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Large Deviations in Switching Diffusion: From Free Cumulants to Dynamical Transitions.

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Summary
This summary is machine-generated.

We analyzed particle diffusion with a switching diffusion coefficient D(t). Our renewal approach exactly calculated particle position moments, showing linear growth and validating predictions with high precision.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Stochastic Processes

Background:

  • Particle diffusion is fundamental in many physical systems.
  • Understanding anomalous diffusion requires analyzing time-dependent diffusion coefficients.
  • Previous models often assumed constant or smoothly varying diffusion.

Purpose of the Study:

  • To exactly compute moments of particle position for diffusion with a time-dependent, switching diffusion coefficient.
  • To analyze the long-time behavior of diffusion cumulants.
  • To investigate large deviations and dynamical transitions in such systems.

Main Methods:

  • Utilized a renewal approach for exact calculations.
  • Computed moments of particle position ⟨x^{2n}(t)⟩ for finite times.
  • Analyzed large deviations of particle position for specific diffusion coefficient distributions W(D).

Main Results:

  • Derived exact expressions for particle position moments ⟨x^{2n}(t)⟩.
  • Demonstrated linear growth of cumulants ⟨x^{2n}(t)⟩_c with time t for t≫1.
  • Identified rich behaviors and dynamical transitions in the large deviation rate function I(y=x/t).
  • Achieved high-precision numerical validation of analytical predictions (up to 10^{-2000}).

Conclusions:

  • The study provides an exact analytical framework for diffusion with switching coefficients.
  • Long-time diffusion exhibits behavior governed by the distribution of the diffusion coefficient.
  • The findings reveal complex dynamical transitions in anomalous diffusion phenomena.