Scaling of hysteresis loops at phase transitions into a quasiabsorbing state

Models undergoing a phase transition to an absorbing state weakly broken by the addition of a very low spontaneous nucleation rate are shown to exhibit hysteresis loops whose width $\mathrm{\Delta }\lambda$ depends algebraically on the ramp rate $r$. Analytical arguments and numerical simulations show that $\mathrm{\Delta }\lambda \sim r^{\kappa }$ with $\kappa =1∕({\beta }^{\prime }+1)$, where ${\beta }^{\prime }$ is the critical exponent governing the survival probability of a seed near threshold. These results explain similar hysteresis scaling observed before in liquid crystal convection experiments. This phenomenon is conjectured to occur in a variety of other experimental systems.

Figure 1

Hysteresis observed in simulations and experiments. (a) Hysteresis of $(2+1)\mathrm{D}$ CP with $h^{\prime }={10}^{-2}$. Black regions denote active sites. The control parameter $\lambda$ is increased and then decreased in the range of $1\le \lambda \le 3.4$ at the ramp rate of $r=0.001{\mathrm{MCS}}^{-1}$. The critical point for the model without nucleation is ${\lambda }_{\mathrm{c}}=1.64877\left(3\right)$ [9]. The arrows after the values of $\lambda$ denote whether they are increasing or decreasing. (b) Hysteresis in the electrohydrodynamic convection, where the same cell as in Ref. 4 is used. The control parameter, applied voltage $V$, is ramped in the range of $22\mathrm{V}\le V\le 75\mathrm{V}$ at the rate of $r=1.71\mathrm{V}∕\mathrm{s}$ with fixed frequency of $250\mathrm{Hz}$. The critical voltage is $V_{\mathrm{c}}\approx 35\mathrm{V}$ [4]. Darker regions correspond to DSM2, the active state. Note that the global intensity and contrast are adjusted for the sake of clarity, and that DSM1 and DSM2 coexist in the two images at the lower left.

Figure 2

(Color online) Typical hysteresis loops for four different ramp rates $r$ in $(2+1)\mathrm{D}$ CP with $h^{\prime }={10}^{-2}$. Note that the ratios of the four values of $r$ are chosen to be approximately the same as in Fig. 1 of Ref. 6 to allow the comparison (see also Ref. [12]). The hysteretic process can be decomposed into three stages as indicated in the bottom figure.

Figure 3

(Color online) Widths of the loops $\mathrm{\Delta }\lambda$ (disk), $\mathrm{\Delta }{\lambda }^{*}$ (triangle), and $\mathrm{\Delta }{\lambda }_t^{*}$ (square) with respect to the ramp rate $r$, in the case of $(2+1)\mathrm{D}$ CP with $h^{\prime }={10}^{-2}$. The symbols and error bars indicate means and standard deviations, respectively, of 50 independent runs. Dashed curves denote the results of the fitting to the power law $\mathrm{\Delta }\lambda$, $\mathrm{\Delta }{\lambda }^{*}$, $\mathrm{\Delta }{\lambda }_t^{*}\sim r^{\kappa }$. The inset shows the same data in logarithmic scales.

Figure 4

(Color online) Hysteresis scaling for $(1+1)\mathrm{D}$ CP (a), $(2+1)\mathrm{D}$ voterlike (b), $(2+1)\mathrm{D}$ CP with quadratic ramping $\epsilon \left(t\right)=r^{\prime }{t}^2$ (c), and $(2+1)\mathrm{D}$ CP with square root ramping $\epsilon \left(t\right)=r^{\prime }{t}^{1∕2}$ (d). The estimates of the exponent $\kappa$ for $\mathrm{\Delta }{\lambda }_t^{*}$ agree with the theoretical values, namely, 0.783 (a), 0.5 (b), 0.462(2) (c), and 0.774(2) (d).