PHYSICAL REVIEW E 93, 012143 (2016) Phase transitions and entropies for synchronizing oscillators Martin Bier, 1, 2 Bartosz Lisowski, 1, 3 and Ewa Gudowska-Nowak 1, 4 1 M. Smoluchowski Institute of Physics, Jagiellonian University, ul. Lojasiewicza 11, 30-348 Krak´ ow, Poland 2 Department of Physics, East Carolina University, Greenville, North Carolina 27858, USA 3 Unit of Pharmacoepidemiology and Pharmacoeconomics, Faculty of Pharmacy, Jagiellonian University Medical College, ul. Medyczna 9, 30-688 Krak´ ow, Poland 4 Mark Kac Center for Complex Systems Research and Malopolska Center of Biotechnology, Jagiellonian University, Gronostajowa 7A, 30-387 Krak´ ow, Poland (Received 6 October 2015; published 25 January 2016) We study a generic model of coupled oscillators. In the model there is competition between phase synchronization and diffusive effects. For a model with a finite number of states we derive how a phase transition occurs when the coupling parameter is varied. The phase transition is characterized by a symmetry breaking and a discontinuity in the first derivative of the order parameter. We quantitatively account for how the synchronized pulse is a low-entropy structure that facilitates the production of more entropy by the system as a whole. For a model with many states we apply a continuum approximation and derive a potential Burgers’ equation for a propagating pulse. No phase transition occurs in that case. However, positive entropy production by diffusive effects still exceeds negative entropy production by the shock formation. DOI: 10.1103/PhysRevE.93.012143 I. INTRODUCTION The biological world offers many instances of systems where identical oscillators synchronize their phases due to a coupling. Examples include large groups of flashing fireflies [1], pacemaker cells in the heart [2], ensembles of neurons [3], a large number of pedestrians on a bridge [4], and metabolic oscillations of yeast cells in a suspension [5]. In these kind of systems analytical treatment commonly becomes manageable only when the coupling is small and can be mathematically treated as a perturbation [5–7]. Going from incoherent behavior to synchronized oscillators often takes the form of a phase transition as the coupling parameter is increased in value [6]. In this article we study the cyclic, four-state system that is depicted in Fig. 1. We assume a large total population N in the system and p i (t ) = n i (t )/N , where n i (t ) represents the population in state i at time t . This means that p i (t ) can be associated with the probability to be in state i at time t . As the transitions are irreversible, the system is far from equilibrium. For the rates k i = k i →i +1 we take k i = k 0 exp[α(p i +1 − p i −1 )], (1) where α is a non-negative parameter that couples the proba- bilities (i.e., populations) in the different states. The competition between diffusion and synchronization in the system depicted in Fig. 1 can be understood as follows. For α = 0 there is no coupling and the stochasticity in the exponentially distributed transition times will result in a relaxation to p 1 = p 2 = p 3 = p 4 = 1/4 whatever the initial conditions are. However, if α> 0 and if we have p 1 = p 2 = 1 4 (1 + ε) and p 3 = p 4 = 1 4 (1 − ε) (see Fig. 2) and at the same time k 4 >k 2 (see Fig. 1), then the diffusive effects can be counteracted. In that case the speeded up rate out of state 4 and the slowed down rate into state 3 will make the pulse in states 1 and 2 keep its shape. How the coupling strength in Eq. (1) provides the counter- action to the diffusion can be seen as follows. Assume a pulse (high probability) concentrated in one state. If there is a more populated state ahead of the pulse as compared to behind, i.e., p i +1 >p i −1 , then the exponent in Eq. (1) is positive and the forward-moving-rate of the pulse is speeded up. The pulse will then swallow the state ahead. If, on the other hand, a less populated state is ahead and a more populated state is behind, i.e., p i +1 <p i −1 , then the forward rate of the pulse is slowed down and the population behind can merge with the pulse. In both cases there is a pull towards the pulse and the pulse acts like an absorber. The mechanism is somewhat mindful of the well-known Burgers’ shock [8]. Ultimately, in a steady state, diffusion and accumulation balance each other out and the pulse travels around the cycle like a soliton in a time of approximately 4/k 0 . A three-state version of the setup in Fig. 1 was numerically and analytically studied in Refs. [9–13]. In these works a mean- field approximation was applied and a Hopf bifurcation to synchronized behavior was found. It was furthermore analyzed how the system responds to disorder introduced on the level of the transition rates, demonstrating the rise of synchrony in a population of nonidentical units. Four-state cycles occur frequently in biological systems. They are seen, for instance, in the ligand-gated and voltage- gated ion channels that maintain homeostasis and facilitate signal transduction [14]. A four-state system similar to the one in Fig. 1 was derived and studied in Ref. [15]. Interestingly, it has been described how different states of an ion channel involve different shapes that imply different stress and strain on the surrounding membrane [16]. Channels that operate in close proximity to each other could thus chemo- mechanically couple their kinetics. Our coupling parameter, α, could then reflect the strength of this interaction. We will come back to this idea in Sec. VI. The setup of Fig. 1 gives rise to a three-dimensional system of coupled ODEs. A linear analysis [7] around the fixed point p 1 = p 2 = p 3 = p 4 = 1/4 leads to eigenvalues λ 1 =−2k 0 and λ 2,3 = 1 2 k 0 {(α − 2) ± i (α + 2)}. (2) 2470-0045/2016/93(1)/012143(9) 012143-1 ©2016 American Physical Society