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Scaling of the dynamics of homogeneous states of one-dimensional long-range interacting systems

Quasi-Stationary States of long-range interacting systems have been studied at length over the last fifteen years. It is known that the collisional terms of the Balescu-Lenard and Landau equations vanish for one-dimensional systems in homogeneous states, thus requiring a new kinetic equation with a proper dependence on the number of particles. Here we show that previous scalings described in the literature are due either to small size effects or the use of improper variables to describe the dynamics. The correct scaling is proportional to the square of the number of particles and deduce the kinetic equation valid for the homogeneous regime and numerical evidence is given for the Hamiltonian Mean Field and ring models.

Classical systems with long range interaction can present unusual properties such as as non-ergodicity, anomalous diffusion, aging, non-Gaussian Quasi-Stationary States, negative microcanonical heat capacity and, owing to the known result of the heat capacity being always positive in the canonical ensemble, also ensemble inequivalence [1-4, [9][10][11]. A pair interaction potential is said to be long ranged if it decays at long distances as r −α such that α ≤ d, with d the spatial dimension [5]. The dynamics of these systems has essentially three stages. The first one corresponds to a violent relaxation into a Quasi Stationary State (QSS) in a short time roughly independent on the number of particles N . The second stage corresponds to the QSS with a very long relaxation time to thermodynamic equilibrium which diverges with N . Finally the third step corresponds to the final thermodynamic equilibrium.
In the thermodynamic limit this final stage is never attained. After the violent relaxation the system may also oscillate around a QSS with an amplitude decreasing with time due to a non-linear Landau damping [12]. The slowly varying state remains Vlasov stable (see Ref. [30] and references therein) and in some cases may loose its stability and rapidly evolves into another QSS and thence resuming the slow evolution towards equilibrium. This slow dynamics of the QSS have been extensively studied in the literature for different systems such as the Hamiltonian Mean Field (HMF) model [13][14][15][16][17] define by the Hamiltonian where θ i is the position angle of particle i on a circle and p i its conjugate momentum, the Ring model with Hamiltonian [18]: and also one [19][20][21][22][23], two [24,27] and three-dimensional self-gravitating systems [25,26,28,29].
The first stage of the dynamics is described by the Vlasov equation which is exact equation satisfied by the one particle distribution function in the N → ∞ limit [30,31]. For finite N this equation is valid only for short times encompassing the initial violent relaxation, a term coined by Lynden-Bell who first developed a statistical theory for predicting its outcome [32], even though a satisfactory theory is still lacking [10,[33][34][35]. After this initial stage collisional effects (graininess) accumulate and the Vlasov equation must be corrected by considering higher order terms in 1/N leading to kinetic equations such as the Landau or Balescu-Lenard equations [36]. A sensible revision of the known kinetic equations for   long-range interacting systems and their deductions, with all proper references, is presented by Chavanis in Refs. [37][38][39][40]. These equations usually can be obtained from the BBGKY hierarchy by taking into account the contribution from the two-body correlation functions, which are of order 1/N [36,37] and therefore lead to a time scale of collisional relaxation proportional to N [37]. For three-dimensional gravity the dynamics scales as N log N (the so-called Chandrasekhar scaling) [41].
The Balescu-Lenard equation for a one-dimensional homogeneous system is written as [36]: where f 1 (p 1 ; t) is the one-particle reduced distribution function, n the particle density, p i the momentum of particle i,Ṽ (k) the Fourier transform of the pair interaction potential and ε(k, kp 1 ) the dielectric function. Setting ε(k, kp 1 ) = 1 amounts to not considering collective effects and yields the Landau equation. The right-hand side of Eq. (3) vanishes identically due to the Dirac delta function [42,43]. Therefore higher order terms must be kept when truncating the BBGKY hierarchy, leading to a different scaling of the time evolution in a homogeneous state. More recently Sato proposed a derivation of a kinetic equation for one-dimensional homogeneous systems by summing contributions of all orders in the hierarchy [44]. Unfortunately his approach is limited to dilute gases and is not relevant for the problems at hand here. It would be natural to expect that in this case the predominant collisional corrections to the Kinetic equation come from higher order proportional to 1/N 2 implying a relaxation scaling proportional to N 2 . On the other hand different scalings proportional to N 1.7 and exp(N ) were reported in Refs. [13,14] for different initial conditions while the a N 2 scaling was estimated numerically for one-dimensional plasmas [45,46]. The authors obtained in Ref. [30] for the HMF model and a different type of initial condition a scaling proportional to N 2 . These facts seems to indicate a dominant contribution from the next term in the 1/N expansion. A possible explanation for these discrepancies comes from the fact that the number of particles in the simulations described in [30] are much greater than in [13,14], and a possible finite size effect should be carefully investigated. In the present letter we show that indeed the observed scaling N 1.7 is due to finite size effect (small N ) and the correct scaling for higher values of N is the more obvious value N 2 . We illustrate this point from numerical simulations for the HMF model for the type of initial conditions considered in Refs. [13,14] and varying the number of particles up to larger values than in previous studies. The exp(N ) scaling was obtained in Ref. [14] by considering the dynamics of the phase of the magnetization vector. We show that that the conclusions in Ref. [14] are misleading and that looking at the time evolution of higher order momenta of the velocity distribution leads to a correct estimation of the scaling in the dynamics. We also show that the N 2 scaling also occurs in homogeneous states for the Ring-model. The scaling of the dynamics of the QSS of the latter model has not been studied before, up to the authors knowledge, due to the difficulty to pinpoint homogeneous QSS with a finite lifetime and the numerical effort necessary for numerical simulations.
