Non-diagonalizable and non-divergent susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions

We investigate 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 the 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 quasista- tionary state is predicted by a linear response theory based on the Vlasov equation. For spatially homogeneous stable states, whose momentum distributions are asymmetric with zero-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 the skew-normal distribution functions.Comment: 10 pages, 8 figure

Relaxation and Diffusion in a Globally Coupled Hamiltonian System

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 non-stationarity. Contrary to previous claims, in quasi-stationary 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.Comment: 9 pages, 23 eps figures, revtex

Low-frequency discrete breathers in long-range systems without on-site potential

We propose a new mechanism of long-range coupling to excite low-frequency discrete breathers without the on-site potential. This mechanism is universal in long-range systems irrespective of the spatial boundary conditions, of topology of the inner degree of freedom, and of precise forms of the coupling functions. The limit of large population is theoretically discussed for the periodic boundary condition. Existence of discrete breathers is numerically demonstrated with stability analysis.Comment: 5 pages, 4 figure

On algebraic damping close to inhomogeneous Vlasov equilibria in multi-dimensional spaces

We investigate the asymptotic damping of a perturbation around inhomogeneous stable stationary states of the Vlasov equation in spatially multi-dimensional systems. We show that branch singularities of the Fourier-Laplace transform of the perturbation yield algebraic dampings. In two spatial dimensions, we classify the singularities and compute the associated damping rate and frequency. This 2D setting also applies to spherically symmetric self-gravitating systems. We validate the theory using a toy model and an advection equation associated with the isochrone model, a model of spherical self-gravitating systems.Comment: 37 pages, 10 figure

Collective 1/f1/f fluctuation by pseudo-Casimir-invariants

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.Comment: 13 pages, 12 figure