Theory of driven nonequilibrium critical phenomena

A system driven in the vicinity of its critical point by varying a relevant field in an arbitrary function of time is a generic system that possesses a long relaxation time compared with the driving time scale and thus represents a large class of nonequilibrium systems. For such a manifestly nonlinear nonequilibrium strongly fluctuating system, we show that there exists universal nonequilibrium critical behavior that is incredibly well described by its equilibrium critical properties. A dynamic renormalization-group theory is developed to account for the behavior. The weak driving may give rise to several time scales depending on its form and thus rich nonequilibrium phenomena of various regimes and their crossovers, negative susceptibilities, as well as a violation of fluctuation-dissipation theorem and hysteresis. An initial condition that can be in either equilibrium or nonequilibrium but has longer correlations than the driving scales also results in a unique regime and complicates the situation. The implication of the results on measurement is also discussed. The theory may shed light on the study of other nonequilibrium systems and even nonlinear science.

Figure 1

(a) Generic $H=R_3{t}^3-R_{1}t$ curve. (b) Schematic picture of its $t_H$ and other time scales vs $t$. (c) Schematic picture of $t_H$ and other time scales vs $t$ for the sinusoidal driving $H=Asin\mathrm{\Omega }t$. The red dash-dot lines correspond to the case in which the initial state is dominant, while the blue dash lines to the case in which equilibrium can be achieved. See the text for details.

Figure 2

Effects of (a) equilibrium initial conditions and (b) nonequilibrium initial conditions of the stepwise linear driving with $t_1{R}_{{\mathrm{st}}_1}^{z/r_1}=0.5$. On the right-hand side of each panel, the black curves are flat showing the irrelevance of the initial state; while the red curves have slopes of $-0.0674\left(3\right)$ in (a) and $-0.0294\left(3\right)$ in (b), consistent with the theoretic value $-1/\delta =-0.0666$ and $-\beta /\nu r_1=-0.0309$, respectively. On the left-hand side, the black curves depend on the initial conditions and the red curves tend to be flat, showing clearly the existence of the initial state dominated regime. Correspondingly, the slopes of the black curves become $0.0538\left(1\right)$ for the leftmost three data in (a) and $0.0311\left(2\right)$ in (b), close to the theoretical values $1/\delta$ and $\beta /\nu r_1$, respectively. The thick blue curve in (a) is a fit to the expansion of the scaling function in Eq. (19) to order 2. Note the different scales of the right and the left vertical axes. Thin lines connecting symbols are only a guide to the eye.

Figure 3

Magnetization of the three-step piecewise linear driving. $t_1{R}_{{\mathrm{st}}_1}^{z/r_1}=0.5$ and $R_{{\mathrm{st}}_1}/R_{{\mathrm{st}}_2}=0.2$. These choices yield $H_1{R}_{{\mathrm{st}}_2}^{-\beta \delta /\nu r_1}=0.2370<1$ and so $H_1$ lies inside the FTS regime of $R_{{\mathrm{st}}_2}$. The numbers list the values of $t_1$. The two vertical dashed lines demarcate the three steps. Inset: Original curves before rescaled.

Figure 4

Hysteresis loops for $H=R_2{t}^2$ (left column) and $H=R_3{t}^3$ (right column) for various listed $R_2$ and $R_3$, respectively. The original loops (top) collapse well after rescaled (bottom), verifying Eq. (25).

Figure 5

$M$ vs $t$ (top) and $M$ vs $H$ (bottom) for $H=R_n{t}^n$ with $n=2$ (left) and $n=3$ (right), respectively. The data are chosen to satisfy $\left|t\right|R_n^{z/r_n}<1$ near the critical point and are fitted to polynomials up to order 3 for $t>0$ and $t<0$, respectively. The goodnesses of the fits in (a) and (b) confirms the analyticity of $f_n^t$ in Eq. (24).

Figure 6

(a) $\chi$ and $C/T$ vs $H$ at equilibrium and (b) to (f) their rescaled for the driving $H=R_n{t}^n$ with (b) $n=0.5$, (c) 1, (d) 2, (e) 3, and (f) 29, respectively. Note that subscripts are absent in $R$ and $r$ of the axis titles. In (b) to (f), the red and blue curves, marked with $C/T$, are the rescaled curves $C{R}^{-\gamma /\nu r}/T$, while the green and black curves, marked with $\chi$, are the rescaled curves $\chi R^{-\gamma /\nu r}$. The solid and dashed curves represent the different rates listed. In (b) and (d), the solid and dashed curves correspond to $t<0$, while the dash-dotted curves to $t>0$ as indicated. Note that $\chi$ is negative for $t$ close to $0^{+}$. As predicted by Eq. (26), $\chi$ vanishes, is finite, and diverges at $H=0$ for $n<1$ (b), $n=1$ (c), and $n>1$ (d) to (f), respectively. The collapses of $C/T$ are not as good as $\chi$ due to large fluctuations. The two curves in (a) are averaged over 10 000 000 MC steps.

