TY - JOUR

T1 - Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects

AU - Corral, Álvaro

AU - Garcia-Millan, Rosalba

AU - Moloney, Nicholas R.

AU - Font-Clos, Francesc

PY - 2018/6/29

Y1 - 2018/6/29

N2 - © 2018 American Physical Society. 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.

AB - © 2018 American Physical Society. 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.

U2 - 10.1103/PhysRevE.97.062156

DO - 10.1103/PhysRevE.97.062156

M3 - Article

VL - 97

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 6

M1 - 062156

ER -