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| In [[non-equilibrium thermodynamics]], '''GENERIC''' is an acronym for '''General Equation for Non-Equilibrium Reversible-Irreversible Coupling'''. It is the general form of dynamic equation for a system with both [[reversible dynamics|reversible]] and [[irreversible dynamics]] (generated by [[energy]] and [[entropy]], respectively). GENERIC formalism is the theory built around the GENERIC equation, which has been proposed in its final form in 1997 by Miroslav Grmela and Hans Christian Öttinger.
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| <ref>
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| {{cite journal
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| |author=M. Grmela and H.C Öttinger
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| |title=Dynamics and thermodynamics of complex fluids. I. Development of a general formalism
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| |journal=Phys. Rev. E
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| |volume=56
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| |pages=6620-6632
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| |year=1997
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| |doi=10.1103/PhysRevE.56.6620}}
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| </ref><ref>
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| {{cite journal
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| |author=H.C Öttinger and M. Grmela
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| |title=Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism
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| |journal=Phys. Rev. E
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| |volume=56
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| |pages=6633-6655
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| |year=1997
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| |doi=10.1103/PhysRevE.56.6633}}
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| </ref><ref>
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| {{cite book
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| |author=H.C Öttinger
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| |title=Beyond Equilibrium Thermodynamics
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| |publisher=Wiley, Hoboken
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| |year=2004}}
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| </ref>
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| ==GENERIC equation==
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| The GENERIC equation is usually written as
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| :<math>\frac{dx}{dt}=L(x)\cdot\frac{\delta E}{\delta x}(x)+M(x)\cdot\frac{\delta S}{\delta x}(x).</math>
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| Here:
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| * <math>x</math> denotes a set of [[Variable (mathematics)|variables]] used to describe the [[state space]]. The vector <math>x</math> can also contain variables depending on a continuous index like a temperature field. In general, <math>x</math> is a function <math>S\rightarrow\mathbb R</math>, where the set <math>S</math> can contain both discrete and continuous indexes. Example: <math>x=(U,V,T(\vec r))</math> for a gas with nonuniform temperature, contained in a volume <math>\Sigma\subset\mathbb R^3</math> (<math>S=\{1,2\}\cup\Sigma</math>)
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| * <math>E(x)</math>, <math>S(x)</math> are the system's total [[energy]] and [[entropy]]. For purely discrete state variables, these are simply functions from <math>\mathbb R^n</math> to <math>\mathbb R</math>, for continuously indexed <math>x</math>, they are [[functional (mathematics)|functional]]s
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| * <math>\delta E/\delta x</math>, <math>\delta S/\delta x</math> are the derivatives of <math>E</math> and <math>S</math>. In the discrete case, it is simply the [[gradient]], for continuous variables, it is the [[functional derivative]] (a function <math>S\rightarrow\mathbb R</math>)
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| * the [[Poisson matrix]] <math>L(x)</math> is an [[antisymmetric matrix]] (possibly depending on the continuous indexes) describing the reversible dynamics of the system according to [[Hamiltonian mechanics]]. The related [[Poisson bracket]] fulfills the [[Jacobi identity]].<ref>
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| {{cite journal
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| |author=M. Kröger and M. Hütter
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| |title=Automated symbolic calculations in nonequilibrium thermodynamics
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| |journal=Comput. Phys. Commun.
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| |volume=181
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| |pages=2149–2157
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| |year=2010
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| |doi=10.1016/j.cpc.2010.07.050
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| |bibcode = 2010CoPhC.181.2149K }}</ref>
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| * the '''friction matrix''' <math>M(x)</math> is a [[positive semidefinite]] (and hence symmetric) matrix describing the system's irreversible behaviour.
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| In addition to the above equation and the properties of its constituents, systems that ought to be properly described by the GENERIC formalism are required to fulfill the [[degeneracy conditions]]
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| :<math>L(x)\cdot\frac{\delta S}{\delta x}(x)=0</math>
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| :<math>M(x)\cdot\frac{\delta E}{\delta x}(x)=0</math>
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| which express the conservation of entropy under reversible dynamics and of energy under irreversible dynamics, respectively. The conditions on <math>L</math> (antisymmetry and some others) express that the energy is reversibly conserved, and the condition on <math>M</math> (positive semidefiniteness) express that the entropy is irreversibly non-decreasing.
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| ==References==
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| {{Reflist|1}}
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| [[Category:Non-equilibrium thermodynamics]]
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