Influence of distinct kinds of temporal disorder in discontinuous phase transitions

Based on mean-field theory (MFT) arguments, a general description for discontinuous phase transitions in the presence of temporal disorder is considered. Our analysis extends the recent findings [C. E. Fiore et al., Phys. Rev. E 98, 032129 (2018)] by considering discontinuous phase transitions beyond those with a single absorbing state. The theory is exemplified in one of the simplest (nonequilibrium) order-disorder (discontinuous) phase transitions with “up-down” $Z_2$ symmetry: the inertial majority vote model for two kinds of temporal disorder. As for absorbing phase transitions, the temporal disorder does not suppress the occurrence of discontinuous phase transitions, but remarkable differences emerge when compared with the pure (disorderless) case. A comparison between the distinct kinds of temporal disorder is also performed beyond the MFT for random-regular complex topologies. Our work paves the way for the study of a generic discontinuous phase transition under the influence of an arbitrary kind of temporal disorder.

Figure 1

(a) Depicts the MFT phase diagram for a RR topology with $k_0=12$. ORD, ME, and DIS denote the ordered, metastable, and disordered phases, respectively. In (b), the steady magnetization $m$ versus $f$ for $\theta =0.45$. Continuous and dashed lines denote the stable and unstable solutions of Eq. (5), respectively. Their main features are exemplified in (c) and (d) by taking the time evolution of $m$ as a function of time for $f=0.016\left(c\right), f=0.10\left(d\right)$ and 0.06 (inset) for distinct initial conditions $m\left(0\right)$.

Figure 2

MFT temporal disorder analysis: For a RR network with $k_0=12, \theta =0.45$, and $\mathrm{\Delta }t=5$, the time evolution of $m$ for distinct sets of $f_{+}$ and $f_{-}$ and distinct independent realizations. (a)–(d) Exemplify the following cases: $(f_{-},f_{+})\in (\mathrm{ORD},\mathrm{ME})$ with $\overline{f}<f_b$ and $\overline{f}>f_b, (f_{-},f_{+})\in (\mathrm{ORD},\mathrm{DIS})$, and $(f_{-},f_{+})\in (\mathrm{ME},\mathrm{DIS})$, respectively. Dashed and symbol curves correspond to the pure versions (for $f=f_{+}$) and $m$ averaged over $N_D={10}^3$ realizations, respectively.

Figure 3

For a RR network with $k_0=12$ and $\theta =0.45$, (a) and (b) show $m$ averaged over $N_D={10}^3$ realizations for $\mathrm{\Delta }t=2,3,4,5$, and 6 for $(f_{-},f_{+})\in$ (ORD,DIS) and (ME,DIS), respectively.

Figure 4

MFT phase diagram for RR network with values $k_0=12$ and $\theta =0.45$ under temporal disorder over the control parameter $f$. The resulting phase is represented by distinct colors. Dotted and dashed lines represent crossovers and discontinuous transition lines, respectively.

Figure 5

MFT analysis for the temporal disorder in the inertia: For RR network with values $k_0=12$ and $f=0.12$, (a)–(c) exemplify the average time evolution of the $m$ for distinct initial configurations and sorts of inertia $({\theta }_{-},{\theta }_{+})\in$: (ORD,ME), (ORD,DIS), (ME,DIS), (ME,ME) (inset), respectively. In (d) the phase diagram with dashed and dotted lines representing discontinuous phase transition lines and crossover between phases, respectively. The resulting phase is represented by distinct colors.

Figure 6

For $k_0=20,f=0.12, {\theta }_{-}=0.334\in$ ME, and ${\theta }_{+}=0.412\in$ DIS, the average $m$ versus $t$ obtained for $N_D={10}^3$ disorder realizations and distinct $\mathrm{\Delta }t$'s.

Figure 7

(a) Depicts, for $N=5000,k_0=20$, and $\theta =0.3$, the order parameter $\left|m\right|$ versus $f$ for the pure system. Continuous and dotted lines denote the forward and backward increase of $f$, respectively. (b)–(d) Show the average time evolution of the order parameter $m$ (over $N_D={10}^4$ realizations) for distinct initial conditions $m\left(0\right)$ and different sorts of $[f_{-},f_{+}]\in$ [ME,ME], [ME,DIS], [ORD,DIS]. respectively. Due to finite-size effects, $m(t\to \infty )$ does not vanish in the disorder phase, but it is proportional to $1/\sqrt{N}$.

Figure 8

For a RR network of size $N={10}^4,k_0=20$, and $\theta =0.3$, panel (a) depicts the average time evolution of the order parameter $m$ starting from the ordered state for $f_{-}=0.10$ (ME) for distinct $f_{+}$'s. (b) Depicts, for $f=0.12$ and $m\left(0\right)=1$, the time evolution of the average $m$ for ${\theta }_{-}=0.30$ (ME) for distinct ${\theta }_{+}$'s. In (c) the same but for ${\theta }_{-}=0.28$ and 0.29 (both belonging to the ORD phase) and distinct ${\theta }_{+}$'s starting from $m\left(0\right)={10}^{-4}$. In all cases, averages are evaluated over $N_D={10}^3–{10}^4$ realizations. Due to finite-size effects, $m(t\to \infty )$ does not vanish in the disorder phase, but it is proportional to $1/\sqrt{N}$.