﻿

# Dissipative system

Another meaning of "dissipative system" is one that dissipates heat, see heat dissipation.

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.

Simple examples include convection, cyclones and hurricanes. More complex examples include lasers, Bénard cells, the Belousov–Zhabotinsky reaction, and living organisms.

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 x(t), inputs u(t) and outputs y(t), which satisfy the so-called "dissipation inequality". Given a function w on U×Y, with finite integral of its modulus for any input function u and initial state x(0) over any finite time t, called the "supply rate", a system is said to be dissipative if there exist a continuous nonnegative function V(x), with x(0) = 0, called the storage function, such that for any input u and initial state x(0) the difference V(x(t)) − V(x(0)) does not exceed the integral of the supply over (0,t) for any t (dissipation inequality). Dissipative systems with supply rate w= u.y, where . denotes the scalar product, are "passive systems"; equivalently such systems satisfy the inequality: dV(x(t))/dt less or equal u(ty(t). The physical interpretation is that V(x) is the energy in the system, whereas u·y 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

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.

## References

• Davies, Paul The Cosmic Blueprint 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Dissipative system — dissipative dis si*pa*tive (d[i^]s s[i^]*p[asl]*t[i^]v), a. Tending to dissipate. [1913 Webster] {Dissipative system} (Mech.), an assumed system of matter and motions in which forces of friction and resistances of other kinds are introduced… …   The Collaborative International Dictionary of English

• dissipative system — disipatyvioji sistema statusas T sritis fizika atitikmenys: angl. dissipative system vok. dissipatives System, n rus. диссипативная система, f pranc. système à dissipation d’énergie, m; système dissipatif, m …   Fizikos terminų žodynas

• dissipative — dis si*pa*tive (d[i^]s s[i^]*p[asl]*t[i^]v), a. Tending to dissipate. [1913 Webster] {Dissipative system} (Mech.), an assumed system of matter and motions in which forces of friction and resistances of other kinds are introduced without regard to …   The Collaborative International Dictionary of English

• Dissipative soliton — Dissipative solitons (DSs) are stable solitary localized structures that arise in nonlinear spatially extended dissipative systems due to mechanisms of self organization. They can be considered as an extension of the classical soliton concept in… …   Wikipedia

• Dissipative particle dynamics — (DPD) is a stochastic simulation technique for simulating the dynamic and rheological properties of simple and complex fluids. It was initially devised by Hoogerbrugge and Koelman [1][2] to avoid the lattice artifacts of the so called lattice gas …   Wikipedia

• Dissipative Struktur — Beispiel dissipativer Strukturen: Granulation auf der Sonnenoberfläche. Bilddurchmesser ca. 35.000 km Mit dem Begriff Dissipative Struktur (engl. dissipative structure ‚zerstreuende Struktur‘) wird das Phänomen sich selbstorganisierender,… …   Deutsch Wikipedia

• System — Organismus; Organisation; Struktur; Anlage; Gebilde; Ordnungsprinzip; Gedankenfolge; Mechanismus; Zusammenhang; Umstand * * * Sys|tem [zʏs te:m], das; s, e …   Universal-Lexikon

• dissipative Struktur — dis|si|pa|ti|ve Struk|tur [zu ↑ Dissipation] bes. Bez. für ein im ↑ stationären Zustand befindliches thermodynamisch offenes System mit einem speziellen Ordnungsgrad, dessen Gesetzmäßigkeiten so unterschiedlichen Erscheinungen wie oszillierenden… …   Universal-Lexikon

• Dynamical system (definition) — This article presents the many ways to define a dynamical system. See the main article, dynamical system, for an overview of the topic. The dynamical system concept is a mathematical formalization for any fixed rule which describes the time… …   Wikipedia

• Complex system — This article largely discusses complex systems as a subject of mathematics and the attempts to emulate physical complex systems with emergent properties. For other scientific and professional disciplines addressing complexity in their fields see… …   Wikipedia