Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. II: construction of SDS with nonlinear force and multiplicative noise

By Ping Ao. 2004

174 citations recorded by the source corpus.

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 $\phi$, 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.

Subfields: Computer Networks and Communications, Genetics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1de1b5a50b0a599a18420ef918478bec0e9a91e7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

1. Karl J. Friston
2. Jin Wang
3. Ping Ao
4. Olivier David

In Genetics

1. Karl J. Friston
2. Jin Wang
3. Ting Lu
4. Xiaona Fang
5. Karsten Kruse

In Statistical and Nonlinear Physics

1. Karl J. Friston
2. Ting Lu
3. Xiaona Fang
4. Karsten Kruse
5. Jin Wang
6. Ping Ao
7. Melody Andrews
8. Andy Clark

Citations in the corpus

• Free Energy, Value, and Attractors (2011, background): “An important decomposition described in [37, 56], formulates the divergence-free part in terms of an antisymmetric matrix, Q(x̃) = −Q(x̃) and the scalar potential, which we will call value, such that”
• The free energy principle made simpler but not too simple (2022, unknown): “This component breaks detailed balance and renders the steady-state density a nonequilibrium steady-state density [31,32].”
• Nonequilibrium physics in biology (2019, background): “One approach aimed at finding a new type of stochastic dynamics (Ao, 2004), although it may be challenging to obtain numerical so-”
• A free energy principle for a particular physics (2019, methodology): “…(a.k.a., the fundamental theorem of vector calculus) and can be formulated in terms of an anti-symmetric matrix TQ Q= − and a scalar potential ( )x (Ao, 2004)5, ( )f Q= − (1.7) Using this standard form (Yuan et al., 2010), it is straightforward to show that ( ) exp( ( ))p x x= − is the…”
• Stochastic model of yeast cell-cycle network (2006, background): “In this language, the biological pathway can be viewed as a valley in the landscape [18,19,17].”