### Abstract

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e. g. Brownian motion. We study the limit where friction effects dominate the inertia, i. e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.

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

Pages (from-to) | 762-773 |

Number of pages | 12 |

Journal | Journal of Statistical Physics |

Volume | 146 |

Issue number | 4 |

DOIs | |

State | Published - Feb 2012 |

### Fingerprint

### Keywords

- Brownian motion
- Einstein mobility-diffusion relation
- Smoluchowski-Kramers approximation
- Stochastic differential equations

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit.** / Hottovy, Scott; Volpe, Giovanni; Wehr, Jan.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 146, no. 4, pp. 762-773. https://doi.org/10.1007/s10955-012-0418-9

}

TY - JOUR

T1 - Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

AU - Hottovy, Scott

AU - Volpe, Giovanni

AU - Wehr, Jan

PY - 2012/2

Y1 - 2012/2

N2 - We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e. g. Brownian motion. We study the limit where friction effects dominate the inertia, i. e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.

AB - We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e. g. Brownian motion. We study the limit where friction effects dominate the inertia, i. e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.

KW - Brownian motion

KW - Einstein mobility-diffusion relation

KW - Smoluchowski-Kramers approximation

KW - Stochastic differential equations

UR - http://www.scopus.com/inward/record.url?scp=84858340353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858340353&partnerID=8YFLogxK

U2 - 10.1007/s10955-012-0418-9

DO - 10.1007/s10955-012-0418-9

M3 - Article

AN - SCOPUS:84858340353

VL - 146

SP - 762

EP - 773

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -