24,383 research outputs found
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
A Geometrical Model for Stagnant Motion in Hamiltonian Systems with Many Degrees of Freedom
We introduce a model of Poincar\'e mappings which represents hierarchical
structure of phase spaces for systems with many degrees of freedom. The model
yields residence time distribution of power type, hence temporal correlation
remains long. The power law behavior is enhanced as the system size increases.Comment: 6 pages, 3 Encapsulated Postscript figures, LaTeX (58 kb
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