The Renormalization group and fractional Brownian motion

We find that in generic field theories the combined effect of fluctuations and interactions leads to a probability distribution function which describes fractional Brownian Motion (fBM) and ``complex behavior''. To show this we use the Renormalization Group as a tool to improve perturbative calculations, and check that beyond the classical regime of the field theory (i. e., when no fluctuations are present) the non--linearities drive the probability distribution function of the system away from classical Brownian Motion and into a regime which to the lowest order is that of fBM. Our results can be applied to systems away from equilibrium and to dynamical critical phenomena. We illustrate our results with two selected examples: a particle in a heat bath, and the KPZ equation.

Note:

14 pages (REVTeX4), no figures Subj-class: Statistical Mechanics: Chaotic Dynamics Journal-ref: Physics Letters A296 (2002) 272

statistics: fractional
renormalization group
fluctuation: effect
interaction: effect
effect: nonlinear
perturbation theory
critical phenomena
asymptotic behavior
correlation function

References(28)

Figures(0)

[1]

Complex Systems

- Science 284 (1999) 2

[2]

H. Scher
,
M. Shlesinger
,
J.T. Bendler

- Phys.Today 44 (1991) 26

[3]

N. Goldenfeld
,
L.P. Kadanoff

- Science 284 (1999) 87

[4]

The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, New York, 1993

S.A. Kauffman

[5]

The Fractal Geometry of Nature, Freeman, San Francisco, 1982

B.B. Mandelbrot

[6]

The Random and the Regular, Santa Fe Institute’s 1999 Ulam’s Lecture Series, Santa Fe, NM, November 1999

M. Gell-Mann

[7]

The Quantum Theory of Fields I & II, Cambridge University Press, Cambridge, England, 1996

S. Weinberg

[8]

K. Wilson
,
M.E. Fisher

- Phys.Rev.Lett. 28 (1972) 248

[9]

Quantum electrodynamics at small distances

Murray Gell-Mann
(
- Illinois U., Urbana
)
,
F.E. Low
(
- Illinois U., Urbana
)

- Phys.Rev. 95 (1954) 1300-1312
•
DOI:
- 10.1103/PhysRev.95.1300

[10]

Lectures on Phase Transitions and the Renormalization Group, AddisonWesley, New York, 1992

N. Goldenfeld

[11]

Quantum Field Theory

C. Itzykson
(
- Saclay
)
,
J.B. Zuber
(
- Saclay
)

- New York, Usa: Mcgraw-hill (1980) 705 P.(International Series In Pure and Applied Physics)

[12]

Path Integral Methods in Quantum Field Theory, Cambridge University Press, Cambridge 1987

R.J. Rivers

[13]

Effective action for stochastic partial differential equations

David Hochberg
(
- Madrid, LAEFF and
- INTA, Madrid
)
,
Carmen Molina-Paris
(
- Los Alamos
)
,
Juan Perez-Mercader
(
- Madrid, LAEFF and
- INTA, Madrid
)
,
Matt Visser
(
- Washington U., St. Louis
)

- Phys.Rev.E 60 (1999) 6343-6360
•
e-Print:
- cond-mat/9904215
•
DOI:
- 10.1103/PhysRevE.60.6343

[14]

Quantum Field Theory and Critical Phenomena, Oxford University Press, Oxford, 1996, 3rd edition

J. Zinn-Justin

[15]

Field Theory, the Renormalization Group and Critical Phenomena Singapore, 1984, 2nd edition

D. Amit

[16]

Quantum Field Theory, Cambridge University Press, Cambridge, 1992

L.S. Brown

[17]

improved

[18]

Two-loop renormalization-group analysis of the Burgers-Kardar-Parisi-Zhang equation

Erwin Frey
(
- Munich, Tech. U.
)
,
Uwe Claus Tauber
(
- Harvard U.
)

- Phys.Rev.E 50 (1994) 1024-1045
•
e-Print:
- cond-mat/9406068
•
DOI:
- 10.1103/PhysRevE.50.1024

[19]

Quantum and Statistical Field Theory, Oxford University Press, Oxford, 1991

M. Le Bellac

[19]

Chap 7

[20]

Burgers equation with correlated noise: Renormalization-group analysis and applications to directed polymers and interface growth

- Phys.Rev.A 39 (1989) 3053-3075
•
DOI:
- 10.1103/PhysRevA.39.3053

[21]

The noise is defined to be gaussian, with zero mean and a two-point correlation function given by η(x, t)η(x′, t′) = 2D(x, x′

[21]

t, t′). The local function D(x, x′

[21]

t, t′) is the double Fourier transform of D(k, ω) ∝ k-2ρω-2θ. The existence of power law behavior for the correlation functions of φ is not due to the choice of noise correlations given above

[21]

it is a generic prediction of the Renormalization Group due to the fact that the RGEs are quasi-linear first order partial differential equations

1-25 of 28
1
2
25 / page