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

Strange Scaling and Temporal Evolution of Finite-Size Fluctuation in Thermal Equilibrium

We numerically exhibit strange scaling and temporal evolution of finite-size fluctuation in thermal equilibrium of a simple long-range interacting system. These phenomena are explained from the view point of existence of the Casimirs and their nonexactness in finite-size systems, where the Casimirs are invariants in the Vlasov dynamics describing the long-range systems in the limit of large population. This explanation expects appearance of the reported phenomena in a wide class of isolated long-range systems. The scaling theory is also discussed as an application of the strange scaling.Comment: 5 pages, 5 figure

Slow Relaxation at Critical Point of Second Order Phase Transition in a Highly Chaotic Hamiltonian System

Temporal evolutions toward thermal equilibria are numerically investigated in a Hamiltonian system with many degrees of freedom which has second order phase transition. Relaxation processes are studied through local order parameter, and slow relaxations of power type are observed at the critical energy of phase transition for some initial conditions. Numerical results are compared with results of a phenomenological theory of statistical mechanics. At the critical energy, the maximum Lyapunov exponent takes the largest value. Temporal evolutions and probability distributions of local Lyapunov exponents show that the system is highly chaotic rather than weakly chaotic at the critical energy. Consequently theories for perturbed systems may not be applied to the system at the critical energy in order to explain the slow relaxation of power type.Comment: 16 pages, LaTeX, 13 Postscript 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

Nonlinear response for external field and perturbation in the Vlasov system

A nonlinear response theory is provided by use of the transient linearization method in the spatially one-dimensional Vlasov systems. The theory inclusively gives responses to external fields and to perturbations for initial stationary states, and is applicable even to the critical point of a second order phase transition. We apply the theory to the Hamiltonian mean-field model, a toy model of a ferromagnetic body, and investigate the critical exponent associated with the response to the external field at the critical point in particular. The obtained critical exponent is nonclassical value 3/2, while the classical value is 3. However, interestingly, one scaling relation holds with another nonclassical critical exponent of susceptibility in the isolated Vlasov systems. Validity of the theory is numerically confirmed by directly simulating temporal evolutions of the Vlasov equation.Comment: 15 pages, 8 figures, accepted for publication in Phys. Rev. E, Lemma 2 is correcte

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