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Convergence to equilibrium distribution. Dirac fields coupled to a particle

T. V. Dudnikova

Abstract

For a system consisting of several Dirac fields and a particle, we study the Cauchy problem with random initial data. We assume that the initial measure has zero mean value, a finite mean charge density, a translation-invariant covariance and satisfies a mixing condition. The main result is the long-time convergence of distributions of the random solutions to a limit Gaussian measure.

Convergence to equilibrium distribution. Dirac fields coupled to a particle

Abstract

For a system consisting of several Dirac fields and a particle, we study the Cauchy problem with random initial data. We assume that the initial measure has zero mean value, a finite mean charge density, a translation-invariant covariance and satisfies a mixing condition. The main result is the long-time convergence of distributions of the random solutions to a limit Gaussian measure.

Paper Structure

This paper contains 15 sections, 25 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Conditions on the system. Cauchy problem
  3. Random solution. Convergence to equilibrium
  4. Main result
  5. Conditions on the initial measure
  6. Convergence to equilibrium for the free Dirac equation
  7. Statement of result
  8. Asymptotic behavior of $Y(t)$ as $t\to\infty$
  9. Proof of Theorem \ref{['tA']}
  10. Compactness of the measures $\mu_t$
  11. Convergence of correlation functions
  12. Convergence of characteristic functionals
  13. Conclusion
  14. Appendix A: Long-time asymptotics
  15. Appendix B: Proof of Theorem \ref{['h-eq']}

Key Result

Lemma 2

Let conditions A1-- A3 hold. Then (i) for every $Y_0 \in {\cal E}$, the Cauchy problem CP has a unique solution $Y(t)\in C(\mathbb{R}, {\cal E})$. (ii) For every $t\in \mathbb{R}$, the operator $U(t):Y_0\mapsto Y(t)$ is continuous on ${\cal E}$. Moreover, for every $R>R_\rho$ and $T>0$,

Theorems & Definitions (41)

  • Definition 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • proof
  • Corollary 7
  • Definition 8
  • Lemma 9
  • ...and 31 more