### Abstract

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists’ intuition. In terms of Freidlin-Wentzell’s LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.

Stochastic approaches to nonlinear dynamics have attracted great interests from physicists, biologists, and mathematicians in current research. More than 70 yr ago, Kramers has developed a diffusion model characterizing the molecular dynamics along a reaction coordinate, via a barrier crossing mechanism, and calculated the reaction rate for an emergent chemical reaction. The work explained the celebrated Arrhenius relation as well as Eyring’s concept of “transition state”. Kramers’ theory, however, is only valid for stochastic dynamics in closed systems with detailed balance (i.e., a gradient flow), where the energy landscape gives the equilibrium stationary distribution via Boltzmann’s law. It is not suitable for models of open systems. Limit cycle oscillation is one of the most important emergent behaviors of nonlinear, non-gradient systems. The large deviation theory (LDT) from probability naturally provides a basis for the concept of a “landscape” in a deterministic nonlinear, non-gradient dynamics. In the present study, we initiate a line of studies on the dynamics of and emergent landscape in open systems. In particular, using singularly perturbed diffusion on a circle as a model, we study systems with stable limit cycle as well as systems with multiple attractors with nonzero flux. A seeming paradox concerning emergent landscape for limit cycle is resolved; a local theory for transitions between two adjacent attractors, à la Kramers, is discussed; and a “

*λ*-surgery” to obtain nonequilibrium steady state (NESS) landscape for multiple attractors is described.

H.Q. thanks Ping Ao, Bernard Deconinck, Gang Hu, Rachel Kuske, Robert O’Malley, and Jin Wang for many fruitful discussions. H.G. acknowledges support by NSFC 10901040, specialized Research Fund for the Doctoral Program of Higher Education (New Teachers) 20090071120003 and the Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 201119).

I. INTRODUCTION

II. GENERAL STATIONARY SOLUTION AND WKB APPROXIMATION

III. THE THEORY OF DIFFUSION ON A CIRCLE

A. A simple example of diffusion on a circle

IV. GENERAL DERIVATION FOR HIGH DIMENSIONAL SYSTEMS WITH A LIMIT CYCLE

A. Beyond limit cycle

V. LOCAL AND GLOBAL LANDSCAPES IN THE CASE OF MULTIPLE ATTRACTORS AND EMERGENT NONEQUILIBRIUM STEADY STATE

VI. CONCLUSIONS

### Key Topics

- Attractors
- 17.0
- Diffusion
- 16.0
- Nonlinear dynamics
- 14.0
- Probability theory
- 8.0
- Chemical dynamics
- 6.0

## Figures

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 .

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 .

(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 .

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.

(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.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content