Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects

Physical Review. E
Álvaro CorralFrancesc Font-Clos

Abstract

We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant reflecting boundary correspond to percolating trees, and those that do not correspond to nonpercolating trees. In fact, both systems display a second-order phase transition between "conducting" and "insulating" phases, controlled by the drift velocity in the Brownian system. In the limit of large system size, we obtain exact expressions for the Laplace transforms of the probability distributions and their first and second moments. These quantities are also shown to obey finite-size scaling laws.

References

Nov 27, 1995·Physical Review Letters·S ZapperiH E Stanley
Dec 31, 2005·Physical Review Letters·Robert Botet, Marek Płoszajczak
Jan 19, 2013·Scientific Reports·Jiangbo PuQingming Luo
May 15, 2015·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Rosalba Garcia-MillanÁlvaro Corral
Oct 16, 2016·Physical Review. E·Francesc Font-Clos, Nicholas R Moloney
Jan 6, 2017·Scientific Reports·Isabel Serra, Álvaro Corral

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Citations

Jan 23, 2020·Physical Review. E·Víctor Navas-PortellaÁlvaro Corral
Dec 9, 2020·Entropy·Álvaro Corral, Montserrat García Del Muro
Mar 20, 2021·Physical Review. E·Álvaro Corral

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