A dissipative system is a thermodynamically
open system which is operating out of, and often far from,
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
in an environment with which it exchanges
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
and
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
. A
tornado
A tornado is a violently rotating column of air that is in contact with both the surface of the Earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. It is often referred to as a twister, whirlwind or cyclone, altho ...
may be thought of as a dissipative system. Dissipative systems stand in contrast to
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink ...
s.
A dissipative structure is a dissipative system that has a dynamical regime that is in some sense in a reproducible
steady state
In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
. 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
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to a ...
structure is characterized by the spontaneous appearance of symmetry breaking (
anisotropy
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
) and the formation of complex, sometimes
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
, structures where interacting particles exhibit long range correlations. Examples in everyday life include
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
,
turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
,
cyclone
In meteorology, a cyclone () is a large air mass that rotates around a strong center of low atmospheric pressure, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere as viewed from above (opposite to an anti ...
s,
hurricane
A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depend ...
s and
living organisms
In biology, an organism () is any life, living system that functions as an individual entity. All organisms are composed of cells (cell theory). Organisms are classified by taxonomy (biology), taxonomy into groups such as Multicellular o ...
. Less common examples include
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s,
Bénard cells
Benard or Bénard is a surname or given name. Notable people with the name include:
Surname
* Abraham-Joseph Bénard (1750–1822), French actor of the Comédie-Française
* Aimé Bénard (1873–1938), Canadian politician
* Alexander Benard, Ame ...
,
droplet cluster
Droplet cluster is a self-assembled levitating monolayer of microdroplets usually arranged into a hexagonally ordered structure over a locally heated thin (about 1 mm) layer of water. The droplet cluster is typologically similar to col ...
, and the
Belousov–Zhabotinsky reaction
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in ...
.
One way of mathematically modeling a dissipative system is given in the article on ''
wandering set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposit ...
s'': it involves the action of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
on a
measurable set
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
.
Dissipative systems can also be used as a tool to study economic systems and
complex systems
A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication s ...
. For example, a dissipative system involving
self-assembly
Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the ...
of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems.
The
Hopf decomposition
In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (''X'', μ) with respect to an invertible non-singular transformation ''T'':''X''→''X'', i.e. a transformation which with its ...
states that
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s can be decomposed into a conservative and a dissipative part; more precisely, it states that every
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
with a
non-singular transformation can be decomposed into an invariant
conservative set and an invariant dissipative set.
Dissipative structures in thermodynamics
Russian-Belgian physical chemist
Ilya Prigogine
Viscount Ilya Romanovich Prigogine (; russian: Илья́ Рома́нович Приго́жин; 28 May 2003) was a physical chemist and Nobel laureate noted for his work on dissipative structures, complex systems, and irreversibility.
Biogra ...
, who coined the term ''dissipative structure,'' received the
Nobel Prize in Chemistry
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then "M ...
in 1977 for his pioneering work on these structures, which have dynamical regimes that can be regarded as thermodynamic steady states, and sometimes at least can be described by suitable
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 extr ...
.
In his Nobel lecture,
Prigogine explains how thermodynamic systems far from equilibrium can have drastically different behavior from systems close to equilibrium. Near equilibrium, the ''local equilibrium'' hypothesis applies and typical thermodynamic quantities such as free energy and entropy can be defined locally. One can assume linear relations between the (generalized) flux and forces of the system. Two celebrated results from linear thermodynamics are the
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.
"Reciprocal relations" occur betw ...
and the principle of minimum
entropy production
Entropy production (or generation) is the amount of entropy which is produced in any irreversible processes such as heat and mass transfer processes including motion of bodies, heat exchange, fluid flow, substances expanding or mixing, anelastic d ...
. After efforts to extend such results to systems far from equilibrium, it was found that they do not hold in this regime and opposite results were obtained.
One way to rigorously analyze such systems is by studying the stability of the system far from equilibrium. Close to equilibrium, one can show the existence of a
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s se ...
which ensures that the entropy tends to a stable maximum. Fluctuations are damped in the neighborhood of the fixed point and a macroscopic description suffices. However, far from equilibrium stability is no longer a universal property and can be broken. In chemical systems, this occurs with the presence of
autocatalytic
A single chemical reaction is said to be autocatalytic if one of the reaction products is also a catalyst for the same or a coupled reaction.Steinfeld J.I., Francisco J.S. and Hase W.L. ''Chemical Kinetics and Dynamics'' (2nd ed., Prentice-Hall 199 ...
reactions, such as in the example of the
Brusselator
The Brusselator is a theoretical model for a type of autocatalytic reaction.
The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles.
It is a portmanteau of Brussels and oscillator.
...
