One-Dimensional Self-Organization and Nonequilibrium Phase Transition in a Hamiltonian System

Self-organization and nonequilibrium phase transitions are well known to occur in two- and three-dimensional dissipative systems. Here, instead, we provide numerical evidence that these phenomena also occur in a one-dimensional Hamiltonian system. To this end, we calculate the heat conductivity by coupling the two ends of our system to two heat baths at different temperatures. It is found that when the temperature difference is smaller than a critical value, the heat conductivity increases with the system size in power law with an exponent considerably smaller than 1. However, as the temperature difference exceeds the critical value, the system’s behavior undergoes a transition and the heat conductivity tends to diverge linearly with the system size. Correspondingly, an ordered structure emerges. These findings suggest a new direction for exploring the transport problems in one dimension.

Figure 1

The heat conductivity as a function of the system size for $h=10$ and for several temperature difference values $\mathrm{\Delta }>0$. As a comparison, the black open triangles connected by the dotted line segments are for the heat conductivity computed in the equilibrium state (corresponding to $\mathrm{\Delta }=0$) by using the Green-Kubo formula [20]. For reference, we show also the case $h=0$ for $\mathrm{\Delta }=0.2$ (open squares) and $\mathrm{\Delta }=1.0$ (plus signs) and the case $h=\infty$ for $\mathrm{\Delta }=0.2$ (open circles) and $\mathrm{\Delta }=1.0$ (crosses). The straight dot-dashed line is the result given by Eq. (3) up to the first order of $\mathrm{\Delta }/T$.

Figure 2

(a)–(c) The average mass of particles at position $x$, at the stationary state, for the same value $h=10$ as in Fig. 1. (a) $\mathrm{\Delta }=0.9>{\mathrm{\Delta }}_{\text{cr}}$, (b) $\mathrm{\Delta }=0.75={\mathrm{\Delta }}_{\text{cr}}$, and (c) $\mathrm{\Delta }=0.6<{\mathrm{\Delta }}_{\text{cr}}$. The corresponding temperature profile is shown in (d)–(f). In all six panels, the orange dotted, the gray dash-dotted, the red dashed, and the blue solid line are, respectively, for the system size $N=1600$, 3200, 6400, and 12800. The green vertical line in (a) indicates the position of the interface of the two species of particles if they are assumed to separate entirely.

Figure 3

The average particle mass $\overline{m}(0)$ at the leftmost end of the system for $h=10$. The dotted, dashed, and dot-dashed lines indicate, respectively, the scaling $\sim N^0$, $\sim N^{-1}$, and $\sim N^{-1.5}$.

Figure 4

The dependence of the critical temperature difference ${\mathrm{\Delta }}_{\text{cr}}$ on the interaction parameter $h$. Inset: Same as the main panel but in log-log scale.