Glass transition and random walks on complex energy landscapes

We present a simple mathematical model of glassy dynamics seen as a random walk in a directed weighted network of minima taken as a representation of the energy landscape. Our approach gives a broader perspective to previous studies focusing on particular examples of energy landscapes obtained by sampling energy minima and saddles of small systems. We point out how the relation between the energies of the minima and their number of neighbors should be studied in connection with the network’s global topology and show how the tools developed in complex network theory can be put to use in this context.

Figure 1

(Color online) Evolution of $P\left(k;t_w\right)$ for an uncorrelated scale-free network. Here $N={10}^6\left(k_c={10}^3\right)$, $\gamma =3$, and $\beta E_0=2$ so that $P\left(k;t_w\right)\sim k^{-2}$ at short times and $P_{eq}\left(k\right)\sim k^0$. Inset: $t_w^{1/2}P\left(k;t_w\right)$ vs $k/t_w^{1/2}$ for $t_w<t_{eq}\sim {10}^6$.

Figure 2

(Color online) Top: average escape time $t_{esc}\left(t_w\right)$ divided by the large time prediction Eq. (7) for various $N$ and $\beta$. Bottom: $C\left(t_w+t,t_w\right)$ vs $t/t_w$ for an uncorrelated scale-free network of $N={10}^6$ minima. Here $\gamma =3$ and $\beta E_0=4$. Inset: $C$ vs $t$.

Figure 3

(Color online) MFPT for scale-free uncorrelated random networks. Here $E_0=1$. Top: $\gamma =2.2$ and $N={10}^5$; both Eq. (9) and a VTF fit (with $T_0\approx 0.023$) are shown. Bottom: $\gamma =3$ and various network sizes. For $\beta <{\beta }_c=1$, $\tau \propto N$, while $\tau \propto N^{2+\left(\beta E_0-\gamma \right)/2}$ for $\beta >{\beta }_c$.