### Abstract

We obtain the stationary probability distribution functions of the order parameter near onset for the one-dimensional real Ginzburg-Landau and Swift-Hohenberg equations with a fluctuating control parameter. A perturbative expansion in the intensity of the fluctuations leads to a hierarchy of Fokker-Planck equations for conditional probability distribution functions that relate components of the order parameter that evolve in different time scales. Successive integration leads to a Fokker-Planck equation for the slowest mode, which we solve analytically for the models studied. In all cases, the probability distribution function above onset is of the form (Formula presented) where (Formula presented) is the slow component of the order parameter and the values of δ and γ depend explicitly on the intensity of the fluctuations. Knowledge of (Formula presented) allows the calculation of an effective bifurcation threshold and of the moments of (Formula presented) above threshold.

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

Article number | 026120 |

Pages (from-to) | 261201-261208 |

Number of pages | 8 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2001 |

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### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*64*(2), 261201-261208. [026120]. https://doi.org/10.1103/PhysRevE.64.026120

**Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter.** / Drolet, François; Viñals, Jorge.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 64, no. 2, 026120, pp. 261201-261208. https://doi.org/10.1103/PhysRevE.64.026120

}

TY - JOUR

T1 - Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter

AU - Drolet, François

AU - Viñals, Jorge

PY - 2001/8

Y1 - 2001/8

N2 - We obtain the stationary probability distribution functions of the order parameter near onset for the one-dimensional real Ginzburg-Landau and Swift-Hohenberg equations with a fluctuating control parameter. A perturbative expansion in the intensity of the fluctuations leads to a hierarchy of Fokker-Planck equations for conditional probability distribution functions that relate components of the order parameter that evolve in different time scales. Successive integration leads to a Fokker-Planck equation for the slowest mode, which we solve analytically for the models studied. In all cases, the probability distribution function above onset is of the form (Formula presented) where (Formula presented) is the slow component of the order parameter and the values of δ and γ depend explicitly on the intensity of the fluctuations. Knowledge of (Formula presented) allows the calculation of an effective bifurcation threshold and of the moments of (Formula presented) above threshold.

AB - We obtain the stationary probability distribution functions of the order parameter near onset for the one-dimensional real Ginzburg-Landau and Swift-Hohenberg equations with a fluctuating control parameter. A perturbative expansion in the intensity of the fluctuations leads to a hierarchy of Fokker-Planck equations for conditional probability distribution functions that relate components of the order parameter that evolve in different time scales. Successive integration leads to a Fokker-Planck equation for the slowest mode, which we solve analytically for the models studied. In all cases, the probability distribution function above onset is of the form (Formula presented) where (Formula presented) is the slow component of the order parameter and the values of δ and γ depend explicitly on the intensity of the fluctuations. Knowledge of (Formula presented) allows the calculation of an effective bifurcation threshold and of the moments of (Formula presented) above threshold.

UR - http://www.scopus.com/inward/record.url?scp=85035263790&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevE.64.026120

DO - 10.1103/PhysRevE.64.026120

M3 - Article

AN - SCOPUS:0035420706

VL - 64

SP - 261201

EP - 261208

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 2

M1 - 026120

ER -