Search

A coupled phase-oscillator model consists of phase oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the nonsynchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality classes is reduced to one and the critical exponent is shared in the considered models having coupling functions up to second harmonics with unimodal and symmetric natural frequency distributions.

Synchronization among rhythmic elements is modeled by coupled phase-oscillators, each of which has the so-called natural frequency. A symmetric natural frequency distribution induces a continuous or discontinuous synchronization transition, or oscillation, from the nonsynchronized state, for example. It has been numerically reported that asymmetry in the natural frequency distribution introduces new types of bifurcation diagrams with, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method for five types of bifurcation diagrams including the new ones, paying attention to the generality of the theory. The oscillation and the jump from partially synchronized states are discussed, respectively, by the linear analysis around the nonsynchronized state and by extending the amplitude equation up to the third leading term. The theoretical classification is examined through comparison with the numerically obtained classification.

We develop a linear response theory by computing the asymptotic value of the order parameter from the linearized equation of continuity around the nonsynchronized reference state using the Laplace transform in time. The proposed theory is applicable to a wide class of coupled phase oscillator systems and allows for any coupling functions, any natural frequency distributions, any phase-lag parameters, and any values for the time-delay parameter. This generality is in contrast to the limitation of the previous methods of the Ott–Antonsen ansatz and the self-consistent equation for an order parameter, which are restricted to a model family whose coupling function consists of only a single sinusoidal function. The theory is verified by numerical simulations.

In this study, we propose a universal scenario explaining the 1/f fluctuation, including pink noises, in Hamiltonian dynamical systems with many degrees of freedom under long-range interaction. In the thermodynamic limit, the dynamics of such systems can be described by the Vlasov equation, which has an infinite number of Casimir invariants. In a finite system, they become pseudoinvariants, which yield quasistationary states. The dynamics then exhibit slow motion over them, up to the timescale where the pseudo-Casimir-invariants are effective. Such long-time correlation leads to 1/f fluctuations of collective variables, as is confirmed by direct numerical simulations. The universality of this collective 1/f fluctuation is demonstrated by taking a variety of Hamiltonians and changing the range of interaction and number of particles.

A mechanism of long-range couplings is proposed to realize low-frequency discrete breathers without on-site potentials. The realization of such discrete breathers requires a gap below the band of linear eigenfrequencies. Under the periodic boundary condition of a one-dimensional lattice and the limit of large population, we show theoretically that the long-range couplings universally open the gap below the band irrespective of the coupling functions, while the short-range couplings cannot. The existence of the low-frequency discrete breathers, spatial localization, and stability are numerically analyzed from long range to short range.

We numerically exhibit two strange phenomena of finite-size fluctuation in thermal equilibrium of a paradigmatic long-range interacting system having a second-order phase transition. One is a nonclassical finite-size scaling at the critical point, which differs from the prediction by statistical mechanics. With the aid of this strange scaling, the scaling theory for infinite-range models conjectures the nonclassical values of critical exponents for the correlation length. The other is relaxation of the fluctuation strength from one level to another in spite of being in thermal equilibrium. A scenario is proposed to explain these phenomena from the viewpoint of the Casimir invariants and their nonexactness in finite-size systems, where the Casimir invariants are conserved in the Vlasov dynamics describing the long-range interacting systems in the limit of large population. This scenario suggests appearance of the reported phenomena in a wide class of isolated long-range interacting systems.

Predicting the long-lasting quasistationary state for a given initial state is one of central issues in Hamiltonian systems having long-range interaction. A recently proposed method is based on the Vlasov description and uniformly redistributes the initial distribution along contours of the asymptotic effective Hamiltonian, which is defined by the obtained quasistationary state and is determined self-consistently. The method, to which we refer as the rearrangement formula, was suggested to give precise prediction under limited situations. Restricting initial states consisting of a spatially homogeneous part and small perturbation, we numerically reveal two conditions that the rearrangement formula prefers: One is a no Landau damping condition for the unperturbed homogeneous part, and the other comes from the Casimir invariants. Mechanisms of these conditions are discussed. Clarifying these conditions, we validate to use the rearrangement formula as the response theory for an external field, and we shed light on improving the theory as a nonequilibrium statistical mechanics.

