In this article, several aspects of the dynamics of a toy model for longrange Hamiltonian systems are tackled focusing on linearly unstable unmagnetized (i.e. force-free) cold equilibria states of the Hamiltonian Mean Field (HMF). For special cases, exact finite-N linear growth rates have been exhibited, including, in some spatially inhomogeneous case, finite-N corrections. A random matrix approach is then proposed to estimate the finite-N growth rate for some random initial states. Within the continuous, $N \rightarrow \infty$, approach, the growth rates are finally derived without restricting to spatially homogeneous cases. All the numerical simulations show a very good agreement with the different theoretical predictions. Then, these linear results are used to discuss the large-time nonlinear evolution. A simple criterion is proposed to measure the ability of the system to undergo a violent relaxation that transports it in the vicinity of the equilibrium state within some linear e-folding times.
Systems of particles interacting via two-body long-range forces are well-known to have peculiar equilibrium and non-equilibrium statistical mechanics (see e.g. Ref. [1] and references therein). As far as their relaxation properties are concerned, much progress originated from numerical simulations of the one-dimensional gravitational system. In 1982, Wright, Miller and Stein [2] observed its reluctance to thermalize due to the existence of quasi-stationary states (QSSs). These observations were later refined by various authors showing that the relaxation of the one-dimensional gravitational system usually proceeds through a rapid approach to a QSS, referred to as violent relaxation, followed by a very slow drift toward equilibrium [3,4]. Such studies initiated a still very active line of research (see e.g. Refs. [5,6,7]) on the intricate interplay between dynamics, ergodic properties and statistical mechanics in self-gravitating Hamiltonian systems.
Moreover, in the case where space dimension is larger than one, the thoroughly investigated gravitational system, as well as the Coulomb system, combine the difficulties of long-range interaction with a short-range divergence. This was a motivation to introduce models in which the potential was truncated to retain only its long-range components. In addition, the periodic boundary conditions considered in such models amount to work with a compact space which is numerically convenient. Various numerical simulations and theoretical arguments [1,8,9,10] gave indications that the corresponding toy models obtained in this way were sharing purely long-range relaxation characteristics similar to the original systems.
The Hamiltonian Mean Field (HMF) model [9] derives from such a truncation procedure as it amounts, in its attractive ferromagnetic-like form, to the one-dimensional gravitational system with periodic boundary conditions where only the lowest Fourier mode is retained. It has become a well-known toy model to address the intricate relationships between dynamics and statistical mechanics of long-range interacting systems. It is defined by the following Hamiltonian
where N is the number of particles, and θ i and p i denote respectively the position and momentum of the i th particle. A useful collective quantity to introduce is the so-called magnetization vector (M x , M y ) with
The average energy per particle U = H/N reads then
where M ≡ M x 2 + M y 2 denotes the modulus of the magnetization vector. Recently, much interest has been devoted to the QSSs which are known to be responsible for the very slow convergence towards the statistical mechanics equilibrium predictions. Far from being difficult to generate, these QSSs naturally emerge in the HMF model from waterbag initial distributions (see e.g. [11,12,14,15,16] and the recent review [17]). It is also known that initial waterbag conditions in momenta, associated to zero or almost zero initial magnetization, induce the longest lasting QSSs. However, when lowering towards zero the initial temperature of the particles, it is possible to exhibit waterbag momenta configurations with vanishing magnetization in which the magnetization eventually converges exponentially towards its Boltzmann-Gibbs equilibrium value. This calls for a linear theory approach.
Linear stability of the HMF model about unmagnetized equilibrium states has been up to now only studied within the Vlasov framework [9,12,28], which assumes in particular an infinite number of particles. Moreover, let alone some very recent publications [28], the linear stability of spatially inhomogeneous, unmagnetized, equilibria has never been considered yet.
The motivation of the present study is then twofold: Firstly and mostly, one wishes to tackle the linear study of the unmagnetized cold HMF equilibria, within a finite-N, therefore exact, framework; secondly, the ensuing nonlinear dynamics is briefly addressed to show that the thermalization of cold unmagnetized HMF systems finely illustrates Lynden-Bell’s concept of violent relaxation for long-range systems.
In Section 2, we shall establish the finite-N framework used for the linear stability derivation. In Section 3, we shall calculate the exact linear growth rates for two finite-N equilibria, both of zero temperature and zero magnetization, and compare them to numerical simulations. Section 4 is dedicated to a random matrix approach for the calculation of symmetric non-deterministic initial states growth rates. In Section 5, we eventually derive the linear theory in the N → ∞ limit using a fluid approach derived from the Vlasov equation for a vanishing temperature. Section 6 ends this study by discussing the connections between the linear features just derived and the HMF thermalization properties. The dynamics of the cold unmagnetized HMF model is proposed as a paradigm of violent relaxation.
The equations of motion can straightforwardly be written from Equation (1) as
Using Equation ( 2), th
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