Landscape and related Kramers’ rate theory for a bistable system. The local minima correspond to stable fixed points of a deterministic dynamics while the maximum corresponds to an unstable fixed (saddle) point. The V(1,2) and V(2,1) represent the energy barriers for exiting energy wells 1 and 2, respectively. For very small , the stationary probability distribution for stochastic dynamics with Brownian motion B(t), , is . The Kramers theory yields transition rates between the two attractors: and . According to Freidlin-Wentzell’s LDT, this theory still applies for every pair of neighbouring attractors of a non-gradient system in terms of a local landscape. However, the stationary probability distribution follows a different, global landscape. Also see Fig. 4 .
(a) The thin solid line is , where f is represented by theslope of the dashed line. The thick solid line is . When combining with , as shown in (b), one obtains given in (c). is periodic but contains non-differentiable points. If is monotonically decreasing, then and .
The limiting distribution according to Eq. (23) for nonlinear dynamics on a circle , with f = 5, 2, 1.1, and 1.05. With , it approaches to . For , the distribution is where .
(a) Pairwise local landscapes; (b) a simple “pasting together” leads to discontinuous matched case; (c) the global landscape is obtained by a “ -surgery and pasting” procedure: The surgery lifts the well-2 with respect to well-1 an amount of , which is precisely the free energy difference for well-2 with respect to well-1 in nonequilibrium steady state. 43 Similarly, it lifts the amount of for well-3 with respect to well-2, and for well-1 with respect to well-3. Therefore, the total lift is ; (d)the final global landscape. Note that –V(x) in Fig. 2(c) is just one example of such a global landscape. It is a piecewise smooth function with “flat regions” at its local maxima.
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