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Register a new userExact correlations in the nonequilibrium stationary state of the noisy Kuramoto model
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We obtain exact results on autocorrelation of the order parameter in the nonequilibrium stationary state of a paradigmatic model of spontaneous collective synchronization, the Kuramoto model of coupled oscillators, evolving in presence of Gaussian, white noise. The method relies on an exact mapping of the stationary-state dynamics of the model in the thermodynamic limit to the noisy dynamics of a single, non-uniform oscillator, and allows to obtain besides the Kuramoto model the autocorrelation in the equilibrium stationary state of a related model of long-range interactions, the Brownian mean-field model. Both the models show a phase transition between a synchronized and an incoherent phase at a critical value of the noise strength. Our results indicate that in the two phases as well as at the critical point, the autocorrelation for both the model decays as an exponential with a rate that increases continuously with the noise strength.
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Synchronization is a ubiquitous phenomenon occurring in social, biological, and technological systems when the internal rhythms of their constituents are adapted to be in unison as a result of their coupling. This natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research in recent decades. The Kuramoto model constitutes the most studied and paradigmatic framework in which to study synchronization. In particular, it shows how synchronization appears as a phase transition from a dynamically disordered state at some critical value for the coupling strength between the interacting units. The critical properties of the synchronization transition of this model have been widely studied and many variants of its formulations have been considered to address different physical realizations. However, the Kuramoto model has been studied only within the domain of classical dynamics, thus neglecting its applications for the study of quantum synchronization phenomena. Based on a system-bath approach and within the Feynman path-integral formalism, we derive equations for the Kuramoto model by taking into account the first quantum fluctuations. We also analyze its critical properties, the main result being the derivation of the value for the synchronization onset. This critical coupling increases its value as quantumness increases, as a consequence of the possibility of tunneling that quantum fluctuations provide.
We study two interacting identical run and tumble particles (RTPs) in one dimension. Each particle is driven by a telegraphic noise, and in some cases, also subjected to a thermal white noise with a corresponding diffusion constant $D$. We are interested in the stationary bound state formed by the two RTPs in the presence of a mutual attractive interaction. The distribution of the relative coordinate $y$ indeed reaches a steady state that we characterize in terms of the solution of a second-order differential equation. We obtain the explicit formula for the stationary probability $P(y)$ of $y$ for two examples of interaction potential $V(y)$. The first one corresponds to $V(y) sim |y|$. In this case, for $D=0$ we find that $P(y)$ contains a delta function part at $y=0$, signaling a strong clustering effect, together with a smooth exponential component. For $D>0$, the delta function part broadens, leading instead to weak clustering. The second example is the harmonic attraction $V(y) sim y^2$ in which case, for $D=0$, $P(y)$ is supported on a finite interval. We unveil an interesting relation between this two-RTP model with harmonic attraction and a three-state single RTP model in one dimension, as well as with a four-state single RTP model in two dimensions. We also provide a general discussion of the stationary bound state, including examples where it is not unique, e.g., when the particles cannot cross due to an additional short-range repulsion.
We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling $K$ is larger than a threshold $K_E$, there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at $K_E$. We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase ($K>K_E$), vortices move in fixed paths around clusters, while in the unentrained phase ($K<K_E$), vortices sometimes wander off. These deviant vortices are responsible for the disappearance of the macroscopic cluster. The regularity of vortex motion is determined by whether clusters behave as single effective oscillators. The unentrained phase is also characterized by time-dependent cluster structure and the presence of chaos. Thus, the entrainment transition is actually an order-chaos transition. We present an analytical argument for the scaling $K_Esim K_L$ for small lattices, where $K_L$ is the threshold for phase-locking. By also deriving the scaling $K_Lsimlog N$, we thus show that $K_Esimlog N$ for small $N$, in agreement with numerics. In addition, we show how to use the linearized model to predict where vortices are generated.
We analyze the influence of classical Gaussian noise on Landau-Zener transitions during a two-level crossing in a time-dependent regular external field. Transition probabilities and coherence factors become random due to the noise. We calculate their two-time correlation functions, which describe the response of this two-level system to a weak external pulse signal. The spectrum and intensity of the magnetic response are derived. Although fluctuations are of the same order of magnitude as averages, the results is obtained in an analytic form.
A stochastic dynamics has a natural decomposition into a drift capturing mean rate of change and a martingale increment capturing randomness. They are two statistically uncorrelated, but not necessarily independent mechanisms contributing to the overall fluctuations of the dynamics, representing the uncertainties in the past and in the future. A generalized Einstein relation is a consequence solely because the dynamics being stationary; and the Green-Kubo formula reflects a balance between the two mechanisms. Equilibrium with reversibility is characterized by a novel covariance symmetry.
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