To determine the correct scaling for the dynamics we look at the fourth momentum M 4 = p 4 of the momentum distribution starting from a homogeneous waterbag initial condition define by  Figure 1 shows the time evolution of M 4 with two different scalings: N 1.7 and N 2 for the same simulation parameters as in Ref. [13]. Clearly the data collapse is better for the former scaling. Nevertheless the number of particles used in Ref. [13] is not so expressive, so we show in Fig. 2 simulations with much larger values of N . This time the N 2 scaling clearly gives a better data collapse. Figure 3 shows the observed lifetimes of the QSS for the initial conditions in Figs. 1 and 2. In the left pane a scaling close to N 1.7 is obtained for N < 10, 000 while the N 2 scaling is obtained for N > 10, 000 in the right hand panel. Note also that the error bars for smaller N are still non-negligible. To address the scaling exp(N ) as presented in Ref. [14] we redo the same simulations with the same type of semi-elliptic initial conditions (see [14] for a description). important fluctuations are still present the N 2 scaling is also observed in this case. All the numeric data here presented points out to a kinetic equation obtained from the next order correction in N −1 , which implies the observed N 2 scaling for homogeneous states with N sufficiently large. Other values for the scaling exponent are due either to small size effects or a misinterpretation of numerical data.
The kinetic equation for a homogeneous one-dimensional system can be obtained starting from the BBGKY hierarchy [48]: where f s (1, . . . , s; t) ≡ f s (x 1 , p 1 , . . . , x s , p s ; t) is the normalized s-particle reduced distribution, x i and p i the position and momentum variables of particle i, respectively, di ≡ dx i dp i and G i,j = −∂V (x i − x j )/∂x i the force on particle i due to particle j, with V (x i − x j ) the pair interaction potential. The Liouvillian operatorL s in Eq. (5) is defined bŷ By using the Kac prescription [49] the pair interaction potential can be written as  then be written as: In order to obtain a closed equation for f 1 we must determine the two-particle correlation function. Two-particle correlations require the interaction of the two particles, which is of order N −1 . For three-particle correlations a two particle interaction between say particles 1 and 2, and then between particles 2 and 3 is required, and therefore this correlation function if or order N −2 , and so on. The irreducible cluster representation (correlation expansion) for the reduced distribution functions f s up to s = 3 is given by [36,48]: where P (1,2,3) stands for the sum of the different permutations between particles 1, 2 and 3.
Note that since the reduced distributions are taken as fully symmetric by the permutation of any two particles, the same is valid for the pure correlation functions C 2 and C 3 . We thus write: The Vlasov and Landau equations are obtained at order N 0 and N −1 , respectively. Plugging Eqs. (12), (13) and (14) in Eq. (8) for s = 1, 2, and equating equal powers of N on each side yields in a cumbersome but quite straightforward way the equations for the coefficients in Eq. (13) and (14). We first note that some preliminary simplifications are possible. Since the states are always homogeneous here we have that On the other hand, the vanishing of the collisional integral in the Landau equation is equivalent toÎ 2 C (1) 2 (1, 2) = 0. The mean field force always vanishes in a homogeneous state, and therefore the action ofÎ s on any product of any number of functions f (n) 1 (i) and C (1) 2 (j, l) also vanishes. In this way, the dynamics of the one-particle distribution function is given by the second order correction in N −1 : At order N −1 we obtain: It is easily checked out that Eq. (18) ensues from Eq. (17). At order N −2 we have: Another equation is obtained at this order, but involves C 3 (1, 2, 3) and as we are going to show the determination of the three-particle correlation is not required here. From Eq. (16) and the definition ofÎ 2 in Eq. (10) it also straightforward to show that the third and fifth terms in Eq. (19) cancel mutually (the error is of order N 3 ) and we are left with: The strategy to obtain the kinetic equation up to order N −2 is thus to solve equation (17) for C (1) 2 (1, 2; t).