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Hitting probability for anomalous diffusion processes.

Satya N Majumdar1, Alberto Rosso, Andrea Zoia

  • 1CNRS-Université Paris-Sud, LPTMS, UMR8626-Bât. 100, 91405 Orsay Cedex, France.

Physical Review Letters
|April 7, 2010
PubMed
Summary
This summary is machine-generated.

We found universal scaling laws for hitting probabilities in stochastic processes. This probability, describing a process reaching an upper boundary before a lower one, scales with process exponents, verified in disordered systems.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Dynamical Systems

Background:

  • Understanding the behavior of stochastic processes within confined domains is crucial in various scientific fields.
  • The hitting probability, a key metric, quantifies the likelihood of a process reaching a boundary first.

Purpose of the Study:

  • To determine the universal features of the hitting probability Q(x,L) for generic stochastic processes in a bounded interval [0, L].
  • To establish a scaling relationship for hitting probability in self-affine processes and verify it with exact calculations and numerical simulations.

Main Methods:

  • Analytical derivation of scaling laws for the hitting probability Q(x,L) for generic self-affine processes.
  • Exact calculations for specific cases, including particle diffusion in disordered potentials.
  • Numerical simulations to support the derived analytical results.

Main Results:

  • Identified a universal scaling law for hitting probability: Q(z) approximately z^{phi}, where z = x/L.
  • The scaling exponent phi is determined by the process's persistence exponent (theta) and Hurst exponent (H), specifically phi = theta/H.
  • The derived scaling law was confirmed through exact calculations and numerical simulations.

Conclusions:

  • The study reveals universal scaling properties of hitting probabilities in stochastic processes, independent of specific details beyond self-affinity.
  • The findings provide a fundamental understanding of boundary-hitting phenomena in complex systems, applicable to areas like disordered systems and diffusion.