Hamiltonian Dynamics Reveals the Existence of Quasistationary States for Long-Range Systems in Contact with a Reservoir

We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian mean field model and the equilibrium ensemble equivalence. We find that long-lasting quasistationary states persist in the presence of the interaction with the environment. Our results indicate that quasistationary states are indeed reproducible in real physical experiments.

Figure 1

Sketch of the interactions considered in our canonical setup. The dashed lines mimic the interactions between the HMF (solid circles) and the TB (empty circles) spins. The solid (dotted) lines represent the HMF (TB) couplings.

Figure 2

(a) Time evolution of the $T_{\mathrm{HMF}}$ and $T_{\mathrm{TB}}$ temperatures for $M={10}^3$, $ϵ=0.01$, and $T_0=0.5$. Initially, $p_{\mathrm{HMF}}(l,0)=\exp (-l^2/2T(t^{*}))/\sqrt{2\pi T(t^{*})}$, with $T(t^{*})=0.7$. (b) Velocity PDF at $t\gg t^{*}$. The solid line is $p_{\mathrm{TB}}(l,0)$. (c) Caloric curve. The solid line is the BG equilibrium, and the dashed line is the prolongation of the ordered phase to subcritical energies. The empty symbols are the average value of $e_{\mathrm{HMF}}(t)$ at equilibrium. The stars refer to Nosé-Hoover simulations. The solid circles correspond to the QSS studied in the Letter and to the microcanonical and canonical equilibrium obtained as $t\to \infty$.

Figure 3

(a) Microcanonical and (b),(c) canonical QSSs in terms of $m_{\mathrm{HMF}}^2$. The curves have been obtained by averaging over a number $I$ of initial conditions following the probability distributions described in the text. $I$ varies from a maximum of 20 for small $M$ down to 5 for big $M$. (d) Log-log plot of $t_x$ as a function of the system size $M$. The QSS lifetime $t_x$ has been defined as the time at which a phase function ($T_{\mathrm{HMF}}$ for ${ϵ}^{*}=0$ and $e_{\mathrm{HMF}}$ for ${ϵ}^{*}\ne 0$) changes 10% of its stationary average value.

Figure 4

Time dependence of the $H$ function in the microcanonical (${ϵ}^{*}=0$) and canonical (${ϵ}^{*}\ne 0$) ensembles for $M={10}^4$ and $T_0=0.38$. The dashed line indicates the common maximum of $H$ for ${ϵ}^{*}\ne 0$ reached during the QSS.