Figure 7

Hysteresis loops for $H=R_{1}t+R_3{t}^3$ with $R_1{R}_3^{-r_1/r_3}=0.1$. The legend lists $R_1$ used. (a) is original data and (b) is rescaled by $R_1$. The collapses get somehow poor when the absolute values of horizontal axis are large, possibly due to corrections to scaling.

Figure 8

Different FTS regimes and their crossover for (a) $H=R_{1}t+R_3{t}^3$ and (b) $H=R_3{t}^3-R_{1}t$ near $t_{-}$. Black and red curves are the same data rescaled by $R_1$ [$2R_1$ in (b)] and $R_3$, respectively. Each rescaled curve consists of a leading $R_1$ section (where $R_1{R}_3^{-r_1/r_3}\gg 1$), a leading $R_3$ section (where $R_1{R}_3^{-r_1/r_3}\ll 1$) with different slopes and a crossover between them (where $R_1{R}_3^{-r_1/r_3}\sim 1$). The slope of the black (red) curve in the $R_3$ ($R_1$) dominated regime is $-0.0305\left(1\right)$ [$0.0309\left(1\right)$] in (a) and $-0.0309\left(1\right)$ [$0.0299\left(4\right)$] in (b), consistent with theoretical absolute value of $\beta /\nu r_1$. No error bars appear in (b) as $M_0$ is obtained by interpolating the averaged magnetization curves at the first $H=0$. However, they cannot be appreciably larger than those displayed in Fig. 9 below. Lines connecting symbols are only a guide to the eye.

Figure 9

Different regimes and their crossover for $H=R_3{t}^3-R_{1}t$ near $t_0$. Black and red curves are the same data rescaled by $R_1$ and $R_3$, respectively. When $R_1{R}_3^{-r_1/r_3}\gg 1$, the process can be treated as a pure $R_1$ driving. As a result, the black curve is flat, whereas the slope of the red curve is $0.0310\left(5\right)$, in agreement with the theoretical value of $\beta /\nu r_1$. When $R_1{R}_3^{-r_1/r_3}\ll 1$, the red curve becomes flat, while the black curve has a slope $-0.0320\left(2\right)$, consistent again with $-\beta /\nu r_1$ from Eqs. (33) and (34). Lines connecting symbols are only a guide to the eye. Insets: Generic $M$ vs $H$ curves in the two corresponding regimes. Black solid, blue dashed, and red dash-dotted curves represent the 1, 2, and 3 parts of the process, respectively. In (a), the blue curve begins at $H>0$ and $M<0$, which is evidently nonequilibrium. In (b), the curves of the first (black) and the third (red) parts nearly coincide.

Figure 10

Hysteresis loops of the sinusoidal driving $H=Asin\mathrm{\Omega }t$. (a) For $A^{-1}{\mathrm{\Omega }}^{\beta \delta /\nu z}=0.1<1, M$ saturates at large $H$. (b) For $A^{-1}{\mathrm{\Omega }}^{\beta \delta /\nu z}=10>1, M$ does not saturate. (c) The hysteresis loops in (b) rescaled by $\mathrm{\Omega }$. The legend in (c) lists $\mathrm{\Omega }$ used in (b) and (c).

Figure 11

Different regimes and their crossover for $H=Asin\mathrm{\Omega }t$. Rescaled of the same data (a) by $A\mathrm{\Omega }$ (black squares) and $A$ (red circles) and (b) by $A\mathrm{\Omega }$ (black squares) and $\mathrm{\Omega }$ (magenta triangles). The black curves are identical in (a) and (b). The magenta curve is not flat in any regime, implying that $\mathrm{\Omega }$ does not dominate. The black curves are flat at ${\mathrm{\Omega }}^{-1}{A}^{\nu z/\beta \delta }\gg 1$, indicating the dominance of $t_{A\mathrm{\Omega }}$ in this regime. Correspondingly, the slopes of red and magenta curves are $-0.0312\left(2\right)$ and $0.0559\left(4\right)$, in agreement with the theoretic values of $-\beta /\nu r_1$ and $\beta /\nu z=0.0577$ according to Eqs. (42) and (44) and Eqs. (40) and (44), respectively. For ${\mathrm{\Omega }}^{-1}{A}^{\nu z/\beta \delta }\ll 1$, the red curve becomes flat and the slopes of the black and magenta curves are $0.0289\left(5\right)$ and $0.0269\left(2\right)$, consistent with the theoretic values of $\beta /\nu r_1$ and $(1/z-1/r_1)\beta /\nu =0.0268$, respectively, showing the importance of the initial condition. Lines connecting symbols are only a guide to the eye.