2008/03/31 by P. Ao · 3 citations
Physics and Astronomy · Mathematics · Biochemistry, Genetics and Molecular Biology · #cond-mat.other #astro-ph #math-ph #math.DS #math.MP #nlin.AO #physics.bio-ph #q-bio.PE #q-bio.SC
paper · pdf · doi:10.1088/0305-4470/37/3/l01
published as Potential in Stochastic Differential Equations: Novel Construction. P. Ao, J. Phys. A37 L25-L30 (2004) · Latex, 9 pages
arxiv created 2008/03/31 · arxiv updated 2016/09/08
There is a whole range of emergent phenomena in non-equilibrium behaviors can be well described by a set of stochastic differential equations. Inspired by an insight gained during our study of robustness and stability in phage lambda genetic switch in modern biology, we found that there exists a classification of generic nonequilibrium processes: In the continuous description in terms of stochastic differential equations, there exists four dynamical elements: the potential function ϕ, the friction matrix S, the anti-symmetric matrix T , and the noise. The generic feature of absence of detailed balance is then precisely represented by T. For dynamical near a fixed point, whether or not it is stable or not, the stochastic dynamics is linear. A rather complete analysis has been carried out (Kwon, Ao, Thouless, cond-mat/0506280; PNAS, \bf 102 (2005) 13029), referred to as SDS I. One important and persistent question is the existence of a potential function with nonlinear force and with multiplicative noise, with both nice local dynamical and global steady state properties. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. First, we provide the construction. One of most important ingredient is the generalized Einstein relation. We then present an approximation scheme: The gradient expansion which turns every order into linear matrix equations. The consistent of such methodology with other known stochastic treatments will be discussed in next paper, SDS III; and the explicitly connection to statistical mechanics and thermodynamics will be discussed in a forthcoming paper, SDS IV.