We investigate the response to an external magnetic field in the Hamiltonian mean-field model, which is a paradigmatic toy model of a ferromagnetic body and consists of plane rotators like XY spins. Due to long-range interactions, the external field drives the system to a long-lasting quasistationary state before reaching thermal equilibrium, and the susceptibility tensor obtained in the quasistationary state is predicted by a linear response theory based on the Vlasov equation. For spatially homogeneous stable states, whose momentum distributions are asymmetric with 0 means, the theory reveals that the susceptibility tensor for an asymptotically constant external field is neither symmetric nor diagonalizable, and the predicted states are not stationary accordingly. Moreover, the tensor has no divergence even at the stability threshold. These theoretical findings are confirmed by direct numerical simulations of the Vlasov equation for skew-normal distribution functions.

An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two nonclassical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends the universality class of the nonclassical exponents to spatially periodic one-dimensional systems and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.

Mean-field theory tells us that the classical critical exponent of susceptibility is twice that of magnetization. However, linear response theory based on the Vlasov equation, which is naturally introduced by the mean-field nature, makes the former exponent half of the latter for families of quasistationary states having second order phase transitions in the Hamiltonian mean-field model and its variances, in the low-energy phase. We clarify that this strange exponent is due to the existence of Casimir invariants which trap the system in a quasistationary state for a time scale diverging with the system size. The theoretical prediction is numerically confirmed by N-body simulations for the equilibrium states and a family of quasistationary states.

Response to small external forces is investigated in quasistationary states of Hamiltonian systems having long-range interactions. Quasistationary states are recognized as stable stationary solutions to the Vlasov equation, and, hence, the linear response theory to the Vlasov equation is proposed for spatially one-dimensional systems with periodic boundary condition. The proposed theory is applicable both to homogeneous and to inhomogeneous quasistationary states and is demonstrated in the Hamiltonian mean-field model. In the homogeneous case magnetic susceptibility is explicitly obtained, and the Curie-Weiss like law is suggested in a high-energy region. The linear response is also computed in the inhomogeneous case, and resonance absorption is investigated to extract nonequilibrium dynamics in the unforced system. Theoretical predictions are examined by direct numerical simulations of the Vlasov equation.

Traveling clusters are ubiquitously observed in the Hamiltonian mean-field model for a wide class of initial states, which are not predicted to become spatially inhomogeneous states by nonequilibrium statistical mechanics and by nonlinear Landau damping. To predict such a cluster state from a given initial state, we combine nonequilibrium statistical mechanics and a construction method of Bernstein-Greene-Kruskal (BGK) waves with the aid of phenomenological assumptions. The phenomenological theory is partially successful, and the theoretically constructed cluster states are in good agreement with N-body simulations. Robustness of the theory is also discussed for unsuccessful initial states.

Long-lasting small traveling clusters are studied in the Hamiltonian mean-field model by comparing between attractive and repulsive interactions. Nonlinear Landau damping theory predicts that a Gaussian momentum distribution on a spatially homogeneous background permits the existence of traveling clusters in the repulsive case, as in plasma systems, but not in the attractive case. Nevertheless, extending the analysis to a two-parameter family of momentum distributions of Fermi-Dirac type, we theoretically predict the existence of traveling clusters in the attractive case; these findings are confirmed by direct N -body numerical simulations. The parameter region with the traveling clusters is much reduced in the attractive case with respect to the repulsive case.

The Lynden-Bell statistics has been proposed to explain common features among galaxies, which are not in thermal equilibrium. The statistics is not successful to reproduce energy distribution in the one-dimensional self-gravitating sheet model except for initial states near the virial equilibrium. The breakdown is caused by dynamically accelerated high-energy sheets, and hence a modified statistics is examined by focusing on low-energy sheets in order to clarify validity of the basic idea of the Lynden-Bell statistics. The modification improves agreement between the theoretical and numerical energy distributions in a wide interval of the initial virial ratio.

The relation between relaxation and diffusion is investigated in a Hamiltonian system of globally coupled rotators. Diffusion is anomalous if and only if the system is going towards equilibrium. The anomaly in diffusion is not anomalous diffusion taking a power-type function, but is a transient anomaly due to nonstationarity. For a certain type of initial condition, in quasistationary states, diffusion can be explained by a stretched exponential correlation function, whose stretching exponent is almost constant and correlation time is linear as functions of degrees of freedom. The full time evolution is characterized by varying stretching exponent and correlation time.