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# Dissipative system

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

A dissipative system (or dissipative structure) is a thermodynamically open system which is operating far from thermodynamic equilibrium in an environment with which it exchanges energy and matter.

## Overview

A dissipative system 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 Belgian scientist Ilya Prigogine, who pioneered research in the field and won the Nobel Prize in Chemistry in 1977.

Simple examples include convection, cyclones and hurricanes. More complex examples include lasers, Bénard cells, the Belousov-Zhabotinsky reaction and at the most sophisticated level, life itself.

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 Systems and control theory, dissipative systems are dynamical systems with a state x(t), inputs u(t) and outputs y(t), such that there exist so-called storage functions V(x,t) and supply rates w(u,y) such that: V(.,.) is a nonnegative function, and for any time t one has dV(x(t),t)/dt less than u(t).y(t), where . is the scalar product. 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 in Systems and Control is a physically-based notion that extends to more abstract systems, and that is extremely useful and powerful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been settled 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-Yakubovic-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.