Noise and Fluctuations: An Introduction

By D. K. C. MacDonald

An understanding of fluctuations and their role is both useful and fundamental to the study of physics. This concise study of random processes offers graduate students and research physicists a survey that encompasses both the relationship of Brownian Movement with statistical mechanics and the problem of irreversible processes. It outlines the basics of the physics involved, without the strictures of mathematical rigor.
The three-part treatment starts with a general survey of Brownian Movement, including electrical Brownian Movement and "shot-noise," Part two explores correlation, frequency spectrum, and distribution function, with particular focus on application to Brownian Movement. The final section examines noise in electric currents, including noise in vacuum tubes and a random rectangular current. Frequent footnotes amplify the text, along with an extensive selection of Appendixes.

• Language: English
• Publisher: Dover Publications
• Release date: Jan 23, 2013
• ISBN: 9780486174464

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Bibliographical Note

This Dover edition, first published in 2006, is an unabridged republication of the work originally published by John Wiley & Sons, New York, in 1962.

9780486174464

Manufactured in the United States of America

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Preface

It is probably fair comment to say that to many physicists the subject of fluctuations (or noise, to put it bluntly) appears rather esoteric and perhaps even pointless; spontaneous fluctuations seem nothing but an unwanted evil which only the unwise experimenter would encounter! I will agree that perhaps it is only a rather funny (funny-peculiar) type of mind that likes to deal with more or less random processes, but at the same time I am convinced that some basic understanding of fluctuations and their role is both useful and rather fundamental in physics. I hope that this short book may suggest in part the fascination that I have always found in the study of random processes, and I have tried to indicate the intimate relationship of the Brownian Movement with statistical mechanics on the one hand, and with the problem of irreversible processes on the other.

My aim has been to outline the fundamentals of the physics involved without attempting any real mathematical rigor, but I hope that there are no gross omissions or errors which will seriously offend the specialist. I hope the book may be useful to graduating students, or physicists starting research, who might wish for some survey of this subject without becoming embroiled in too much technical detail. Let me mention that I have tried to use words as far as possible with their normal colloquial meaning, and to avoid rather specialized adjectives such as stochastic. Thus, for example a more or less random process is meant to suggest things which on the one hand could vary fairly smoothly in time, but with some degree of irregularity, to the other extreme of an extremely ragged fluctuation whose behavior is almost entirely unpredictable from instant to instant.

Perhaps I ought to apologize for my frequent footnotes; they are meant to amplify, and not to interrupt the text. If the book is worth reading twice, then the footnotes might well be omitted on first reading.

I am grateful to Mrs. T. F. Armitage for her tireless patience in typing drafts of the manuscript, and to Miss I. Assing and Mrs. Armitage for their help in correcting proofs, etc. I am also grateful to various colleagues in the past for their patience in listening to my arguments about Brownian Movement, and particularly to Dr. A. M. Guénault and Dr. T. H. K. Barron for reading the manuscript. I would also like to thank Dr. R. Fürth, F.R.S.E., for his encouragement and stimulation when I first became ensnared by the charms of noise problems, and finally my daughter, Aileen, for providing the design-motif for the dust-jacket.

May 1962

D. K. C. MACDONALD

Table of Contents

Title Page

Copyright Page

Preface

1 - General survey

2 - Correlation, frequency spectrum, and distribution function

3 - Noise in electric currents

Appendix I - The mean and fluctuations of a Bernoulli distribution

Appendix II - Alternative derivation of Einstein’s Brownian Movement equation

Appendix III - Proof of Campbell’s theorem

Appendix IV - The autocorrelation function for a random rectangular current, and the Poisson distribution

References

Index

1

General survey

1.1 INTRODUCTION

In this first Chapter, I would like to try to outline some of the main problems that arise from physical fluctuations or noise, and to try to dispel some of the mystery that often seems to surround the significance of fluctuation phenomena. First, let us clear up the matter of terminology. Acoustically a regular tone or sequence of tones that may either convey obvious information in some sort of code, or simply present a musical melody, can usually be distinguished quite readily from what we call noise.¹ In the limit, audible noise is just a more or less random jumble of tones of a wide range of frequencies having no particular connection with one another, and such that if displayed on an oscillograph the noise would give rise to a more or less jagged and very irregular pattern whose amplitude varies randomly between fairly wide limits. This is in strong contrast to the pattern produced by a regular and unchanging single musical tone (see Figs. 1a and 1b).

This contrast between a regular pure tone and noise can also be seen in the degree of predictability. For a regular musical tone the pattern on an oscillograph and, of course, the corresponding audible sound are almost perfectly predictable, whereas with noise both the visual pattern and perhaps the audible content are largely unpredictable. It may be partly the latter aspect that makes audible noise sometimes so objectionable, although I am not suggesting that perfectly predictable experiences are necessarily the acme of human enjoyment.

