Order-by-disorder in classical oscillator systems

Abstract

We consider classical nonlinear oscillators on hexagonal lattices. When the coupling between the elements is repulsive, we observe coexisting states, each one with its own basin of attraction. These states differ by their degree of synchronization and by patterns of phase-locked motion. When disorder is introduced into the system by additive or multiplicative Gaussian noise, we observe a non-monotonic dependence of the degree of order in the system as a function of the noise intensity: intervals of noise intensity with low synchronization between the oscillators alternate with intervals where more oscillators are synchronized. In the latter case, noise induces a higher degree of order in the sense of a larger number of nearly coinciding phases. This order-by-disorder effect is reminiscent to the analogous phenomenon known from spin systems. Surprisingly, this non-monotonic evolution of the degree of order is found not only for a single interval of intermediate noise strength, but repeatedly as a function of increasing noise intensity. We observe noise-driven migration of oscillator phases in a rough potential landscape.

Physical Aging of Classical Oscillators

“…1. The intermediate larger values of the potential indicate an unstable oscillatory state (unstable in the deterministic limit) for which we found numerical indications in [16]. After passing such an unstable state after one or more cycles in its vicinity, the system approaches another metastable state that was stable without noise.…”

“…Without noise, already for a system as small as a 4 × 4 hexagonal lattice, we observe a multitude of coexisting attractors [16]. The attractors correspond to fixed-point solutions, frequency-synchronized solutions, or quasiperiodic solutions.…”

“…We use additive noise to represent stochastic fluctuations, while the volume is finite and can be actually quite small. Based on our former results [16], we know that we have multiple inherent time scales in this kind of rotators. Moreover, in [16], we found indications for hierarchies in the potential's barriers of a similar kind as they are implemented in the trap model and the random energy model to explain aging in spin glasses [17].…”

“…Based on our former results [16], we know that we have multiple inherent time scales in this kind of rotators. Moreover, in [16], we found indications for hierarchies in the potential's barriers of a similar kind as they are implemented in the trap model and the random energy model to explain aging in spin glasses [17]. Therefore, based on these previous results, we expected to observe aging, as from an abstract point of view, the underlying mechanism is the same as in spin glasses.…”

“…=ð1 − λ min =N Þ (λ min denoting the minimal eigenvalue of the adjacency matrix and N the number of nearest neighbors) with multistable solutions, differing by the patterns of phase-locked motion, which were studied in more detail in [16]. Negative κ in combination with the hexagonal coupling pattern turns all bonds into frustrated bonds [19].…”