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

Thermodynamic equilibrium

In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

 in an environment with which it exchanges energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 and matter

Matter

Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

.

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

Dissipation

In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...

 structure is characterized by the spontaneous appearance of symmetry breaking (anisotropy

Anisotropy

Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...

) and the formation of complex, sometimes chaotic

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

, structures where interacting particles exhibit long range correlations. The term dissipative structure was coined by Russian-Belgian physical chemist Ilya Prigogine

Ilya Prigogine

Ilya, Viscount Prigogine was a Russian-born naturalized Belgian physical chemist and Nobel Laureate noted for his work on dissipative structures, complex systems, and irreversibility.-Biography :...

, who was awarded the Nobel Prize in Chemistry

Nobel Prize in Chemistry

The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...

 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

Extremal principles in non-equilibrium thermodynamics

Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics...

.

Simple examples include convection

Convection

Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

, cyclone

Cyclone

In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

s and hurricane

Tropical cyclone

A tropical cyclone is a storm system characterized by a large low-pressure center and numerous thunderstorms that produce strong winds and heavy rain. Tropical cyclones strengthen when water evaporated from the ocean is released as the saturated air rises, resulting in condensation of water vapor...

s. More complex examples include laser

Laser

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

s, Bénard cells, the Belousov–Zhabotinsky reaction, and living organisms

Life

Life is a characteristic that distinguishes objects that have signaling and self-sustaining processes from those that do not, either because such functions have ceased , or else because they lack such functions and are classified as inanimate...

.

One way of mathematically modeling a dissipative system is given in the article on wandering set

Wandering set

In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system...

s: it involves the action of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 on a measurable set

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

.

In control theory

In systems and control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

, 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(t)·y(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

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

, 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

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, and any classical dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

, relies heavily on Hamiltonian mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

 for which time is reversible

Time reversibility

Time reversibility is an attribute of some stochastic processes and some deterministic processes.If a stochastic process is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which...

, 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

Master equation

In physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...

 which is a special case of a more general setting called the Lindblad equation

Lindblad equation

In quantum mechanics, the Kossakowski–Lindblad equation or master equation in the Lindblad form is the most general type of markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix \rho that is trace preserving and completely positive for any initial...

 that is the quantum equivalent of the classical Liouville equation

Liouville's theorem (Hamiltonian)

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...

. 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

H-theorem

In Classical Statistical Mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process. H-theorem follows from considerations of Boltzmann's equation...

 and constructive role for time.

See also

• Non-equilibrium thermodynamics
  Non-equilibrium thermodynamics
  Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...
  
  
• Extremal principles in non-equilibrium thermodynamics
  Extremal principles in non-equilibrium thermodynamics
  Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics...
  
  
• Self-organization
  Self-organization
  Self-organization is the process where a structure or pattern appears in a system without a central authority or external element imposing it through planning...
  
  
• Autocatalytic reactions and order creation
• Dynamical system
  Dynamical system
  A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
  
  
• Autopoiesis
• Relational order theories
  Relational order theories
  A number of independent lines of research depict the universe, including the social organization of living creatures which is of particular interest to humans, as systems, or networks, of relationships....
  
  
• Loschmidt's paradox
  Loschmidt's paradox
  Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics...
  
  

External links

• The dissipative systems model The Australian National University