Generalized maximum entropy approach to quasistationary states in long-range systems

Systems with long-range interactions display a short-time relaxation towards quasistationary states (QSSs) whose lifetime increases with the system size. In the paradigmatic Hamiltonian mean-field model (HMF) out-of-equilibrium phase transitions are predicted and numerically detected which separate homogeneous (zero magnetization) and inhomogeneous (nonzero magnetization) QSSs. In the former regime, the velocity distribution presents (at least) two large, symmetric bumps, which cannot be self-consistently explained by resorting to the conventional Lynden-Bell maximum entropy approach. We propose a generalized maximum entropy scheme which accounts for the pseudoconservation of additional charges, the even momenta of the single-particle distribution. These latter are set to the asymptotic values, as estimated by direct integration of the underlying Vlasov equation, which formally holds in the thermodynamic limit. Methodologically, we operate in the framework of a generalized Gibbs ensemble, as sometimes defined in statistical quantum mechanics, which contains an infinite number of conserved charges. The agreement between theory and simulations is satisfying, both above and below the out-of-equilibrium transition threshold. A previously unaccessible feature of the QSSs, the multiple bumps in the velocity profile, is resolved by our approach.

Figure 1

(a) Time evolution of the first few moments ${\eta }_n$ with $n=2,3,4$. After a violent relaxation the moments converges to their asymptotic values ${\epsilon }_n$, shown by dashed lines. The results refer to a choice of the parameter which yields homogeneous QSSs. Similar observation holds for the magnetized QSS phase. Notice that in the homogeneous QSS $\langle \eta {(\theta ,p;t)}^n\rangle \equiv \langle p^{2n}\rangle /2^n$. Imposing the conservation of ${\eta }_n$ is equivalent to assuming the (rescaled) even velocity moments are invariant. (b) Running averages ${\eta }_n{|}_{t_0}^t={\left(t-t_0\right)}^{-1}{\int }_{t_0}^t{\eta }_n\left(t^{\prime }\right)d{t}^{\prime }$ in the time interval $(250,1000)$ (i.e., $t_0=250)$.

Figure 2

The velocity distribution profiles are displayed for $e=0.9$ and different choices of the initial magnetization (from left to right, $M_0=0.3,0.5,0.7$). In all cases the system is predicted to converge to a homogeneous QSS. The result of the Vlasov simulations is depicted by a black solid line. It is averaged over the latter half of the simulation time. The standard Lynden-Bell theory yields the gray dashed curve. The modified maximum entropy scheme with the inclusion of the additional pseudoconserved quantities yields the profiles represented by red solid lines.

Figure 3

The velocity distribution profiles are plotted for $e=0.6$ and different choices of the initial magnetization (from left to right, $M_0=0.3,0.5,0.7$). The system converges to a magnetized QSS. The result of the Vlasov simulations are drawn with a black solid line (averaged, as in Fig. 2). The standard Lynden-Bell theory yields the gray dashed curve, whose deviations from the simulation results are clearly visible. Predictions based on the modified maximum entropy scheme with the inclusion of the additional pseudoconserved quantities are depicted by red solid lines. The latter are almost identical to the simulated profiles for $M_0=0.3$ and $0.5$ and still improve on the standard Lynden-Bell result for $M_0=0.7$.