In non-equilibrium thermodynamics

Non-equilibrium thermodynamics

Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...

, 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

Reversible dynamics

- Mathematics :In mathematics, a dynamical system is invertible if the forward evolution is one-to-one, not many-to-one; so that for every state there exists a well-defined reverse-time evolution operator....

 and irreversible dynamics (generated by energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 and entropy

Entropy

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

, 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


Here:

• denotes a set of variables
  Variable (mathematics)
  In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
  
   used to describe the state space
  State space
  In the theory of discrete dynamical systems, a state space is a directed graph where each possible state of a dynamical system is represented by a vertex, and there is a directed edge from a to b if and only if ƒ = b where the function f defines the dynamical system.State spaces are...
  
  . The vector can also contain variables depending on a continuous index like a temperature field. In general, is a function , where the set can contain both discrete and continuous indexes. Example: for a gas with nonuniform temperature, contained in a volume ()
• , are the system's total energy
  Energy
  In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
  
   and entropy
  Entropy
  Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
  
  . For purely discrete state variables, these are simply functions from to , for continuously indexed , they are functional
  Functional
  Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
  
  s
• , are the derivatives of and . In the discrete case, it is simply the gradient
  Gradient
  In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
  
  , for continuous variables, it is the functional derivative
  Functional derivative
  In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
  
   (a function )
• the Poisson matrix  is an antisymmetric matrix (possibly depending on the continuous indexes) describing the reversible dynamics of the system according to Hamiltonian mechanics
  Hamiltonian mechanics
  Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
  
  
• the friction matrix is a positive semidefinite
  Positive semidefinite
  In mathematics, positive semidefinite may refer to:* positive-semidefinite matrix* positive-semidefinite function...
  
   (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



which express the conservation of entropy under reversible dynamics and of energy under irreversible dynamics, respectively. The conditions on (antisymmetry and some others) express that the energy is reversibly conserved, and the condition on (positive semidefiniteness) express that the entropy is irreversibly non-decreasing.