dissipative system

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. Examples in everyday life include convection, turbulent flow, cyclones, hurricanes and living organisms. Less common examples include lasers, Bénard cells, droplet cluster, and the Belousov–Zhabotinsky reaction.{{cite journal|last1=Li|first1=HP|title=Dissipative Belousov–Zhabotinsky reaction in unstable micropyretic synthesis|journal=Current Opinion in Chemical Engineering|date=February 2014|volume=3|pages=1–6|doi=10.1016/j.coche.2013.08.007|bibcode=2014COCE....3....1L }}

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.

Dissipative systems can also be used as a tool to study economic systems and complex systems.{{Cite book|title = The Unity of Science and Economics: A New Foundation of Economic Theory|last = Chen|first = Jing|publisher = Springer|year = 2015|url=https://www.springer.com/us/book/9781493934645}} For example, a dissipative system involving self-assembly of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems.{{cite journal|last1=Hubler|first1=Alfred|last2=Belkin|first2=Andrey|last3=Bezryadin|first3=Alexey|title=Noise induced phase transition between maximum entropy production structures and minimum entropy production structures?|journal=Complexity|date=2 January 2015|volume=20|issue=3|pages=8–11|doi=10.1002/cplx.21639|bibcode=2015Cmplx..20c...8H}}

The Hopf decomposition states that dynamical systems can be decomposed into a conservative and a dissipative part; more precisely, it states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant dissipative set.

Russian-Belgian physical chemist Ilya Prigogine, who coined the term dissipative structure, received the Nobel Prize in Chemistry 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.

In his Nobel lecture,{{cite journal|last1=Prigogine|first1=Ilya|title=Time, Structure and Fluctuations|url=https://www.nobelprize.org/nobel_prizes/chemistry/laureates/1977/prigogine-lecture.html|journal=Science|year=1978|volume=201|issue=4358|pages=777–785|doi=10.1126/science.201.4358.777|pmid=17738519|bibcode=1978Sci...201..777P |s2cid=9129799 }} 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 and the principle of minimum entropy production.{{cite journal|last1=Prigogine|first1=Ilya|title=Modération et transformations irréversibles des systèmes ouverts|journal=Bulletin de la Classe des Sciences, Académie Royale de Belgique|date=1945|volume=31|pages=600–606}} 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 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 reactions, such as in the example of the Brusselator. 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 where increasing one of the parameters beyond a certain value leads to limit cycle behavior. If spatial effects are taken into account through a reaction–diffusion equation, long-range correlations and spatially ordered patterns arise,{{cite journal|last1=Lemarchand|first1=H.|last2=Nicolis|first2=G.|title=Long range correlations and the onset of chemical instabilities|journal=Physica|date=1976|volume=82A|issue=4|pages=521–542|doi=10.1016/0378-4371(76)90079-0|bibcode=1976PhyA...82..521L}} such as in the case of the Belousov–Zhabotinsky reaction. 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.{{cite journal|last1=England|first1=Jeremy L.|title=Dissipative adaptation in driven self-assembly|journal=Nature Nanotechnology|date=4 November 2015|volume=10|issue=11|pages=919–923|doi=10.1038/NNANO.2015.250|pmid=26530021|bibcode=2015NatNa..10..919E}}

Willems first introduced the concept of dissipativity in systems theory{{cite journal |last1=Willems |first1=J.C. |title=Dissipative dynamical systems part 1: General theory |journal=Arch. Rational Mech. Anal. |date=1972 |volume=45 |issue=5 |page=321 |doi=10.1007/BF00276493 |bibcode=1972ArRMA..45..321W |hdl=10338.dmlcz/135639 |s2cid=123076101 |url=https://homes.esat.kuleuven.be/~sistawww/smc/jwillems/Articles/JournalArticles/1972.1.pdf }} to describe dynamical systems by input-output properties. Considering a dynamical system described by its state $x(t)$, its input $u(t)$ and its output $y(t)$, the input-output correlation is given a supply rate $w(u(t),y(t))$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $V(x(t))$ such that $V(0)=0$, $V(x(t))\backslash ge\; 0$ and

:$\backslash dot\{V\}(x(t))\; \backslash le\; w(u(t),y(t))$.{{cite book |last1=Arcak |first1=Murat |last2=Meissen |first2=Chris |last3=Packard |first3=Andrew |title=Networks of Dissipative Systems |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-29928-0 }}

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 $w(u(t),y(t))\; =\; u(t)^Ty(t)$.

The physical interpretation is that $V(x)$ is the energy stored in the system, whereas $w(u(t),y(t))$ 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{{clarify|reason=Is this the same as a "linear time-invariant system" as in the Wikipedia articles "LTI system theory"?|date=April 2015}}, 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{{clarify|reason=What is a positive real system?|date=April 2015}}.{{cite book|url=https://www.springer.com/978-1-84628-892-0|title=Process Control - The Passive Systems Approach| last1=Bao| first1=Jie| last2=Lee| first2=Peter L.| author-link2=Peter Lee (engineer)| publisher=Springer-Verlag London|year=2007|doi=10.1007/978-1-84628-893-7|isbn=978-1-84628-892-0}} Dissipative systems are still an active field of research in systems and control, due to their important applications.

{{main|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.

