A simple example is the Bénard cells. More complex examples include lasers, Belousov-Zhabotinsky reaction, or even life itself.
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 yield a master equation which is a special case of a more general setting called the Lindblad equation.
In physics, dissipation embodies the concept of a dynamical system where important mechanical modes, such as waves or oscillations, lose energy over time, typically due to the action of friction or turbulence.
The precise nature of the effects depends on the nature of the wave: an atmospheric wave, for instance, may dissipate close to the surface due to friction with the land mass, and at higher levels due to radiative cooling.
A formal, mathematical definition of dissipation, as commonly used in the mathematical study of measure-preserving dynamical systems, is given in the article wandering set.
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