Yoshiyuki Y. Yamaguchi

Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field model, a prototype for long-range interactions in N-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N→∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N, dynamics. We then propose, and verify numerically, a scenario for the relaxation process, relying on the Vlasov equation. When starting from a nonstationary or a Vlasov unstable stationary state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via nonstationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann–Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite N system is initialized in a Vlasov stable homogeneous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N1.7. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.

Algebraic damping in the one-dimensional Vlasov equation

We investigate the asymptotic behaviour of a perturbation around a spatially non-homogeneous stable stationary state of a one-dimensional Vlasov equation. Under general hypotheses, after transient exponential Landau damping, a perturbation evolving according to the linearized Vlasov equation decays algebraically with the exponent −2 and a well-defined frequency. The theoretical results are successfully tested against numerical N-body simulations, corresponding to the full Vlasov dynamics in the large N limit, in the case of the Hamiltonian mean-field model. For this purpose, we use a weighted particles code, which allows us to reduce finite size fluctuations and to observe the asymptotic decay in the N-body simulations.

Dynamics of perturbations around inhomogeneous backgrounds in the HMF model

We investigate the dynamics of perturbations around inhomogeneous stationary states of the Vlasov equation corresponding to the Hamiltonian mean-field model. The inhomogeneous background induces a separatrix in the one-particle Hamiltonian system, and branch cuts generically appear in the analytic continuation of the dispersion relation in the complex frequency plane. We test the theory by direct comparisons with N-body simulations, using two families of distributions: inhomogeneous water-bags, and inhomogeneous thermal equilibria. In the water-bag case, which is not generic, no branch cut appears in the dispersion relation, whereas in the thermal equilibrium case, when looking for the root of the dispersion relation closest to the real axis, we have to consider several Riemann sheets. In both cases, we show that the roots of the continued dispersion relation give information that is useful for understanding the dynamics of a perturbation, although it is not complete.

Algebraic correlation functions and anomalous diffusion in the Hamiltonian mean field model

We numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions in the quasi-stationary states of the Hamiltonian mean field model. This is an example, in an N-body Hamiltonian system, of anomalous transport properties characterized by non-exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. It is noteworthy that, at statistical equilibrium also, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties.

The Vlasov equation and the Hamiltonian mean-field model

We show that the quasi-stationary states of homogeneous (zero magnetization) states observed in the N-particle dynamics of the Hamiltonian mean-field (HMF) model are nothing but Vlasov stable homogeneous states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis q-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite N effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann–Gibbs equilibrium.

Slow relaxation at critical point of second order phase transition in a highly chaotic Hamiltonian system

Temporal evolution toward thermal equilibria is numerically investigated in a Hamiltonian system with many degrees of freedom which exhibits a second order phase transition. Relaxation processes are studied through a local order parameter, and slow relaxations of the power type are observed at the critical energy of the 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 assumes a maximal value. Temporal evolution and probability distributions of local Lyapunov exponents indicate that the system is highly chaotic rather than weakly chaotic at the critical energy. Consequently theories for perturbed systems may not be applicable at the critical energy for the purpose of explaining the slow relaxation of the power type.

Geometric approach to Lyapunov analysis in Hamiltonian dynamics

As is widely recognized in Lyapunov analysis, linearized Hamiltons equations of motion have two marginal directions for which the Lyapunov exponents vanish. Those directions are the tangent one to a Hamiltonian flow and the gradient one of the Hamiltonian function. To separate out these two directions and to apply Lyapunov analysis effectively in directions for which Lyapunov exponents are not trivial, a geometric method is proposed for natural Hamiltonian systems, in particular. In this geometric method, Hamiltonian flows of a natural Hamiltonian system are regarded as geodesic flows on the cotangent bundle of a Riemannian manifold with a suitable metric. Stability/instability of the geodesic flows is then analyzed by linearized equations of motion which are related to the Jacobi equations on the Riemannian manifold. On some geometric setting on the cotangent bundle, it is shown that along a geodesic flow in question, there exist Lyapunov vectors such that two of them are in the two marginal directions and the others orthogonal to the marginal directions. It is also pointed out that Lyapunov vectors with such properties can not be obtained in general by the usual method which uses linearized Hamiltons equations of motion. Furthermore, it is observed from numerical calculation for a model system that Lyapunov exponents calculated in both methods, geometric and usual, coincide with each other, independently of the choice of the methods.

Second Order Phase Transition in a Highly Chaotic Hamiltonian System with Many Degrees of Freedom

Second order phase transition is numerically investigated in a Hamiltonian system with many degrees of freedom. Slow relaxations of power type are observed for some initial conditions at critical energy of phase transition. This is consisent with a result of a phenomenological theory of statistical mechanics. On the other hand, the slow relaxations show that the system stays in non-equilibrium states for a while, and that phenomenon does not agree with a result of the theory. To understand the slow relaxation, theories for perturbed systems cannot be applied since near the critical energy the system is highly chaotic rather than nearly integrable. The thresholds of the highly chaotic systems is different from the critical energy of phase transition.

Absence of thermalization for systems with long-range interactions coupled to a thermal bath

We investigate the dynamics of a small long-range interacting system, in contact with a large long-range thermal bath. Our analysis reveals the existence of striking anomalies in the energy flux between the bath and the system. In particular, we find that the evolution of the system is not influenced by the kinetic temperature of the bath, as opposed to what happens for short-range collisional systems. As a consequence, the system may get hotter also when its initial temperature is larger than the bath temperature. This observation is explained quantitatively in the framework of the collisionless Vlasov description of toy models with long-range interactions and shown to be valid whenever the Vlasov picture applies, from cosmology to plasma physics..

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.