. If the system is driven beyond a certain threshold, oscillations are no longer damped out, but may be amplified. Mathematically, this corresponds to a
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses ...
where increasing one of the parameters beyond a certain value leads to
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
behavior. If spatial effects are taken into account through a
reaction–diffusion equation, long-range correlations and spatially ordered patterns arise,
such as in the case of the
Belousov–Zhabotinsky reaction
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in ...
. Systems with such dynamic states of matter that arise as the result of irreversible processes are dissipative structures.
Recent research has seen reconsideration of Prigogine's ideas of dissipative structures in relation to biological systems.
Dissipative systems in control theory
Willems Willems is a patronymic surname of Dutch origin, equivalent to Williams. In 2008, it was the 6th most common surname in Belgium (18,604 peopleand in 2007 it was the 39th most common surname in the Netherlands (17,042 people
People with this surnam ...
first introduced the concept of dissipativity in systems theory to describe dynamical systems by input-output properties. Considering a dynamical system described by its state
, its input
and its output
, the input-output correlation is given a supply rate
. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function
such that
,
and
:
.
As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate
.
The physical interpretation is that
is the energy stored in the system, whereas
is the energy that is supplied to the system.
This notion has a strong connection with
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
, 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 The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number \gamma > 0, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair (A,B) is completely controllable, then a symmetric matrix ...
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 is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and any classical
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
, relies heavily on
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
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
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determine ...
which is a special case of a more general setting called the
Lindblad equation
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindb ...
that is the quantum equivalent of the classical
Liouville equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
. 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.
Recent research has seen the quantum extension
of
Jeremy England
Jeremy England is an American physicist who uses statistical physics arguments to explain the spontaneous Abiogenesis, emergence of life, and consequently, the Modern synthesis (20th century), modern synthesis of evolution. England terms this pr ...
's theory of dissipative adaptation
(which generalizes Prigogine's ideas of dissipative structures to far-from-equilibrium statistical mechanics, as stated above).
Applications on dissipative systems of dissipative structure concept
The framework of dissipative structures as a mechanism to understand the behavior of systems in constant interexchange of energy has been successfully applied on different science fields and applications, as in optics, population dynamics and growth and chemomechanical structures.
See also
*
Autocatalytic reactions and order creation
A single chemical reaction is said to be autocatalytic if one of the reaction products is also a catalyst for the same or a coupled reaction.Steinfeld J.I., Francisco J.S. and Hase W.L. ''Chemical Kinetics and Dynamics'' (2nd ed., Prentice-Hall 199 ...
*
Autopoiesis
The term autopoiesis () refers to a system capable of producing and maintaining itself by creating its own parts.
The term was introduced in the 1972 publication '' Autopoiesis and Cognition: The Realization of the Living'' by Chilean biologist ...
*
Autowave
Autowaves are self-supporting non-linear waves in active media (i.e. those that provide distributed energy sources). The term is generally used in processes where the waves carry relatively low energy, which is necessary for synchronization or ...
*
Conservation equation
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...
*
Complex system
A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication ...
*
Dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
*
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 extr ...
*
Information metabolism
Information metabolism, sometimes referred to as informational metabolism or energetic-informational metabolism, is a psychological theory of interaction between biological organisms and their environment, developed by Polish psychiatrist Antoni K ...
*
Loschmidt's paradox
Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or ', is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (al ...
*
Non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an ext ...
*
Relational order theories
Relationalism is any theoretical position that gives importance to the relational nature of things. For relationalism, things exist and function only as relational entities. Relationalism may be contrasted with relationism, which tends to emphasize ...
*
Self-organization
Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when suffi ...
*
Viable system theory Viable system theory (VST) concerns cybernetic processes in relation to the development/evolution of dynamical systems. They are considered to be living systems in the sense that they are complex and adaptive, can learn, and are capable of maintain ...
*
Vortex Engine
The concept of a vortex engine or atmospheric vortex engine (AVE), independently proposed by Norman Louat and Louis M. Michaud, aims to replace large physical chimneys with a vortex of air created by a shorter, less-expensive structure. The AVE i ...
Notes
{{Reflist
References
* B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative Systems Analysis and Control. Theory and Applications. Springer Verlag, London, 2nd Ed., 2007.
Davies, Paul ''The Cosmic Blueprint''Simon & Schuster, New York 1989 (abridged— 1500 words) (abstract— 170 words) — self-organized structures.
* Philipson, Schuster, ''Modeling by Nonlinear Differential Equations: Dissipative and Conservative Processes'', World Scientific Publishing Company 2009.
* Prigogine, Ilya
''Time, structure and fluctuations'' Nobel Lecture, 8 December 1977.
* J.C. Willems. Dissipative dynamical systems, part I: General theory; part II: Linear systems with quadratic supply rates. Archive for Rationale mechanics Analysis, vol.45, pp. 321–393, 1972.
External links
The dissipative systems modelThe Australian National University
Thermodynamic systems
Systems theory
Non-equilibrium thermodynamics