Geometric structure and information change in phase transitions

We propose a toy model for a cyclic order-disorder transition and introduce a geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of forward and backward processes (FPs and BPs) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent probability density functions (PDFs) and the information length $\mathcal{L}$, which is the total number of different states that a system undergoes during the transition. Time-dependent PDFs during transient relaxation exhibit strikingly different behavior in FPs and BPs. In particular, FPs driven by instability undergo the broadening of the PDF with a large increase in fluctuations before the transition to the ordered state accompanied by narrowing the PDF width. During this stage, we identify an interesting geodesic solution accompanied by the self-regulation between the growth and nonlinear damping where the time scale $\tau$ of information change is constant in time, independent of the strength of the stochastic noise. In comparison, BPs are mainly driven by the macroscopic motion due to the movement of the PDF peak. The total information length $\mathcal{L}$ between initial and final states is much larger in BPs than in FPs, increasing linearly with the deviation $\gamma$ of a control parameter from the critical state in BPs while increasing logarithmically with $\gamma$ in FPs. $\mathcal{L}$ scales as $|lnD|$ and $D^{-1/2}$ in FPs and BPs, respectively, where $D$ measures the strength of the stochastic forcing. These differing scalings with $\gamma$ and $D$ suggest a great utility of $\mathcal{L}$ in capturing different underlying processes, specifically, diffusion vs advection in phase transition by geometry. We discuss physical origins of these scalings and comment on implications of our results for bistable systems undergoing repeated order-disorder transitions (e.g., fitness).

Figure 1

The ratio $\sqrt{\langle x^4\rangle }/\langle x^2\rangle$ as a function of time, for the three values $D={10}^{-3}, {10}^{-5}$, and ${10}^{-7}$, from left to right as indicated. Solid lines denote FP, dashed lines BP. The dots on the lines correspond to the particular snapshots shown in Figs. 2 and 3. The dots on the $D={10}^{-3}$ curves (red) are at $t=1.26$, 2.26, 3.26, 4.26, 6.26, and 8.26; the dots on the $D={10}^{-5}$ curves (blue) are at $t=4.48$, 5.48, 6.48, 7.48, 9.48, and 11.48; the dots on the $D={10}^{-7}$ curves (black) are at $t=7.77$, 8.77, 9.77, 10.77, 12.77, and 14.77. The times 3.26, 6.48, and 9.77 are the values where corresponding solid and dashed lines cross; the other times are then related by fixed offsets either before or after these crossing times. Finally, $\gamma =0.7$ for all six curves.

Figure 2

The PDFs for FP, for (a) $D={10}^{-3}$, (b) $D={10}^{-5}$, (c) $D={10}^{-7}$. The initial condition is given by the central peak. At later times this central value $p(0,t)$ monotonically decreases. The particular times shown are as indicated by the dots in Fig. 1. The thick (black) lines where the central region is flattest correspond to the crossing points in Fig. 1, at times 3.26 in (a), 6.48 in (b), 9.77 in (c); the times for other lines are offset relative to these values as in Fig. 1. Note how the curves at intermediate times are independent of $D$, once this $cln{D}^{-1}$ shift in time is taken into account.

Figure 3

The PDFs for BP, for (a) $D={10}^{-3}$, (b) $D={10}^{-5}$, (c) $D={10}^{-7}$, at the times indicated in Fig. 1. At the earliest time shown there are still two distinct peaks, which subsequently merge into one single peak. As in Fig. 2, the thick (black) lines where the central region is relatively flat correspond to the crossing points in Fig. 1, at times 3.26 in (a), 6.48 in (b), 9.77 in (c). The times for other lines are again offset relative to these values as in Fig. 1. Note how the curves are independent of $D$, once the shift in time is taken into account, as well as the rescaling of $x$ and $p$.

Figure 4

The first row is for FP, with (a) $D={10}^{-3}$, (b) $D={10}^{-5}$, (c) $D={10}^{-7}$; the second row is for BP, with (d) $D={10}^{-3}$, (e) $D={10}^{-5}$, (f) $D={10}^{-7}$. Dashed curves show $|\frac{d}{dt}(\langle x^2\rangle /2)|$, solid curves $\gamma \langle x^2\rangle$, and dash-dotted curves $\langle x^4\rangle$. The dotted lines show the residuals $\frac{d}{dt}(\langle x^2\rangle /2)\mp \gamma \langle x^2\rangle +\langle x^4\rangle$, with $-$ for FP and $+$ for BP. Notice how the residuals are always exactly $D$.

Figure 5

Panels (a) and (b) show $\mathcal{E}$ and $\mathcal{L}$, respectively, as functions of time, for $\gamma =0.7$. Panel (c) shows ${\mathcal{L}}_{\infty }$ as a function of $\gamma$. All three panels are for FP only. $D={10}^{-3}$ to ${10}^{-7}$ as indicated. Note the different combinations of linear and logarithmic scales to emphasize different features in different quantities.

Figure 6

Panels (a) and (b) show $\mathcal{E}$ and $\mathcal{L}$, respectively, as functions of time, for $\gamma =0.7$. Panel (c) shows ${\mathcal{L}}_{\infty }$ as a function of $\gamma$. All three panels are for BP only. $D={10}^{-3}$ to ${10}^{-7}$ as indicated. Note the different combinations of linear and logarithmic scales to emphasize different features in different quantities.

Figure 7

The entropy $S$ as a function of time, for the three values $D={10}^{-3}, {10}^{-5}$, and ${10}^{-7}$, as indicated. Solid lines denote FP, dashed lines BP.