Temperature inversion in long-range interacting systems

Temperature inversions occur in nature, e.g., in the solar corona and in interstellar molecular clouds: Somewhat counterintuitively, denser parts of the system are colder than dilute ones. We propose a simple and appealing way to spontaneously generate temperature inversions in systems with long-range interactions, by preparing them in inhomogeneous thermal equilibrium states and then applying an impulsive perturbation. In similar situations, short-range systems would typically relax to another thermal equilibrium, with a uniform temperature profile. By contrast, in long-range systems, the interplay between wave-particle interaction and spatial inhomogeneity drives the system to nonequilibrium stationary states that generically exhibit temperature inversion. We demonstrate this mechanism in a simple mean-field model and in a two-dimensional self-gravitating system. Our work underlines the crucial role the range of interparticle interaction plays in determining the nature of steady states out of thermal equilibrium.

Figure 1

HMF: Time evolution of the magnetization $m$ with $N={10}^7$ (solid red line). Inset: $m\left(t\right)$ compared to the equilibrium value $m_{\text{eq}}=0$ (dotted black line) for longer times.

Figure 2

HMF: Temperature profile $T\left(\vartheta \right)$ (blue solid line) and density profile $n\left(\vartheta \right)$ (red dashed line), measured in the QSS at $t={10}^4$.

Figure 3

HMF: Time evolution of the distance from equilibrium temperature $\xi$, Eq. (7), with $N$ increasing from bottom to top: $N=5\times {10}^2$ (red), $N={10}^3$ (blue), $N=2.5\times {10}^3$ (black), $N=5\times {10}^3$ (purple). Each curve is the average over $n_r$ realizations with $n_r$ ranging from 20 for $N=5\times {10}^3$ to ${10}^3$ for $N=5\times {10}^2$. Inset: $\xi$ as a function of $t/N$.

Figure 4

2DSGS: Temperature profile $T\left(r\right)$ (blue squares) and density profile $n\left(r\right)/n_0$ (red circles) measured in the QSS at $t=90{\tau }_{\text{dyn}}$ while starting from a thermal state with uniform temperature $T_0=0.5$, in natural units such that $r_{*}\left(t\right)\simeq 7$ and $r_0={10}^{-3}$.

Figure 5

HMF. Left: Momentum distribution $f\left(p\right)$ at $t=0$ (black dashed line), $t=101$ (red plus signs), $t=110$ (blue asterisks), $t=150$ (magenta open squares), $t=200$ (gray solid triangles), $t={10}^3$ (orange open triangles), and $t={10}^4$ (magenta solid rhombi). Right: Distribution function $f(\vartheta ,p)$ measured in the QSS as in Fig. 2, at $\vartheta =0$ (red squares) and $\vartheta =\pi$ (blue triangles). The distribution functions are plotted against $p^2\text{sgn}\left(p\right)$ to better show the difference with respect to the initial Maxwellian.