We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) 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 non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary 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 homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N 1.7 . 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.

Systems with long-range interactions display a short-time relaxation towards quasistationary states whose lifetime increases with system size. With reference to the Hamiltonian mean field model, we here show that a maximum entropy principle, based on Lynden-Bell's pioneering idea of "violent relaxation," predicts the presence of out-of-equilibrium phase transitions separating the relaxation towards homogeneous (zero magnetization) or inhomogeneous (nonzero magnetization) quasistationary states. When varying the initial condition within a family of "water bags" with different initial magnetization and energy, first- and second-order phase transition lines are found that merge at an out-of-equilibrium tricritical point. Metastability is theoretically predicted and numerically checked around the first-order phase transition line.

The dynamical behavior and the relaxation to equilibrium of long range interacting systems of particles still offer several open problems. It is possible to sketch the general theory as follows: the short time behavior, on the order of the dynamical time scale, is described by the Vlasov, or a Vlasov-like equation. The system then settles in a quasi-stationary state (hereafter called QSS), which is a stationary solution of the Vlasov equation. On time scales diverging with the number of particles, the system evolves towards the thermodynamic equilibrium, following the analog of a Lenard-Balescu equation (for a recent review, see [1]). The open questions include for instance the selection of the QSS among the stationary states of the Vlasov equation; the possible selection of periodic solutions of the Vlasov equation instead of a QSS; the complete understanding of the timescales for relaxation to equilibrium, especially around inhomogeneous QSS and close to dynamical transitions. We may mention also the rigorous derivation of the Lenard-Balescu equation, which, in contrast to the Vlasov equation, does not have a mathematical foundation. The stationary solutions of the Vlasov equation may be homogeneous in space, as is always the case in plasma physics, or inhomogeneous, as is the rule in self gravitating systems. In general, there is less understanding of the inhomogeneous cases, even at the level of linear perturbations of the Vlasov stationary state, because of the much greater technical difficulties. This linear understanding is an essential ingredient to derive a Lenard-Balescu like equation, and also to characterize possible undamped periodic solutions [2]. The linearization around a homogeneous stationary state corresponds to the usual theory of Landau damping in plasma physics [3]. Technically, the equation are solved by a Fourier transform in space and a Laplace transform in time. The equations for different Fourier modes decouple, and the computation results in the complex dielectric function (k, ω) for the kth Fourier mode. Because (k, ω) is defined in the upper half ω-plane due to the convergence condition of the Laplace transform, one needs to analytically continue (k, ω) in the lower half ω-plane to obtain the Landau frequency and damping rate. Linearization around a non homogeneous stationary state brings additional technical difficulties. First, one needs to use angle-action variables, to simplify the particle dynamics; but now the equations for different Fourier modes in the angle variables do not decouple any more; the "dielectric function" now becomes the determinant of an infinite matrix. Linearization around inhomogeneous stationary states has been studied in the context of self-gravitating systems, and, despite the technical difficulties, the above procedure has been carried out entirely to study the instability of some galactic models. However, studying the analog of Landau damping requires the analytic continuation part, and this is, in the words of Weinberg [4], a "daunting task". This dif...

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.

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.

Hepatitis C virus (HCV) has been reported to potentially replicate in peripheral blood mononuclear cells (PBMCs), but direct information on the pathogenic implication of HCV infection in PBMCs is still limited. To investigate this issue, we compared the complexity of HCV quasispecies in serum, PBMCs, and livers of 13 patients with type C chronic liver disease. Hypervariable region 1 (HVR 1) was amplified by reverse-transcription polymerase chain reaction (RT-PCR), and the PCR products were subcloned and sequenced. Considerable differences in the complexity of HVR 1 quasispecies were found in the serum, PBMCs, and liver in all patients, and the predominant sequences from each source were mutually different in 3 (23%) patients. An amino acid sequence unique to each source existed as well as a sequence common to serum and PBMCs, common to serum and livers, or common to PBMCs and liver. These results suggest infection of PBMCs by HCV, and that HCV in PBMCs may be differently exposed to host immunity from that in liver. (HEPATOLOGY 1999;29:217-222.)Hepatitis C virus (HCV), a positive-strand RNA virus, is assumed to replicate via the production of a complementary negative-strand template, like that of the closely related flaviviruses. 1,2 Several lines of evidence suggest that this virus can also infect peripheral blood mononuclear cells (PBMCs) in persistently infected individuals on the basis of specific detection of negative-strand HCV RNA in PBMCs by reversetranscription polymerase chain reaction (RT-PCR) and in situ hybridization 3-7 ; however, this issue is still controversial. [8][9][10]

We show that the quasi-stationary states observed in the N -particle dynamics of the Hamiltonian Mean-Field (HMF) model are nothing but Vlasov stable homogeneous (zero magnetization) 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. I. INTRIGUING NUMERICAL RESULTS The Hamiltonian Mean-Field model (HMF) [1]describes the motion of globally coupled particles on a circle: θ i refers to the angle of the i-th particle and p i to its conjugate momentum, while N is the total number of particles. The 1/N prefactor, which has been historically introduced to obtain an extensive energy, can be absorbed in a time rescaling (we shall however keep it to compare with previous results). From the fundamental point of view, this is an ideal toy model. Indeed, although it is simple and the mean-field interaction allows us to perform analytical calculations, it has several features of long-range interactions. Moreover, it is a simplification of physical systems like charged or gravitational sheet models. Finally, in some cases, waveparticle Hamiltonians can be reduced to it. In particular, for what the equilibrium properties are concerned, the HMF Hamiltonian (1) can be mapped onto the Colson-Bonifacio model of the single-pass Free Electron Laser [2].In this short note, we would like to emphasize that several interesting numerical facts which have been reported in the literature can be accurately explained by considering the limit of infinite number of particles, namely the Vlasov equation corresponding to the HMF model. The first numerical fact is the strong disagreement which was reported in Refs. [1,3] between constant energy molecular dynamics simulations and canonical statistical mechanics calculations. This unexpected and striking result, found for energies slightly below the second order phase transition energy (see Fig. 1), was first thought to be the fingerprint of inequivalence between microcanonical and canonical ensemble. It was known that such inequivalence might have been present because of the long-range nature of the interaction. However, it has been later proved that inequivalence occurs only if the system has a first order canonical phase transition, which is not the case for the HMF, which has instead a second order phase transition. Moreover, the microcanonical entropy of the HMF model has been recently derived using large deviation theory [2], showing that the two ensembles give the same predictions.It then became clear that the disagreement must have a dynamical origin. In order to characterize the dynamical properties of the HMF model, the behavio...

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.