thermodynamically steady states, and sometimes at least can be described by suitable extremal principles in non-equilibrium thermodynamics.
Examples in every day life include convection, cyclones, hurricanes and living organisms. Less common examples include lasers, Bénard cells, and the Belousov–Zhabotinsky reaction.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
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.
In control theory
In systems and control theory, dissipative systems are dynamical systems with a state , inputs and outputs , which satisfy the so-called "dissipation inequality".
Given a function on , with finite integral of its modulus for any input function and initial state over any finite time , called the "supply rate", a system is said to be dissipative if there exist a continuous nonnegative function , with , called the storage function, such that for any input and initial state , the following inequality, known as dissipation inequality, always holds:
Dissipative systems with supply rate
where denotes the scalar product,
Dissipative systems satisfy the inequality:
The physical interpretation is that is the energy 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, 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. Dissipative systems are still an active field of research in systems and control, due to their important applications.
Quantum dissipative systems
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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.
See also
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- Non-equilibrium thermodynamics
- Extremal principles in non-equilibrium thermodynamics
- Autowave
- Self-organization
- Autocatalytic reactions and order creation
- Dynamical system
- Autopoiesis
- Relational order theories
- Loschmidt's paradox
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References
- Davies, Paul The Cosmic BlueprintTemplate:Dead link Simon & Schuster, New York 1989 (abridged— 1500 words) (abstract— 170 words) — self-organized structures.
- B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative Systems Analysis and Control. Theory and Applications. Springer Verlag, London, 2nd Ed., 2007.
- J.C. Willems. Dissipative dynamical systems, part I: General theory; part II: Linear systems with quadratic supply rates. Archive for Rationale mechanics Analysis, vol.45, pp. 321–393, 1972.
External links
- The dissipative systems model The Australian National University