If now we extend these ideas to other fields of physical behavior such as electrical oscillations, then the term noise can be applied (if one is not too squeamish) to any physical variable that is not behaving in an entirely regular and predictable manner. If we turn up the volume control of an ordinary broadcast receiver when no station is tuned in, then sooner or later a general mush or rasping noise will be heard from the loudspeaker. If there is no antenna connected, a radio engineer might then say that we were listening to the electrical noise in the resistors and tubes of the first stage. A more respectable physicist might prefer to say that the spontaneous fluctuations of electric charge were being amplified to produce a more or less random, audible sound. Or, if we listen to rain on a metal roof, we hear a general rattling noise caused by the impacts of the individual raindrops which are by and large random and unpredictable in detail; a physicist will point out that the time-average of the impulses gives rise to a steady pressure on the roof, whereas the fluctuations set up random oscillatory vibrations in the roof which in turn we hear as the noise.

Fig. 1. (a) Sketch of oscillograph trace produced by typical random noise. (b) Sketch of oscillograph trace produced by regular periodic monotone.

Why is the whole field of noise or fluctuation phenomena of considerable importance in physics? There is no single reason, and perhaps that is part of the fascination² of the subject. Here are one or two examples. First, assume we are designing a radar set, or alternatively that we are trying to make a very sensitive radio receiver to pick up any radio emissions (in themselves a form of electrical noise!) that may be coming to the earth from outer space (e.g., the sun or some distant stellar object). With a radar set the useful range of the equipment will depend on how much power we send out from the transmitter, and also on the smallest reflected power from the distant object that we can detect with our receiver. From that point of view any improvement that we can make in the ability of our radar receiver to detect a weak signal is precisely equivalent to a corresponding power increase in our radar transmitter, and may therefore represent a very large economic saving. If we are engaged in radio-astronomy, an improvement in our receiver may enable us to observe with certainty more distant galactic objects. In either case the fundamental limit for the ability of our receiver to detect very small incoming signals will be set primarily by the magnitude of the noise or random fluctuations of electric charge in the input circuit and in the first amplifying stage of the receiver. Hence a theoretical understanding of the sources of such noise may enable us directly to design a better receiver. And this, of course, applies to many other fields where sensitive amplifiers and detectors may be required. In various branches of physics very sensitive measurements are often now being made to detect small effects predicted by developments of quantum theory. For example, today (mid-1961) experiments are reported on small superconducting cylinders to demonstrate that magnetic flux is quantized in units of hc/2e(∼2 × 10−7 gauss cm²). Such experiments demand very careful attention to detail, and naturally care must be taken at many points to ensure that noise and random fluctuations in the apparatus are kept to an absolute minimum if we are to observe with any confidence such small effects.

Secondly, consider what we might observe with a sensitive radio receiver when the antenna is pointed at the sun for example. In general, if our antenna is sufficiently sensitive (and can discriminate within a sufficiently small solid angle), we shall notice increased noise at the output of our receiver arising primarily from incoherent electromagnetic radiation of one kind and another from the sun and covering a wide frequency spectrum. If we have a basic theory of how hot bodies of one kind and another emit such noise radiation, we may then be able to use our observations of noise to obtain some direct information about the properties of the sun. A rather related sort of application is to use directly the electrical noise, or spontaneous fluctuation of some other physical quantity in a body, to measure the temperature at which that body finds itself. When the noise from a passive electrical resistor is used in this way to indicate the absolute temperature, the set-up is usually known as a noise thermometer.

Thirdly, let us consider a dissipative process involving a parameter such as viscosity. Consider a small sphere falling steadily through a bath of oil under the force of gravity. The viscous force retarding the motion of the sphere is, of course, determined by some average of the vast number of rapidly varying individual forces exerted by the surrounding oil molecules. At the same time, however, precisely because the average (and irreversible) viscous friction does arise from a large number of individual and more or less independent components, this viscous force must also have a fluctuating component. Einstein (1905, 1906) first showed that there was an essential and fundamental connection between the average force (viscosity or mobility) and the fluctuating component (giving rise to the so-called Brownian Movement of the particle);³ moreover it may be quite possible under suitable conditions to observe directly this fluctuating component. There are in fact situations where it might be simpler to observe the Brownian Movement and hence, using Einstein’s analysis, to deduce the viscosity; alternatively a direct measurement of viscosity may enable us to predict the diffusion of a typical atom, which in effect is just Brownian Movement. Apart from this it may also be that theoretically we can estimate rather