Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNon-standard scaling law of fluctuations in finite-size systems of globally coupled oscillators
119
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Universal scaling laws form one of the central issues in physics. A non-standard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a non-standard scaling law with respect to system size in a synchronized state of globally coupled non-identical phase oscillators [Nishikawa et al., Chaos $boldsymbol{22}$, 013133 (2012)]. However, it is still unclear how widely this non-standard scaling law is observed. In the present paper, we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems, and we validate the conditions by numerical simulations of several different models.
rate research
Read More
We investigate a critical exponent related to synchronization transition in globally coupled nonidentical phase oscillators. The critical exponents of susceptibility, correlation time, and correlation size are significant quantities to characterize fluctuations in coupled oscillator systems of large but finite size and understand a universal property of synchronization. These exponents have been identified for the sinusoidal coupling but not fully studied for other coupling schemes. Herein, for a general coupling function including a negative second harmonic term in addition to the sinusoidal term, we numerically estimate the critical exponent of the correlation size, denoted by $ u_+$, in a synchronized regime of the system by employing a non-conventional statistical quantity. First, we confirm that the estimated value of $ u_+$ is approximately 5/2 for the sinusoidal coupling case, which is consistent with the well-known theoretical result. Second, we show that the value of $ u_+$ increases with an increase in the strength of the second harmonic term. Our result implies that the critical exponent characterizing synchronization transition largely depends on the coupling function.
We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D sim O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D sim O(1/N^a) with a certain constant a>0 in the coherent regime and D sim O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.
The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.
We consider scaling of the entanglement entropy across a topological quantum phase transition in one dimension. The change of the topology manifests itself in a sub-leading term, which scales as $L^{-1/alpha}$ with the size of the subsystem $L$, here $alpha$ is the R{e}nyi index. This term reveals the universal scaling function $h_alpha(L/xi)$, where $xi$ is the correlation length, which is sensitive to the topological index.
In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced by a modified hyperscaling relation. As a result, standard formulations of finite size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free energy cost Delta F of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, Delta F proportional to $L^theta$, with $theta$ the violation of hyperscaling critical exponent, and L the linear extension of the system. This modified behavior facilitates a number of new numerical approaches that can be used to locate critical points in random field systems from finite size simulation data. We test and confirm the new approaches on two random field systems in three dimensions, namely the random field Ising model, and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
