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The well-known Noether theorem in Lagrangian and Hamiltonian mechanics associates symmetries in the evolution equations of a mechanical system with conserved quantities. In this work, we extend this classical idea to problems of non-equilibrium thermodynamics formulated within the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) framework. The geometric meaning of symmetry is reviewed in this formal setting and then utilized to identify possible conserved quantities and the conditions that guarantee their strict conservation. Examples are provided that demonstrate the validity of the proposed definition in the context of finite and infinite dimensional thermoelastic problems.

Classical mechanics, developed in the last few centuries by Galileo, Newton, Lagrange, Hamilton and many others, possesses a rich geometric structure, which was originally unraveled by Poincaré (see, e.g., [

Non-equilibrium thermodynamics lacks a universally accepted formalism, but it will undoubtedly gain from a similar geometric/qualitative approach, as the one developed for classical mechanics (see [

Since GENERIC generalizes Hamiltonian mechanics, it seems natural that the basic notion of symmetry and its associated conserved quantity (as a result of Noether’s theorem) carry on to this extended framework. While symmetry itself has already been addressed in the context of GENERIC ([

In the article, we will provide a definition of a momentum map that applies to both finite and infinite dimensional, non-equilibrium thermodynamical problems expressed in the GENERIC formalism. The main use of such a definition will be a theorem proving the preservation of these momentum maps under certain invariance conditions on the energy, that is, the appropriate statement of Noether’s theorem in the context of GENERIC. As expected, the proposed notion of first integral reduces to that of classical Hamiltonian mechanics for the purely reversible case.

While investigating the notion of conserved quantity, we will study the different roles played by the energy and the entropy in generating the reversible and irreversible dynamics of a given system and their relation with the possible symmetries of the evolution equations. We will show that symmetry transformations must lead only to reversible trajectories in the state space, remaining at the same time “orthogonal” to the irreversible dynamics, in a sense made precise below. This geometric insight is explored first in the context of finite dimensional thermodynamical systems and then extended to the more abstract setting of infinite dimensions. It must be noted that the kind of systems we envision are closed and that energy is thus preserved in them. This choice does not preclude the (non-equilibrium) transfer of energy among their subsystems.

To verify that the definition of a momentum map provided is correct, we will work out in detail two examples. Since non-equilibrium thermodynamics has a wide range of applications, including simple rheological models, field theories, cosmology, material modeling,

The GENERIC framework (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) provides a common mathematical structure for the evolution equations of many thermodynamical systems. As explained in the monograph [

For well-known thermodynamical models, GENERIC might be considered simply as a methodology for writing their evolution equations, which clearly identifies its reversible and irreversible parts. For less understood models, this formalism can be used to check the thermodynamic soundness of the equations, to rule out incorrect terms or even to suggest possible functional expressions for unknown ones. In the present article, we will employ GENERIC only for the first reason, and we will discuss examples whose thermodynamic correctness is not questioned.

We next summarize the basic ingredients of GENERIC in the context of finite dimensional thermodynamical problems. Let

The Poisson and friction matrices model, respectively, the reversible and irreversible mechanisms of the system dynamics. The matrix,

One advantage of the formulation (

Hamilton’s equations for conservative systems are similar to (

In this section, we explore the geometric characterization of symmetries within the structure afforded by GENERIC. By exploiting the properties of the operators,

Let

Furthermore, let us assume that the action, Φ, preserves the GENERIC structure, that is:

Extending the classical concept for Hamiltonian systems, we propose the following:

Let

In Hamiltonian mechanics, the first of these two conditions uniquely defines the momentum, because the matrix,

The identities (

The definition shows that the function,

In Equation (

The previous result is the appropriate statement of Noether’s theorem in the framework of GENERIC and clearly reveals the different roles played by the energy and the entropy and their relations with the symmetries of the thermodynamic system.

The first example we consider is a simple planar thermoelastic pendulum moving inside an isolated box, as depicted in

The state space for this problem is the finite dimensional set:

Thermoelastic pendulum in an insulated cavity.

