[Paper Review] Weakly nonlocal non-equilibrium thermodynamics - variational principles and Second Law

P. Ván|arXiv (Cornell University)|Feb 18, 2009

Thermoelastic and Magnetoelastic Phenomena97 references3 citations

TL;DR

This paper presents a unified, rigorous framework for weakly nonlocal non-equilibrium thermodynamics using variational principles and the Second Law. It derives evolution equations for internal variables, classical irreversible thermodynamics, and Korteweg fluids by applying Liu’s theorem to higher-order gradient-based constitutive spaces, ensuring thermodynamic consistency through constrained entropy inequalities and multiplier methods, with key results including the derivation of Schrödinger-Madelung fluid constitutive relations.

ABSTRACT

A general, uniform, rigorous and constructive thermodynamic approach to weakly nonlocal non-equilibrium thermodynamics is reviewed. A method is given to construct and restrict the evolution equations of physical theories according to the Second Law of thermodynamics and considering weakly nonlocal constitutive state spaces. The evolution equations of internal variables, the classical irreversible thermodynamics and Korteweg fluids are treated.

Motivation & Objective

• To develop a general, uniform, and constructive thermodynamic approach for weakly nonlocal systems beyond local equilibrium.
• To ensure thermodynamic consistency of evolution equations by applying the Second Law as a constrained inequality with derivative-based constraints.
• To unify variational and thermodynamic methods in deriving evolution equations for internal variables and continuum fields.
• To extend classical irreversible thermodynamics and Korteweg fluid models to weakly nonlocal forms using higher-order gradients.
• To derive explicit constitutive relations for complex systems like Schrödinger-Madelung fluids within a consistent thermodynamic framework.

Proposed method

• Applies Liu’s theorem to transform the Second Law into a constrained inequality involving evolution equations and their derivatives, using Lagrange multipliers to enforce thermodynamic consistency.
• Uses vectorial generalizations of Liu’s theorem to handle tensor-valued constraints arising in higher-order gradient theories.
• Treats evolution equations of internal variables as constraints in the entropy inequality, enabling systematic derivation of flux-force relations.
• Constructs weakly nonlocal constitutive state spaces by including spatial derivatives of state variables up to second order, enabling higher-order gradient extensions.
• Applies the method to derive the Ginzburg-Landau equation as a second-order weakly nonlocal extension of relaxation dynamics.
• Derives the pressure tensor and constitutive functions for one-component heat-conducting Korteweg fluids, ensuring compatibility with the Second Law.

Experimental results

Research questions

• RQ1 How can the Second Law of thermodynamics be consistently applied to weakly nonlocal systems with higher-order spatial derivatives?
• RQ2 What is the general variational and thermodynamic framework for deriving evolution equations in weakly nonlocal non-equilibrium thermodynamics?
• RQ3 How can classical irreversible thermodynamics be extended to weakly nonlocal systems using gradient-based constitutive spaces?
• RQ4 Can the Ginzburg-Landau equation be derived as a natural second-order weakly nonlocal extension of relaxation dynamics?
• RQ5 What are the thermodynamically consistent constitutive relations for Korteweg fluids and Schrödinger-Madelung fluids in the weakly nonlocal framework?

Key findings

• The paper establishes a general method to derive evolution equations from the Second Law by treating the evolution equations as constraints in a variational inequality, using Liu’s theorem to ensure thermodynamic consistency.
• A first-order weakly nonlocal theory of internal variables leads to relaxation-type ordinary differential equations, while second-order nonlocality yields the Ginzburg-Landau equation.
• Dual internal variables emerge naturally in second-order weakly nonlocal theories, unifying variational and thermodynamic derivations of evolution equations.
• Classical irreversible thermodynamics is naturally incorporated, with gradients of intensive variables serving as thermodynamic forces in flux-force relations.
• For one-component heat-conducting Korteweg fluids, nontrivial pressure tensor forms are derived that ensure compatibility with the Second Law.
• The method successfully derives the constitutive functions for Schrödinger-Madelung fluids, demonstrating its applicability to quantum hydrodynamic models.