Acceleration effect caused by the Onsager reaction term in a frustrated coupled oscillator system

The Onsager reaction term is one of the most important notions for frustrated random systems. The role of the Onsager reaction term, however, has not been properly elucidated for non-equilibrium random systems with large degrees of freedoms. By analyzing oscillator associative memory models involving two types of diluted couplings, we make clear that the acceleration (deceleration) effect is caused by the Onsager reaction term. Both a symmetric diluted system and an asymmetric diluted system are made to have the same order parameter equations except for the Onsager reaction term. From this, the term is found to lead to a difference in the rotation speed of the oscillators. This non-trivial phenomenon definitely shows the contribution of the Onsager reaction term to the acceleration (deceleration) effect. PACS numbers: 87.10.+e, 05.90.+m, 05.45.-a, 89.70.+c Typeset using REVTEX 1 In coupled oscillator systems, mutual interaction accelerates (decelerates) the rotation speed of all of the oscillators, and this is called the “acceleration (deceleration) effect”. The acceleration (deceleration) effect is one of the most important notions for coupled oscillator systems. Many researchers have energetically investigated the acceleration (deceleration) effect in coupled oscillator systems with few elements [1,2]. In particular, Mizuno and Kurata [2] theoretically showed the mechanism for the acceleration (deceleration) effect in the case of the Curved Isochron Clock (CIC) with weak diffusional coupling. An Isochron is defined as a set of initial states mapped on itself with the same period. The Radial Isochron Clock (RIC) has a straight Isochron with the same direction as a radial one. CIC, on the other hand, consists of a curved Isochron. Accordingly, the CIC features a difference between the Isochron and the radial direction on its orbit, and this difference causes the acceleration (deceleration) effect in weakly diffusional coupled CICs. This is the essential mechanism for the acceleration (deceleration) effect in the general case of weakly diffusional coupled oscillators. We discuss the essential mechanism for the acceleration (deceleration) effect in a large population of frustrated coupled oscillators by analyzing the following simplified model, dφi dt = ωi + N ∑ j 6=i Jij sin(φj − φi + βij + β0), (1) which is famous as a typical model of coupled oscillator systems [4,6]. We can exactly derive Eq. 1 from weakly coupled CICs by using the multi-scale perturbation method [4,5,2]. Here, N is the total number of oscillators, and φi is the phase of the i−th oscillator. ωi stands for the natural frequency assumed to be randomly distributed over the whole population with a density denoted by g(ω). We do not restrict g(ω) to a special case, e.g., a symmetric distribution with average 0. The theory presented below can treat systems with a more general asymmetric distribution g(ω). Jij and βij +β0 denote the amplitude of a coupling and its delay, respectively. In particular, β0 represents a uniform bias, which can be interpreted as the minimum time required for the propagation of information via the coupling. In the context of coupled CICs, β0 is derived from the difference between the Isochron and the radial direction on its orbit [2]. Accordingly, a system with a uniform phase bias, namely Eq. 1, is equivalent to a large population of weakly coupled CICs. If β0 = 0, Eq. 1 consists of weakly coupled RICs. In this paper, we have selected the following generalized Hebb learning rule with diluted couplings [7] to determine Jij and βij , Kij = Jij exp(iβij) = cij cN p