NDLI: Modes of a Gaussian Random Walk

Content Provider	Springer Nature Link
Author	Rawlings, Philip K.
Copyright Year	2003
Abstract	It is demonstrated that a one-dimensional gaussian random walk (GRW) possesses an underlying structure in the form of random oscillatory modes. These modes are not sinusoids, but can be isolated by a well-defined procedure. They have average wavelengths and amplitudes, both of which can be determined by experiments or by theoretical calculations. This paper reports such determinations by both methods and also develops a theory that is ultimately shown to agree with experiments. Both theory and simulations show that the average wavelength and the average amplitude scale with the order of the mode in exactly the same way that the modes of the well-known Weierstrass fractal scale with mode order. This is remarkable since the wave generated by the Weierstrass function, $$W(x) = \sum\nolimits_{m = 1}^\infty {(\tfrac{1}{a}} )^m \cos (g^m x)$$ , is fully determined for the variable x whereas the GRW is stochastic. By increasing the size of the steps in the GRW, it is possible to selectively remove the fastest modes, while leaving the remaining modes almost unchanged. For a GRW, the parameters corresponding to a and g in the Weierstrass function are found to be 2.0 and 4.0, respectively. These values are independent of the variance associated with the GRW. Application of the random modes is reserved for a later paper.
Starting Page	769
Ending Page	788
Page Count	20
File Format	PDF
ISSN	00224715
Journal	Journal of Statistical Physics
Volume Number	111
Issue Number	3-4
e-ISSN	15729613
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2003-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Physical Chemistry Quantum Physics Mathematical and Computational Physics Statistical Physics
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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