Condensate-induced organization of the mass profile and emergent power laws in the Takayasu aggregation model

TL;DR

This paper investigates the formation and effects of condensates in the Takayasu aggregation model, revealing how they influence mass distribution and cause intermittent fluctuations in open systems.

Contribution

It provides an exact analysis of condensate occurrence, their impact on mass profiles, and the resulting power-law behaviors and fluctuations in the Takayasu model.

Findings

Condensates cause a power-law decay in mass distribution with an exponent of 5/3 near the condensate.

Farther sites follow the traditional Takayasu exponent 4/3, with a crossover region.

Open boundaries lead to intermittent mass fluctuations and divergence of flatness.

Abstract

Large-mass condensates, which coexist with a power-law-decaying distribution in the one-dimensional Takayasu model of mass aggregation with input, were recently found in numerical simulations. Here, we establish the occurrence of condensates by analyzing exact recursions for finite systems and further show that they have a strong effect on the properties of the system. In the steady state of a large but finite system, there is a single condensate, whose random movement through the system leads to a reorganization of the mass profile on a macroscopic scale. A scaling analysis of the mean mass and standard deviation at different distances from the condensate leads to the surprising conclusion that the mass distribution on a macroscopic number of sites around the condensate follows a power-law decay with an exponent 5/3, while farther-away sites show the customary Takayasu exponent 4/3, with a crossover in between. Finally, the exit of condensates from a system with open boundaries has a strong effect on the temporal fluctuations of the total mass in the steady state. Their departure is followed by a buildup of mass and subsequent departures, leading to strong intermittency, established through a divergence of the flatness as the scaled time approaches zero.