Attractor nonequilibrium stationary states in perturbed long-range interacting systems

Isolated long-range interacting particle systems appear generically to relax to nonequilibrium states (“quasistationary states” or QSSs) which are stationary in the thermodynamic limit. A fundamental open question concerns the “robustness” of these states when the system is not isolated. In this paper we explore, using both analytical and numerical approaches to a paradigmatic one-dimensional model, the effect of a simple class of perturbations. We call them “internal local perturbations” in that the particle energies are perturbed at collisions in a way which depends only on the local properties. Our central finding is that the effect of the perturbations is to drive all the very different QSSs we consider towards a unique QSS. The latter is thus independent of the initial conditions of the system, but determined instead by both the long-range forces and the details of the perturbations applied. Thus in the presence of such a perturbation the long-range system evolves to a unique nonequilibrium stationary state, completely different from its state in absence of the perturbation, and it remains in this state when the perturbation is removed. We argue that this result may be generic for long-range interacting systems subject to perturbations which are dependent on the local properties (e.g., spatial density or velocity distribution) of the system itself.

Figure 1

Model A. Top panel: Dimensionless energy $\langle E\rangle /E_0$ ($E_0$ is the initial energy). Bottom panel: dimensionless variance $(\langle E^2\rangle -{\langle E\rangle }^2)/E_0^2$ vs dimensionless time $t/{\tau }_{dyn}$ averaged over 100 realizations with ${\gamma }_A=0.03$, of rectangular waterbag initial conditions for $N=128,256,512$ particles.

Figure 2

Model A. Top panel: virial ratio $R$ as a function of time $t/{\tau }_{dyn}$ averaged over 100 realizations with equilibrium initial conditions and for ${\gamma }_A=0.03$ and $N=512$. Bottom panel: variance of the virial ratio for $N=512$ and $N=256$ with the same initial conditions.

Figure 3

Model A: Evolution of the entanglement parameter ${\varphi }_{11}$ (top panel) and of the dimensionless kurtosis ${\beta }_2$ averaged over 100 realizations with ${\gamma }_A=0.1$ and equilibrium initial conditions for $N=1024$. The black curves correspond to a modified simulation in which the perturbation is switched off for $t\ge 300{\tau }_{dyn}$.

Figure 4

Model A: Entanglement parameter ${\varphi }_{11}$, dimensionless kurtosis of the velocity distribution ${\beta }_2$ and $\langle \left|v\right|\rangle /v_{dyn}$, averaged over 300 realizations, as functions of time $t/{\tau }_{dyn}$ for ${\gamma }_A=0.03$ and $N=128,256,512$ (from top to bottom).

Figure 5

Model A: ${\varphi }_{11}$ (top) and of $\langle \left|v\right|\rangle /v_{dyn}$ (bottom) as a function of time $t/{\tau }_{dyn}$, with ${\gamma }_A=0.03$ and $N=512$ for different initial conditions: thermal equilibrium (averaged over 300 realizations), rectangular waterbags with $R_0=1$ (100 realizations), and $R_0=0.01$ (200 realizations).

Figure 6

Model A: Velocity distribution at different times ($t=500,1500,3500{\tau }_{dyn}$) averaged over 100 realizations for ${\gamma }_A=0.1, N=1024$ with initial thermal distribution. Velocities are normalized by $\overline{v}\left(t\right)$, the standard deviation of the velocity distribution. The left panel is a linear plot and the right panel is a log-log plot. The two straight lines indicate the range of the exponents of power law fits to the tails of the distribution.

Figure 7

Model A: Position distribution at different times ($t=500,1500,3500{\tau }_{dyn}$) averaged over 100 realizations for ${\gamma }_A=0.1, N=1024$ with initial thermal distribution. Positions are normalized by $\overline{x}\left(t\right)$, the standard deviation of the position distribution. The left panel is a linear plot and the right panel is a log-log plot. The two straight lines indicate the range of the exponents of power law fits to the tails of the distribution.

Figure 8

Model B: Dimensionless energy $E/E_0$ ($E_0$ is the initial energy), virial ratio $R$, entanglement parameter ${\varphi }_{11}$, and dimensionless kurtosis ${\beta }_2$ vs $t/{\tau }_{dyn}$ for $u_B=1, {\gamma }_B=0.01,0.005, N=512$, and initial rectangular waterbag conditions $R_0=0.01$. Simulation results are averaged over 20 realizations.

Figure 9

Model B: $E/E_0$ (top) and ${\varphi }_{11}$ (bottom) vs time $t/{\tau }_{dyn}$ for $N=512, {\gamma }_B=0.01, u_B=1$ with rectangular waterbag initial conditions, $R_0=0.01,1$.

Figure 10

Model B: $E/E_0$ (top) and ${\varphi }_{11}$ (bottom) for ${\gamma }_B=0.01, u_B=1$, and for rectangular waterbag initial conditions, $R_0=0.01$ with different system sizes, $N=128,256,512$.

Figure 11

Model B: Phase space snapshots in dimensionless units ($x/x_0$ and $v/v_0$) for $N=128, {\gamma }_B=0.01, u_B=1$, rectangular waterbag initial conditions, $R_0=0.01$ and 20 realizations at different times $t$ ranging from $t=0$ to $t=4000{\tau }_{dyn}$.

Figure 12

Model B: Phase space snapshots in dimensionless units ($x/x_0$ and $v/v_0$) for ${\gamma }_B=0.01, u_B=1$, rectangular waterbag initial conditions, $R_0=0.01$, 20 realizations and at two different times $t=1000{\tau }_{dyn}$ (left panels), $t=3500{\tau }_{dyn}$ (right panels). We compare three different system sizes $N=128,256,512$ (from top to bottom). Red dots correspond to one chosen realization, small black dots correspond to the other realizations.