### Abstract

Original language | English |
---|---|

Pages (from-to) | 569–574 |

Number of pages | 6 |

Journal | Chaotic Modelling & Simulation International Journal |

Volume | 2012 |

Issue number | 4 |

Publication status | Published - 2012 |

Externally published | Yes |

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### Keywords

- detailed balance
- non-equilibrium
- chaos
- complex systems

### Cite this

*Chaotic Modelling & Simulation International Journal*,

*2012*(4), 569–574.

**A master equation approach to deciphering non-detailed balance systems.** / Nicholson, S.; Kim, Eun-jin; Schulman, L. S.

Research output: Contribution to journal › Article

*Chaotic Modelling & Simulation International Journal*, vol. 2012, no. 4, pp. 569–574.

}

TY - JOUR

T1 - A master equation approach to deciphering non-detailed balance systems

AU - Nicholson, S.

AU - Kim, Eun-jin

AU - Schulman, L. S.

PY - 2012

Y1 - 2012

N2 - The world is filled with complex systems whether it is the traffic patterns in cities, weather patterns, information flow in the internet, or turbulence in fusion reactors. These complex systems are not often amenable to simple analytic solutions, understanding these systems requires a new statistical method beyond traditional equilibrium theory, i.e. Boltzmann Gibbs statistics. We present a novel method for understanding complex dynamics of such systems by using the Observable Representation which has been successfully applied to complex systems in detailed balance. Specifically we generalise it to non-equilibrium systems where detailed balance does not hold, i.e. the system has non zero currents. We construct a new transition matrix by accounting for this current and compute the eigenvalues and eigenvectors. From these, we define a metric whose distance provides a useful measure of correlation among variables. This is a very general method of understanding correlation in various systems, in particular, long-range correlation, or chaotic properties. As an example we show that these distances can be utilized to control chaos in a simple dynamical system given by the logistic map.

AB - The world is filled with complex systems whether it is the traffic patterns in cities, weather patterns, information flow in the internet, or turbulence in fusion reactors. These complex systems are not often amenable to simple analytic solutions, understanding these systems requires a new statistical method beyond traditional equilibrium theory, i.e. Boltzmann Gibbs statistics. We present a novel method for understanding complex dynamics of such systems by using the Observable Representation which has been successfully applied to complex systems in detailed balance. Specifically we generalise it to non-equilibrium systems where detailed balance does not hold, i.e. the system has non zero currents. We construct a new transition matrix by accounting for this current and compute the eigenvalues and eigenvectors. From these, we define a metric whose distance provides a useful measure of correlation among variables. This is a very general method of understanding correlation in various systems, in particular, long-range correlation, or chaotic properties. As an example we show that these distances can be utilized to control chaos in a simple dynamical system given by the logistic map.

KW - detailed balance

KW - non-equilibrium

KW - chaos

KW - complex systems

M3 - Article

VL - 2012

SP - 569

EP - 574

JO - Chaotic Modelling & Simulation International Journal

JF - Chaotic Modelling & Simulation International Journal

IS - 4

ER -