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Structure of stochastic dynamics near fixed points.
Chulan Kwon1, Ping Ao, David J Thouless
1Department of Physics, Myongji University, Namdong San 38-2, Yongin, Kyonggi-Do, 449-728, Republic of Korea.
Summary
A new cost function defines a Boltzmann-like stationary distribution for stochastic dynamics near fixed points. This distribution may not satisfy detailed balance, instead featuring a divergence-free probability current.
Area of Science:
- Physics
- Physical Chemistry
- Statistical Mechanics
Background:
- Stochastic dynamics near fixed points are fundamental in various scientific fields.
- Traditional analysis often relies on the detailed balance condition for stationary distributions.
Purpose of the Study:
- To investigate the structure of stochastic dynamics near stable or unstable fixed points.
- To explore the existence and properties of stationary distributions beyond detailed balance.
Main Methods:
- Linear approximation of the force near fixed points.
- Definition and construction of a cost function.
- Application of Jordan transformation for the force matrix.
Main Results:
- A Boltzmann-like stationary distribution can always be defined near a fixed point.
- This distribution may exhibit a divergence-free probability current instead of detailed balance.
- An explicit construction of the cost function is provided via Jordan transformation.
Conclusions:
- The identified cost function and resulting stationary distribution offer a generalized framework for stochastic dynamics.
- Singularities in the transformation and nonlinear effects can influence the uniqueness and properties of the stationary distribution.