The internal energy of the spring and the box are, respectively, the functions

Let

We identify the Lie algebra

To prove the existence of a momentum map associated with this action, we first verify that the maps,

Finally, we verify that the thermoelastic pendulum possesses a momentum map in the sense of Definition 1, which coincides with the classical notion of angular momentum for a purely mechanical pendulum. Let

To verify whether or not the angular momentum is a conserved quantity of the thermoelastic solution, we verify that the action, Φ, preserves the energy. A simple calculation shows that:

In the previous sections, we have discussed conserved quantities in thermodynamical systems with finite dimensional state spaces. The concept of momentum map and the corresponding Noether theorem are extended next to cover systems with infinite dimensional state spaces. For that, the GENERIC formalism is first generalized, and later, the most general definition of a momentum map and Noether’s theorem are stated and proven, respectively.

Let

A Poisson bracket is an operation mapping two functionals

A dissipative bracket is another bilinear operation taking again two functionals,

Additionally, in the finite dimensional case, the time evolution of the thermodynamic system requires for its mathematical description a total energy,

The GENERIC framework establishes that every thermodynamically sound model must have evolution equations of the form (

Without any further conditions on the mathematical structure of the brackets, it is simple to prove that such thermodynamical systems possess the following property:

We generalize the notions of symmetry and momentum maps introduced before for the finite dimensional case. Using the same notation as above, we consider a Lie group,

To define the concept of momentum in infinite dimensional spaces, we will require some additional notation. If

We can now propose the proper extension of Definition 1 to infinite dimensional problems:

Let

Similarly, the generalization of Theorem 2 is provided by the following:

In Equation (

The second example we study is the infinite dimensional problem of finite strain thermoelasticity. A different approach to this problem in the context of GENERIC has been presented in [

A thermoelastic body,

The thermoelastic behavior of the body is described by means of the internal energy density

The configuration space of the thermoelastic body is defined as:

The time evolution of the state variables

The complete description of the evolution problem of thermoelasticity requires initial conditions for the state variables at every point

To study momentum maps on a thermoelastic body within the context of GENERIC, we first postulate the Poisson and dissipative operators and then verify that they give rise to the initial value problem previously described. For any two functionals,

The functional derivatives of the total energy and entropy can be calculated in a straightforward manner to be:

The symbol,

To study the symmetries of the coupled thermoelastic problem, we consider two different actions on the state space,

As a first step, we calculate the functional derivatives of the composed map: for every functional,

The scalar function:

The second relevant action on the state space is Ψ, defined as the action of the special orthogonal group, SO(3):

The analysis of the rotational action (

Since the dissipative bracket only depends on the functional derivatives with respect to the specific entropy and the action, Ψ, preserves them, the same argument employed in Equation (

To conclude that

Finally, to verify that the angular momentum is a conserved quantity of the thermoelastic solution, it suffices, according to Theorem 4, to verify the invariance of the energy with respect to Ψ. Using, again, the property

We have proposed a definition of the concept of momentum associated with any symmetry on a thermodynamical system expressed in the GENERIC formalism. This mathematical description generalizes the commonly known expressions of conserved quantities in finite or infinite dimensional Hamiltonian systems. Associated with the proposed definition, a version of Noether’s theorem is provided, which directly applies to the momentum maps, as defined.

The proposed definition of momentum map reduces to the classical one for purely mechanical systems and reveals the different roles that the reversible and irreversible dynamics play in the identification of symmetries and the specific expression for the conserved quantities. As is the case in Hamiltonian mechanics, we showed that the momentum map serves as a functional that drives the symmetry orbits. For general thermodynamical systems, we identified that the symmetry dynamics must be purely reversible, meaning that the irreversible part of the GENERIC dynamics should be “orthogonal” to the flow generated by the momentum map.

The second important result of this article is a generalization of Noether’s theorem to non-equilibrium thermodynamical systems. The statement relies on the proposed definition of momentum maps and proves that such quantities are preserved by the thermodynamic solution if the symmetries preserve the total energy.

To verify the validity of the proposed definitions, we have studied a finite and an infinite dimensional problem. The first one is a simple example of a thermoelastic spring moving within an insulated box, which possesses rotational symmetry about a fixed axis. The corresponding angular momentum is shown to be described by the proposed definition and the associated conservation law, as well. In the second example, the infinite dimensional problem of nonlinear, finite strain thermoelasticity is studied. The corresponding equations are cast in the GENERIC form first. Then, as in the former example, the symmetries of the equations are identified, which turn out to be the full set of spatial translations and rotations. The linear and angular momentum are identified as the associated maps and their conservation proven.

Financial support for this work has been provided by grant DPI2012-36429 from the Spanish Ministry of Economy and Competitiveness.

The authors declare no conflict of interest.