TY - JOUR

T1 - Synchronization and fluctuations

T2 - Coupling a finite number of stochastic units

AU - Rosas, Alexandre

AU - Cisternas, Jaime

AU - Escaff, Daniel

AU - Pinto, Italo'Ivo Lima Dias

AU - Lindenberg, Katja

N1 - Funding Information:
A.R. acknowledges the financial support of CNPq (Grant No. 308344/2018-9). I.P. acknowledges the financial support of FACEPE (Grant No. BFP-0146-1.05/18). D.E. and J.C. thank funding from Fondecyt-Chile (Grant No. 1170669).
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/6

Y1 - 2020/6

N2 - It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization. And yet it is worth noting that finite-number effects should be seriously taken into account since, in general, the limits N→∞ (where N is the number of units) and t→∞ (where t is time) do not commute. Mean-field theory implements the particular choice first N→∞ and then t→∞. Here we analyze an ensemble of three-state coupled stochastic units, which has been widely studied in the thermodynamic limit. We formally address the finite-N problem by deducing a Fokker-Planck equation that describes the system. We compute the steady-state solution of this Fokker-Planck equation (that is, finite N but t→∞). We use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions.

AB - It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization. And yet it is worth noting that finite-number effects should be seriously taken into account since, in general, the limits N→∞ (where N is the number of units) and t→∞ (where t is time) do not commute. Mean-field theory implements the particular choice first N→∞ and then t→∞. Here we analyze an ensemble of three-state coupled stochastic units, which has been widely studied in the thermodynamic limit. We formally address the finite-N problem by deducing a Fokker-Planck equation that describes the system. We compute the steady-state solution of this Fokker-Planck equation (that is, finite N but t→∞). We use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions.

UR - http://www.scopus.com/inward/record.url?scp=85088352040&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.101.062140

DO - 10.1103/PhysRevE.101.062140

M3 - Article

C2 - 32688580

AN - SCOPUS:85088352040

SN - 2470-0045

VL - 101

JO - Physical Review E

JF - Physical Review E

IS - 6

M1 - 062140

ER -