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Observation and Analysis of Blinking Surface-enhanced Raman Scattering

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Published on: January 11, 2018

Top rank statistics for Brownian reshuffling.

Zdzislaw Burda¹, Mario Kieburg²

¹AGH University of Krakow, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Kraków, Poland.

Physical Review. E

|August 19, 2025

Summary

This summary is machine-generated.

We introduce the overlap ratio (Ω(t)) to study top-ranked particles in Brownian motion. This new metric quantifies the stability of top rankings and offers a simple approximation for large systems.

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Area of Science:

Statistical physics
Stochastic processes
Dynamical systems

Background:

Understanding particle dynamics and ranking stability is crucial in various scientific fields.
Traditional methods often require tracking all particles, which can be computationally intensive.

Purpose of the Study:

To introduce and analyze a new observable, the overlap ratio (Ω(t)), for studying the dynamical aspects of top-ranked particles.
To derive an analytical formula for the average overlap ratio in a specific Brownian motion model.
To investigate the universality of the overlap ratio in different stochastic systems.

Main Methods:

Derivation of an analytical formula for the average overlap ratio in the stationary state.
Analysis of a system of N particles undergoing Brownian motion with a reflecting wall and drift.
Numerical studies of various dynamical systems to observe overlap ratio behavior.

Main Results:

An analytical formula for the average overlap ratio was derived for a system of N particles.
For large N, the overlap ratio simplifies to 〈Ω(t)〉=erfc(asqrt[t]), a highly accurate approximation even for moderate top-n list sizes.
The overlap ratio demonstrated universal behavior across diverse dynamical systems.

Conclusions:

The overlap ratio is an effective and easily measurable observable for assessing the stability of top rankings in dynamical systems.
The observed universality suggests the overlap ratio can be a powerful tool for analyzing a wide range of one-dimensional stochastic processes.