A dissipative system (or dissipative structure) is an open system which is operating far from thermodynamic equilibrium within an environment that exchanges energy, matter or entropy. A dissipative system is characterized by the spontaneous appearance of a complex, sometimes chaotic, structure. The term dissipative structures was coined by Ilya Prigogine. It is also called steady-state open system and nonequilibrium open system. An open system is one whose border is permeable to energy (or mass) as distinct from a closed system in which the border is not permeable. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... Matter is commonly defined as the substance of which physical objects are composed. ... The Thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... Ilya Prigogine (January 25, 1917 – May 28, 2003) was a Belgian physicist and chemist noted for his work on dissipative structures, complex systems, and irreversibility. ...

A simple example is the Bénard cells. More complex examples include lasers, Belousov-Zhabotinsky reaction, or even life itself. Bénard cells are obtained in a simple experiment that Bénard, a French physicist, conducted in 1900. ... Lasers range in size from microscopic diode lasers (top) with numerous applications, to football field sized neodymium glass lasers (bottom) used for inertial confinement fusion, nuclear weapons research and other physics experiments. ... A Belousov-Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that result in the establishment of a nonlinear chemical oscillator. ... Look up life and living in Wiktionary, the free dictionary. ...

A formal, mathematical definition of a dissipative system as the action of a group on a measurable set is given in the article on wandering sets. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a wandering set is a concept in dynamical systems and ergodic theory, formalizing a certain idea of movement and mixing in such systems. ...

Quantum dissipative systems

As quantum mechanics relies heavily on Hamiltonian mechanics, it is not intrinsically able to describe dissipative systems. 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. A simple introduction to this subject is provided in Basics of quantum mechanics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... A master equation (also known as the Chapman-Kolmogorov equation in probability) is a phenomenological first order differential equation describing the time-evolution of the probability of a system to occupy each one of a discrete set of states: where Pk is the probability for the system to be in... The Lindblad equation or master equation in the Lindblad form is the most general type of master equation allowed by Quantum mechanics to describe non-unitary (dissipative) evolution of the density matrix (such as ensuring normalisation and hermiticity of ). It reads: where is the density matrix, is the hamiltonian part...

See also

• Self-organization
• Autopoiesis
• Dynamical systems and chaos theory
• Ilya Prigogine