Recent research has seen the quantum extension{{cite journal|last1=Valente|first1=Daniel|last2=Brito|first2=Frederico|last3=Werlang|first3=Thiago|title=Quantum dissipative adaptation|journal=Communications Physics|date=19 January 2021|volume=4|issue=11|page=11 |doi=10.1038/s42005-020-00512-0 |arxiv=2111.08605 |bibcode=2021CmPhy...4...11V |doi-access=free}} of Jeremy England's theory of dissipative adaptation (which generalizes Prigogine's ideas of dissipative structures to far-from-equilibrium statistical mechanics, as stated above).

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,{{cite journal |last1=Lugiato |first1=L. A. |last2=Prati |first2=F. |last3=Gorodetsky |first3=M. L. |last4=Kippenberg |first4=T. J. |title=From the Lugiato–Lefever equation to microresonator-based soliton Kerr frequency combs |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20180113 |doi=10.1098/rsta.2018.0113|pmid=30420551 |arxiv=1811.10685 |bibcode=2018RSPTA.37680113L |s2cid=53289963 }}{{cite journal |last1=Andrade-Silva |first1=I. |last2=Bortolozzo |first2=U. |last3=Castillo-Pinto |first3=C. |last4=Clerc |first4=M. G. |last5=González-Cortés |first5=G. |last6=Residori |first6=S.|author6-link=Stefania Residori |last7=Wilson |first7=M. |title=Dissipative structures induced by photoisomerization in a dye-doped nematic liquid crystal layer |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170382 |doi=10.1098/rsta.2017.0382|pmid=30420545 |pmc=6232603 |bibcode=2018RSPTA.37670382A }} population dynamics and growth{{cite journal |last1=Zykov |first1=V. S. |title=Spiral wave initiation in excitable media |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170379 |doi=10.1098/rsta.2017.0379|pmid=30420544 |pmc=6232601 |bibcode=2018RSPTA.37670379Z |doi-access=free }}{{cite journal |last1=Tlidi |first1=M. |last2=Clerc |first2=M. G. |last3=Escaff |first3=D. |last4=Couteron |first4=P. |last5=Messaoudi |first5=M. |last6=Khaffou |first6=M. |last7=Makhoute |first7=A. |title=Observation and modelling of vegetation spirals and arcs in isotropic environmental conditions: dissipative structures in arid landscapes |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20180026 |doi=10.1098/rsta.2018.0026|pmid=30420548 |pmc=6232604 |bibcode=2018RSPTA.37680026T |doi-access=free }}{{cite journal |last1=Gunji |first1=Yukio-Pegio |last2=Murakami |first2=Hisashi |last3=Tomaru |first3=Takenori |last4=Basios |first4=Vasileios |title=Inverse Bayesian inference in swarming behaviour of soldier crabs |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170370 |doi=10.1098/rsta.2017.0370|pmid=30420541 |pmc=6232598 |bibcode=2018RSPTA.37670370G }} and chemomechanical structures.{{cite journal |last1=Bullara |first1=D. |last2=De Decker |first2=Y. |last3=Epstein |first3=I. R. |title=On the possibility of spontaneous chemomechanical oscillations in adsorptive porous media |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170374 |doi=10.1098/rsta.2017.0374|pmid=30420542 |pmc=6232597 |bibcode=2018RSPTA.37670374B }}{{cite journal |last1=Gandhi |first1=Punit |last2=Zelnik |first2=Yuval R. |last3=Knobloch |first3=Edgar |title=Spatially localized structures in the Gray–Scott model |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170375 |doi=10.1098/rsta.2017.0375|pmid=30420543 |pmc=6232600 |bibcode=2018RSPTA.37670375G |doi-access=free }}{{cite journal |last1=Kostet |first1=B. |last2=Tlidi |first2=M. |last3=Tabbert |first3=F. |last4=Frohoff-Hülsmann |first4=T. |last5=Gurevich |first5=S. V. |last6=Averlant |first6=E. |last7=Rojas |first7=R. |last8=Sonnino |first8=G. |last9=Panajotov |first9=K. |title=Stationary localized structures and the effect of the delayed feedback in the Brusselator model |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170385 |doi=10.1098/rsta.2017.0385|pmid=30420547 |arxiv=1810.05072 |bibcode=2018RSPTA.37670385K |s2cid=53289595 }}

{{Div col}}

• Autocatalytic reactions and order creation
• Autopoiesis
• Autowave
• Conservation equation
• Complex system
• Dynamical system
• Extremal principles in non-equilibrium thermodynamics
• Information metabolism
• Loschmidt's paradox
• Non-equilibrium thermodynamics
• Relational order theories
• Self-organization
• Viable system theory
• Vortex engine

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{{Reflist}}

• B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative Systems Analysis and Control. Theory and Applications. Springer Verlag, London, 2nd Ed., 2007.
• [https://web.archive.org/web/20070312032928/http://sciphilos.info/doc%20PAGES%20/docDaviesSelfOrgStru.html 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, [https://www.nobelprize.org/uploads/2018/06/prigogine-lecture.pdf 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.

• [http://press-files.anu.edu.au/downloads/press/p128271/pdf/part-ch11.pdf The dissipative systems model] The Australian National University

Category:Thermodynamic systems

Category:Systems theory

Category:Non-equilibrium thermodynamics