A field-theoretic approach to the Wiener Sausage
File(s)
Accepted version
Published version
Author(s)
Nekovar, S
Pruessner, G
Type
Journal Article
Abstract
The Wiener Sausage, the volume traced out by a sphere attached
to a Brownian particle, is a classical problem in statistics and mathematical
physics. Initially motivated by a range of field-theoretic, technical questions,
we present a single loop renormalised perturbation theory of a stochastic
process closely related to the Wiener Sausage, which, however, proves to be
exact for the exponents and some amplitudes. The field-theoretic approach is
particularly elegant and very enjoyable to see at work on such a classic problem.
While we recover a number of known, classical results, the field-theoretic
techniques deployed provide a particularly versatile framework, which allows
easy calculation with different boundary conditions even of higher momenta
and more complicated correlation functions. At the same time, we provide a
highly instructive, non-trivial example for some of the technical particularities
of the field-theoretic description of stochastic processes, such as excluded
volume, lack of translational invariance and immobile particles. The aim of
the present work is not to improve upon the well-established results for the
Wiener Sausage, but to provide a field-theoretic approach to it, in order to
gain a better understanding of the field-theoretic obstacles to overcome.
to a Brownian particle, is a classical problem in statistics and mathematical
physics. Initially motivated by a range of field-theoretic, technical questions,
we present a single loop renormalised perturbation theory of a stochastic
process closely related to the Wiener Sausage, which, however, proves to be
exact for the exponents and some amplitudes. The field-theoretic approach is
particularly elegant and very enjoyable to see at work on such a classic problem.
While we recover a number of known, classical results, the field-theoretic
techniques deployed provide a particularly versatile framework, which allows
easy calculation with different boundary conditions even of higher momenta
and more complicated correlation functions. At the same time, we provide a
highly instructive, non-trivial example for some of the technical particularities
of the field-theoretic description of stochastic processes, such as excluded
volume, lack of translational invariance and immobile particles. The aim of
the present work is not to improve upon the well-established results for the
Wiener Sausage, but to provide a field-theoretic approach to it, in order to
gain a better understanding of the field-theoretic obstacles to overcome.
Date Issued
2016-03-09
Date Acceptance
2016-02-15
Citation
Journal of Statistical Physics, 2016, 163 (3), pp.604-641
ISSN
0022-4715
Publisher
Springer Verlag (Germany)
Start Page
604
End Page
641
Journal / Book Title
Journal of Statistical Physics
Volume
163
Issue
3
Copyright Statement
© The Author(s) 2016. This article is published with open access at Springerlink.com
License URL
Subjects
Wiener Sausage problem
field theory
random walks
Publication Status
Published