2. Noise or Random Fluctuations in Physical Systems: A Review

‘Noise’ or random fluctuations characterize all physical systems in nature ranging from biology, botany, physiology, meteorology, astronomy, etc. The apparently irregular or chaotic fluctuations were considered as ‘noise’ in all fields except in astronomy, where the fluctuations from astronomical sources were referred to as signal. Noise and fluctuation has been a field of study since 1826 with the study of Brownian motion which indirectly confirmed the existence of atoms and molecules. The measured characteristics of noise contain recognizable patterns or signal and convey useful information about the system. Statistical data analysis techniques are used to extract the signal, i.e. recognizable patterns in the apparently random fluctuations of physical systems. The analysis of data sets and broad quantification in terms of probabilities belongs to the field of statistics. Early attempts resulted in identification of the following two quantitative (mathematical) distributions which approximately fit data sets from a wide range of scientific and other disciplines of study. The first is the well-known statistical normal distribution and the second is the power-law distribution associated with the recently identified ‘fractals’ or self-similar characteristic of data sets in general. Abraham de Moivre, an eighteenth-century statistician and consultant to gamblers made the first recorded discovery of the normal curve of error (or the bell curve because of its shape) in 1733. The importance of the normal curve stems primarily from the fact that the distributions of many natural phenomena are at least approximately normally distributed. This normal distribution concept underlies how we analyse experimental data over the last 200 years. Most quantitative research involves the use of statistical methods presuming independence among data points and Gaussian ‘normal’ distributions. The Gaussian distribution is reliably characterized by its stable mean and finite variance. Normal distributions place a trivial amount of probability far from the mean and hence the mean is representative of most observations. Even the largest deviations, which are exceptionally rare, are still only about a factor of two from the mean in either direction and are well characterized by quoting a simple standard deviation. However, apparently rare real-life catastrophic events such as major earth quakes, stock market crashes, heavy rainfall events, etc., occur more frequently than indicated by the normal curve, i.e. they exhibit a probability distribution with a fat tail. Fat tails indicate a power-law pattern and interdependence. The ‘tails’ of a power-law curve—the regions to either side that correspond to large fluctuations—fall off very slowly in comparison with those of the bell curve. The normal distribution is therefore an inadequate model for extreme departures from the mean. For well over a century evidence had been mounting that real-world behaviour in particular, behaviour of systems, whether natural, social, economic, or financial does not follow normal distribution characteristics. There is increased evidence for non-normality in real-world settings and in its place an alternative distribution, namely the power-law distribution is shown to be exhibited by real-world systems in all fields of science and other areas of human interest. In this chapter, the following are discussed. (i) A brief history of the two chief quantitative methods of statistical data analysis, namely the statistical normal distribution and the power-law distribution. (ii) The association of power-law distributions with complex systems, scale invariance, self-similarity, fractals, 1/f noise, long-term memory, phase transitions, critical phenomena, and self-organized criticality. (iii) Current status of power-law distributions. (iv) Power-law relations (bivariate) and power-law (probability) distributions. (v) Allometric scaling and fractals. (vi) Fractals and the golden section in plant growth. (vii) Turbulent fluid flow structure, fractals, and the golden ratio (≈1.618). (viii) Fractal space-time and the golden ratio. (ix) Power-law (probability) distributions in the meteorological parameters precipitation, temperature, quaternary ice volume fluctuations and atmospheric pollution. (x) General systems theory model for self-organized criticality (SOC) in atmospheric flows with universal quantification for power-law distribution in terms of the golden ratio.