### Abstract

The incidence of rare events in fast–slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior—such fluctuations are rare on short time-scales but become ubiquitous eventually. Classical results prove that this LDP involves an Hamilton–Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta—in other words, the LDP for the slow variables in fast–slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.

Original language | English (US) |
---|---|

Pages (from-to) | 793-812 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 162 |

Issue number | 4 |

DOIs | |

State | Published - Feb 1 2016 |

### Fingerprint

### Keywords

- Hamilton–Jacobi equation
- Limit theorems
- Quasipotential
- Rare events

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*162*(4), 793-812. https://doi.org/10.1007/s10955-016-1449-4

**Large Deviations in Fast–Slow Systems.** / Bouchet, Freddy; Grafke, Tobias; Tangarife, Tomás; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 162, no. 4, pp. 793-812. https://doi.org/10.1007/s10955-016-1449-4

}

TY - JOUR

T1 - Large Deviations in Fast–Slow Systems

AU - Bouchet, Freddy

AU - Grafke, Tobias

AU - Tangarife, Tomás

AU - Vanden Eijnden, Eric

PY - 2016/2/1

Y1 - 2016/2/1

N2 - The incidence of rare events in fast–slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior—such fluctuations are rare on short time-scales but become ubiquitous eventually. Classical results prove that this LDP involves an Hamilton–Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta—in other words, the LDP for the slow variables in fast–slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.

AB - The incidence of rare events in fast–slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior—such fluctuations are rare on short time-scales but become ubiquitous eventually. Classical results prove that this LDP involves an Hamilton–Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta—in other words, the LDP for the slow variables in fast–slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.

KW - Hamilton–Jacobi equation

KW - Limit theorems

KW - Quasipotential

KW - Rare events

UR - http://www.scopus.com/inward/record.url?scp=84957435071&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957435071&partnerID=8YFLogxK

U2 - 10.1007/s10955-016-1449-4

DO - 10.1007/s10955-016-1449-4

M3 - Article

AN - SCOPUS:84957435071

VL - 162

SP - 793

EP - 812

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -