# Dissipative system

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.[citation needed]

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 $U \times Y$, 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 following inequality, known as dissipation inequality, always holds:

$V(x(t)) - V(x(0)) \le \int_{0}^{t} u(\tau) \cdot y(\tau) d \tau$,

Dissipative systems with supply rate

$w= u \cdot y$

where $\cdot$ denotes the scalar product,

Dissipative systems satisfy the inequality:

$\frac{dV(x(t))}{dt} \le u(t) \cdot y(t)$

The physical interpretation is that $V(x)$ is the energy in the system, whereas $u \cdot y$ 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

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.