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Gradient and GENERIC time evolution towards reduced dynamics.

M Grmela1, V Klika2, M Pavelka3

  • 1École Polytechnique de Montréal, C.P.6079 suc. Centre-ville, Montréal, Québec, Canada H3C 3A7.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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Summary
This summary is machine-generated.

This study shows how reducing mesoscopic dynamical theories to equilibrium thermodynamics yields the fundamental thermodynamic relation. Entropy drives this process to fixed points, defining thermodynamic entropy as a function of state variables.

Keywords:
GENERICmultiscale thermodynamicspattern recognitionreductionvector fields

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Area of Science:

  • Thermodynamics
  • Statistical Mechanics
  • Non-equilibrium Systems

Background:

  • Mesoscopic dynamical theories describe systems with intermediate complexity.
  • Equilibrium thermodynamics provides a macroscopic description of systems at equilibrium.
  • Bridging these theories is crucial for understanding complex system behavior.

Purpose of the Study:

  • To demonstrate the reduction of mesoscopic dynamical theories to equilibrium thermodynamics.
  • To establish the fundamental thermodynamic relation from dynamical principles.
  • To extend this reduction framework to non-equilibrium systems.

Main Methods:

  • Following mesoscopic time evolution to its fixed points.
  • Utilizing entropy as the driving force for reaching these fixed points.
  • Extending the method to analyze time evolution of fluxes in reduced theories.

Main Results:

  • The thermodynamic entropy is derived as a function of state variables by evaluating entropy at fixed points.
  • The reduction process yields the fundamental thermodynamic relation.
  • A flux-entropy was identified as a potential related to entropy production in non-equilibrium reductions.

Conclusions:

  • The reduction of mesoscopic dynamics to equilibrium thermodynamics fundamentally establishes the thermodynamic relation.
  • The framework is extendable to non-equilibrium thermodynamics, linking dynamics to macroscopic relations.
  • This work provides insights into the fundamental aspects of non-equilibrium thermodynamics.