International Symposium on Mathematical Problems in Theoretical Physics

Abstract

Temporal organization of matter is a widespread phenomenon over a macroscopic world in far from thermodynamic equilibrium. A previous study on chemical instability I) implies that a simplest nontrivial model for a temporally organized system may be represented by a macro-scopic self-sustained oscillator Q obeying the equation of motion = (ira + ~)Q-BIQ]2Q , (13 ~,B > O. Consider a population of such oscillators QI' Q2""QN with various frequencies, and introduce interactions between every pair as follows. ~s = (ims + ~)Qs + Z VrsQ r-81Qs]2Qs , r~s (2) r,s = i, 2,.-.N. We found that it is possible to construct from (2) a soluble model for a community exhibiting mutual synchronization or self-entrainment above a certain threshold value of the coupling strength. Such a type of phase transition has been considered by Winfree 2) without resorting to specialized models but only phenomenologically. Our simplifying assumptions are: (1) Vrs = v/N independently of r and s, (II) a,~+~ but ~/fl, m s , v = finite, (hi) N~ i? s Let us put Qs=Ps e Owing to the assumption (II), the amplitude Ps may be fixed at /~7~. Thus we have only to consider the equation = v ~s mS + N Z sin(~ r-~s) (3) r As an illustration, we summarize the results obtained when the distribution of the native frequency is a Lorentzian with the peak at m 0 and the width ~. In this case the threshold condition is n ~ 21x/vl = 1 (43 For n