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Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...
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Related Experiment Video

Updated: May 16, 2025

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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On pattern formation in the thermodynamically-consistent variational Gray-Scott model.

Wenrui Hao1, Chun Liu2, Yiwei Wang3

  • 1Department of Mathematics, Pennsylvania State University, University Park, 16802, PA, United States.

Mathematical Biosciences
|May 2, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a four-species variational Gray-Scott model, enhancing pattern formation analysis. The model stabilizes classical patterns and reveals new oscillating and traveling-wave dynamics for small reaction rates.

Keywords:
Energetic variational approachPattern formationReaction-diffusion systemsTransient patternVariational Gray-Scott models

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Area of Science:

  • Chemical kinetics
  • Pattern formation
  • Computational modeling

Background:

  • The classical Gray-Scott model is a widely studied reaction-diffusion system known for generating complex spatial patterns.
  • Existing models often lack thermodynamic consistency and do not fully account for birth-death processes.
  • A need exists for a more complete and thermodynamically sound framework to study pattern formation.

Purpose of the Study:

  • To introduce and analyze a four-species variational Gray-Scott model that incorporates reverse reactions and a virtual species.
  • To investigate pattern formation in this enhanced model, particularly how it relates to the classical model.
  • To explore the stability and dynamics of steady states and emergent patterns within the variational framework.

Main Methods:

  • Development of a four-species variational Gray-Scott model, ensuring thermodynamic consistency.
  • Numerical exploration of pattern formation in one spatial dimension.
  • Analysis of the influence of a small parameter ϵ (related to reverse reaction rates) on pattern stability and dynamics.
  • Investigation of energy stability for uniform steady states.

Main Results:

  • Stationary patterns from the classical Gray-Scott model can be stabilized as transient states in the variational model for small ϵ.
  • The variational model exhibits novel oscillating and traveling-wave-like patterns for small ϵ, with persistence times of O(ϵ-1).
  • Energy stability analysis reveals that while two uniform steady states are stable, the model exhibits initial-condition-dependent pattern selection.

Conclusions:

  • The four-species variational Gray-Scott model provides a more physically complete and thermodynamically consistent framework for studying reaction-diffusion systems.
  • This enhanced model expands the observable dynamics beyond the classical model, including stable transient patterns and new wave phenomena.
  • The model's gradient flow dynamics demonstrate a selection mechanism influencing pattern emergence based on initial conditions.