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Quantitative Boltzmann-Gibbs Principles via Orthogonal Polynomial Duality.

Mario Ayala1, Gioia Carinci1, Frank Redig1

  • 1Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.

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Summary
This summary is machine-generated.

This study introduces a quantitative generalization of the Boltzmann-Gibbs principle using orthogonal polynomials to analyze fluctuation fields in particle systems. It provides a systematic decomposition for understanding system dynamics, including independent random walkers.

Keywords:
Boltzmann–Gibbs principleDualityFluctuation fieldOrthogonal polynomials

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Area of Science:

  • Statistical Mechanics
  • Probability Theory
  • Mathematical Physics

Background:

  • The Boltzmann-Gibbs principle is fundamental in statistical mechanics for describing equilibrium states.
  • Understanding fluctuation fields in particle systems is crucial for analyzing complex dynamics.
  • Orthogonal polynomials offer a powerful mathematical framework for decomposition and analysis.

Purpose of the Study:

  • To develop a quantitative generalization of the Boltzmann-Gibbs principle.
  • To systematically decompose fluctuation fields in particle systems with duality.
  • To analyze non-stationary (local equilibrium) fluctuation fields.

Main Methods:

  • Utilizing orthogonal polynomials for fluctuation field decomposition.
  • Applying duality properties of particle systems.
  • Analyzing independent random walkers and interacting particle systems (e.g., symmetric exclusion process).

Main Results:

  • A systematic orthogonal decomposition of fluctuation fields is obtained.
  • The order of each term in the decomposition is quantifiable.
  • A quantitative generalization of the Boltzmann-Gibbs principle is established.
  • The framework is successfully applied to independent random walkers, including non-stationary cases.

Conclusions:

  • The developed method provides a rigorous approach to studying fluctuation fields in diverse particle systems.
  • This work extends the applicability of the Boltzmann-Gibbs principle to more general settings.
  • Further applications to other interacting particle systems with duality are possible under specific conditions.