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 ex ...

, 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 and irreversible dynamics (generated by

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 hea ...

and

entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

, 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.

GENERIC equation

The GENERIC equation is usually written as :

\frac=L(x)\cdot\frac(x)+M(x)\cdot\frac(x).

Here: *

x

denotes a set of variables used to describe the

state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the t ...

. The vector

x

can also contain variables depending on a continuous index like a temperature field. In general,

x

is a function

S\rightarrow\mathbb R

, where the set

S

can contain both discrete and continuous indexes. Example:

x=(U,V,T(\vec r))

for a gas with nonuniform temperature, contained in a volume

\Sigma\subset\mathbb R^3

(

S=\\cup\Sigma

) *

E(x)

S(x)

are the system's total

and

. For purely discrete state variables, these are simply functions from

\mathbb R^n

\mathbb R

, for continuously indexed

x

, they are

functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...

s *

\delta E/\delta x

\delta S/\delta x

are the derivatives of

E

and

S

. In the discrete case, it is simply the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

, for continuous variables, it is the

functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...

(a function

S\rightarrow\mathbb R

) * the Poisson matrix

L(x)

is an antisymmetric matrix (possibly depending on the continuous indexes) describing the reversible dynamics of the system according to

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) ''momen ...

. The related

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...

fulfills the

Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the assoc ...

. * the friction matrix

M(x)

is a positive semidefinite (and hence symmetric) matrix describing the system's irreversible behaviour. 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 :

L(x)\cdot\frac(x)=0

M(x)\cdot\frac(x)=0

which express the conservation of entropy under reversible dynamics and of energy under irreversible dynamics, respectively. The conditions on

L

(antisymmetry and some others) express that the energy is reversibly conserved, and the condition on

M

(positive semidefiniteness) express that the entropy is irreversibly non-decreasing.

References

{{Reflist, 1 Non-equilibrium thermodynamics