Variational principles and nonequilibrium thermodynamics

Abstract

<jats:p>Variational principles play a fundamental role in deriving the evolution equations of physics. They work well in the case of non-dissipative evolution, but for dissipative systems, the variational principles are not unique and not constructive. With the methods of modern nonequilibrium thermodynamics, one can derive evolution equations for dissipative phenomena and, surprisingly, in several cases, one can also reproduce the Euler–Lagrange form and symplectic structure of the evolution equations for non-dissipative processes. In this work, we examine some demonstrative examples and compare thermodynamic and variational techniques. Then, we argue that, instead of searching for variational principles for dissipative systems, there is another viable programme: the second law alone can be an effective tool to construct evolution equations for both dissipative and non-dissipative processes.</jats:p> <jats:p>This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.</jats:p>

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory

“…It is noted that the variational approach discussed in the present work is radically different from the one that is directly applied to thermodynamic variables and is somewhat criticized in [37].…”

Nonequilibrium thermodynamics: emergent and fundamental

“…In nonequilibrium thermodynamics, the deviation from equilibrium can be formulated without considering the detailed structure of the material. Additional fields, frequently called internal variables, is a convenient tool [5,9]. This approach proved to be very powerful in continuum theories, in particular when combined with weak nonlocality.…”

“…Dissipative processes tend to equilibrium; entropy is a kind of Lyapunov functional. This aspect is crucial in [1][2][3], important but less apparent in [4,5]. -Symmetric hyperbolic evolution.…”

“…Dissipation does not depend on the observer. This requirement is evident in the relativistic treatment of [4], and present also in [2,5,9]. Also, the preferred application areas influence modelling methods.…”

“…For example, for discrete systems, that is homogeneous or point-like material models, the space-time aspects are less important. These systems are treated in [1,10,11] and also partially in [3,5]. Quantum systems are discrete, too [1,11].…”