Figure 1

Ising model stochastic dynamics. $A_1$ vs magnetization. On the top is shown $T>1$, on the bottom $T<1$. In this figure an $L^2$ normalization is used for $A_k$ (see Ref. 5).Reuse & Permissions

Figure 2

Plot using the first two left eigenvectors ($A_1$ and $A_2$) of the transition matrix, $R$, for a three-phase system. A circle is placed at each point $\left[A_1\right(x),A_2(x\left)\right]$ for each of the $N$ states, $x$, in $X$. The lines connecting the circles are for visualization. The matrix $R$ is generated by combining four blocks, 3 of which are random matrices, the fourth essentially zero. Then a bit of noise is added throughout, with bigger terms for migration out of the fourth block. Finally the diagonal is adjusted to make the matrix stochastic. This leads to a pair of eigenvalues near 1. This plot using the first two eigenvectors shows the extremal points to be clustering in three regions, corresponding to the phases. The points not at the extremals represent the fourth block, all of which head toward one or another phase under the dynamics. For the particular matrix chosen, they are about as likely to end in one phase as another. For all eigenvector plots, the quantities plotted are pure numbers whose scale is set by our normalization convention, discussed in Sec. 2B.Reuse & Permissions

Figure 3

Detail of the upper left vertex in Fig. 2. In actuality the points in each phase cluster closely together and more than one extremal might, in principle, occur. The precision is limited by the non-negligible magnitudes of the quantities $1-{\lambda }_2$ and ${\lambda }_3$.Reuse & Permissions

Figure 4

The first few eigenvalues of $W$ for the four phase system. For those eigenvalues having an imaginary part (which is not the case for the first four), only the real part is shown.Reuse & Permissions

Figure 5

(Color online) Convex hull of the set of points $\mathit{A}\left(y\right)$ for $y∊X$. This is for a case of four phases and the figure formed in ${\mathit{R}}^3$ is a tetrahedron.Reuse & Permissions

Figure 6

(Color online) For the four-phase case, if one plots only in two dimensions one does not see only three extremal points. The fourth apparently sticks out of the triangle formed by the others (although it did not have to) and in a third dimension actually forms a node of a tetrahedron (Fig. 5).Reuse & Permissions

Figure 7

Phases at successively later times for a hierarchical stochastic matrix. As explained in the text this is a two-dimensional projection of the five-dimensional plot of eigenvectors multiplied by eigenvalue to the power $t$. On the shortest time scale there are 6 metastable phases (circles); subsequently they merge into three and finally into a single stationary state. The axes represent particular linear combinations of eigenvectors and lack physical dimensions (but have a scale determined by the normalization of Sec. 2B).Reuse & Permissions

Figure 8

(Color online) Landscape for a random walk on a 15 by 15 lattice. The distance unit on the lattice is arbitrary and the scale of the potential chosen so as to give a dynamical spectrum illustrating our representation.Reuse & Permissions

Figure 9

Probability distribution for the stationary state of a walk on the landscape shown in Fig. 8.Reuse & Permissions

Figure 10

Observable representation, in ${\mathit{R}}^3$, of the states for the walk on the landscape shown in Fig. 8. Each circle represents a point on the 15 by 15 lattice and its position within the tetrahedron (when expressed in barycentric coordinates with respect to the extremals) gives the probability of starting at that point and arriving at one or another extremal. The plot is very much like that shown in Fig. 5, but includes interior points. (Fig. 5 shows only the convex hull.)Reuse & Permissions

Figure 11

Observable representation, in ${\mathit{R}}^2$, of the states for Brownian motion. The circle is for the one-dimensional case of a walk on a ring. The rectangular figure is for a two-dimensional random walk with nonperiodic boundary conditions.Reuse & Permissions