Pitt Logo LinkContact Us

Analytical and Numerical Results on Escape of Brownian Particles

Caginalp, Carey (2011) Analytical and Numerical Results on Escape of Brownian Particles. Undergraduate Thesis, University of Pittsburgh.

[img] PDF - Primary Text
Restricted to University of Pittsburgh users only until 29 April 2016.
Download (589Kb) | Request a copy

    Abstract

    A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called "window" or "gate") in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is given for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and sufficient amount of sample paths are compared with the exact solution to the Brownian motion (the limit of zero stepsize), providing an empirical formula for the divergence. Histograms of first hitting times are also generated.

    Share

    Citation/Export:
    Social Networking:
    Share |

    Details

    Item Type: University of Pittsburgh ETD
    ETD Committee:
    ETD Committee TypeCommittee MemberEmail
    Committee ChairChen, Xinfuxinfu@pitt.edu
    Committee MemberWang, Dehuadhwang@pitt.edu
    Committee MemberJiang, Huiqianghqjiang@pitt.edu
    Committee MemberSlemrod, Marshallslemrod@math.wisc.edu
    Title: Analytical and Numerical Results on Escape of Brownian Particles
    Status: Unpublished
    Abstract: A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called "window" or "gate") in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is given for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and sufficient amount of sample paths are compared with the exact solution to the Brownian motion (the limit of zero stepsize), providing an empirical formula for the divergence. Histograms of first hitting times are also generated.
    Date: 29 April 2011
    Date Type: Completion
    Defense Date: 28 January 2011
    Approval Date: 29 April 2011
    Submission Date: 17 March 2011
    Access Restriction: 5 year -- Restrict access to University of Pittsburgh for a period of 5 years.
    Patent pending: No
    Institution: University of Pittsburgh
    Thesis Type: Undergraduate Thesis
    Refereed: Yes
    Degree: BPhil - Bachelor of Philosophy
    URN: etd-03172011-191325
    Uncontrolled Keywords: Brownian motion; stochastics
    Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
    University Honors College
    Date Deposited: 10 Nov 2011 14:32
    Last Modified: 02 Mar 2012 12:10
    Other ID: http://etd.library.pitt.edu/ETD/available/etd-03172011-191325/, etd-03172011-191325

    Actions (login required)

    		 View Item

    Document Downloads