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Path-Dependent Energy Lagrangian for Irreversible Thermomechanical Systems

Huilong Ren

TL;DR

A Path-Dependent Energy Lagrangian (PDEL) is proposed to unify reversible mechanics and irreversible thermodynamics within a single variational principle. The action S combines the reversible Helmholtz free energy with a history integral of nonthermal channel powers, and an upper-limit (tangential) variation ensures that the same channel powers feed both the dissipative forces and the entropy/heat source, closing the first law without double counting. Channel-wise power balancing yields nonnegative entropy production and recovers standard dissipative models (e.g., Kelvin–Voigt, diffusion) while clarifying reversible thermo-mechanical cross terms; the framework also accommodates multiphysics extensions such as diffusion, electrochemistry, and hereditary effects. Overall, PDEL offers a compact, multiphysics variational framework that subsumes traditional formalisms (Rayleigh, Onsager, GENERIC) with minimal algebraic overhead and straightforward extension to complex cross-coupled thermomechanical problems.

Abstract

We present a minimal Path-Dependent Energy Lagrangian (PDEL) that generates, from a single action, the balance equations of mechanics and the entropy/heat equation for irreversible thermomechanical systems. The reversible part is the Helmholtz free energy, while irreversible effects enter through a history integral of channel powers. A single upper-limit/tangential variation rule makes the same instantaneous power appear as a dissipative force in the mechanical/internal-variable equations and as a positive source in the entropy/heat equation, closing the first law without double counting and guaranteeing nonnegative entropy production under mild monotonicity assumptions. PDEL preserves the classical Lagrangian mechanics while subsuming standard dissipative models (Kelvin--Voigt viscosity, diffusion) and their viscous heating, and clarifies the reversible character of thermo-mechanical cross terms. The formulation offers a compact alternative to Rayleigh/Onsager appendices and GENERIC/metriplectic brackets, with limited algebraic complexity and straightforward extension to multiphysics.

Path-Dependent Energy Lagrangian for Irreversible Thermomechanical Systems

TL;DR

A Path-Dependent Energy Lagrangian (PDEL) is proposed to unify reversible mechanics and irreversible thermodynamics within a single variational principle. The action S combines the reversible Helmholtz free energy with a history integral of nonthermal channel powers, and an upper-limit (tangential) variation ensures that the same channel powers feed both the dissipative forces and the entropy/heat source, closing the first law without double counting. Channel-wise power balancing yields nonnegative entropy production and recovers standard dissipative models (e.g., Kelvin–Voigt, diffusion) while clarifying reversible thermo-mechanical cross terms; the framework also accommodates multiphysics extensions such as diffusion, electrochemistry, and hereditary effects. Overall, PDEL offers a compact, multiphysics variational framework that subsumes traditional formalisms (Rayleigh, Onsager, GENERIC) with minimal algebraic overhead and straightforward extension to complex cross-coupled thermomechanical problems.

Abstract

We present a minimal Path-Dependent Energy Lagrangian (PDEL) that generates, from a single action, the balance equations of mechanics and the entropy/heat equation for irreversible thermomechanical systems. The reversible part is the Helmholtz free energy, while irreversible effects enter through a history integral of channel powers. A single upper-limit/tangential variation rule makes the same instantaneous power appear as a dissipative force in the mechanical/internal-variable equations and as a positive source in the entropy/heat equation, closing the first law without double counting and guaranteeing nonnegative entropy production under mild monotonicity assumptions. PDEL preserves the classical Lagrangian mechanics while subsuming standard dissipative models (Kelvin--Voigt viscosity, diffusion) and their viscous heating, and clarifies the reversible character of thermo-mechanical cross terms. The formulation offers a compact alternative to Rayleigh/Onsager appendices and GENERIC/metriplectic brackets, with limited algebraic complexity and straightforward extension to multiphysics.

Paper Structure

This paper contains 12 sections, 2 theorems, 24 equations, 2 figures.

Table of Contents

  1. Framework
  2. Action and upper-limit/tangential variations
  3. Conjugacy and cross derivatives
  4. Stationarity: balances and boundary conditions
  5. Temperature form and reversible cross terms
  6. Thermodynamic consistency
  7. Compact examples
  8. Compressible thermo-mechanics (ideal gas)
  9. Kelvin--Voigt viscous solid (PDEL realization)
  10. Diffusion channel
  11. Electrochemistry (battery half–cell)
  12. Relation to existing frameworks and outlook

Key Result

Lemma 1

Let $D = \operatorname{Div}_\mathbf{X} \mathbf{q}_0 - \sum_\alpha \Phi_\alpha$, with each $\Phi_\alpha=Y_\alpha:\dot Z_\alpha$. Under the upper-limit/tangential variation rule Eq.eq:upperlimit, the same $\Phi_\alpha$ appears as $+\Phi_\alpha$ in Eq.eq:entropy and as a force$+Y_\alpha$ in the $Z_\alp so there is no double counting and the first law closes.

Figures (2)

  • Figure 1: Classical two-law route versus the single PDEL route. In PDEL, a single action with an upper-limit/tangential variation rule generates both the mechanical and entropy/heat equations.
  • Figure 2: Channel-wise power balancing: the same instantaneous power enters the heat/entropy equation as a positive source and the mechanical/structural energy as negative power; conduction balances with boundary heat flux.

Theorems & Definitions (2)

  • Lemma 1: Power-balancing projection
  • Proposition 1: Cross terms produce no entropy