Previously we derived the probability density function (PDF) of the zero-crossing interval for 1/fα noise and found that the PDF, L(t) obeys the power law of the form 1/tc whose exponent c relates to
the exponent α of the power spectrum density as c = 3-α when 0 < α < 1 and c = (5 - α)/2 when 1< α < 2. (Proc. SPIE Vol. 5471, p. 29, 2004).
This analytical result agreed with numerical experiments by Mingesz et al. (Proc. SPIE Vol. 5110, p.312, 2003) for 0.7 less than or equivalent to α < 2,
but not for 0 < α less than or equivalent to 0.7;
the experimental PDF deviates from the power law in the latter range.
We present here a discretized version of the previous theory by noting
that the experimental time interval takes discrete numbers.
The present result agrees well with the experiment for
the whole range of α and explains the deviation from the power law of
PDF in the range of small α.
We present an analytic relation between the correlation function of dichotomous (taking two values, ± 1) noise and the probability density function (PDF) of the zero crossing interval. The relation is exact if the values of the zero crossing interval τ are uncorrelated. It is proved that when the PDF has an asymptotic form L(τ) = 1/τc, the power spectrum density (PSD) of the dichotomous noise becomes S(f) = 1/fβ where β = 3 - c. On the other hand it has recently been found that the PSD of the dichotomous transform of Gaussian 1/fα noise has the form 1/fβ with the exponent β given by β = α for 0 < α < 1 and β = (α + 1)/2 for 1 < α < 2. Noting that the zero crossing interval of any time series is equal to that of its dichotomous transform, we conclude that the PDF of level-crossing intervals of Gaussian 1/fα noise should be given by L(τ) = 1/τc, where c = 3 - α for 0 < α < 1 and c = (5 - α)/2 for 1 < α < 2. Recent experimental results seem to agree with the present theory when the exponent α is in the range 0.7 ⪅ α < 2 but disagrees for 0 < α ⪅ 0.7. The disagreement between the analytic and the numerical results will be discussed.
We present a model of financial markets originally proposed for a turbulent flow, as a dynamic basis of its intermittent behavior. Time evolution of the price change is assumed to be described by Brownian motion in a power-law potential, where the 'temperature' fluctuates slowly. The model generally yields a fat-tailed distribution of the price change. Specifically a Tsallis distribution is obtained if the inverse temperature is χ2-distributed, which qualitatively agrees with intraday data of foreign exchange market. The so-called 'volatility', a quantity indicating the risk or activity in financial markets, corresponds to the temperature of markets and its fluctuation leads to intermittency.
Symmetric white Poissonian shot noise with Gaussian distributed amplitudes is shown from an overdamped Langevin equation to induce directed motion of ratchets. The noise becomes white Gaussian in the limit λ→∞, where λ is the average number of the delta pulses per unit time. The current tends to zero as 1/λ→0, which agrees with the fact that the directed motion cannot be induced by the thermal noise alone. However, a finite value of 1/λ yields a finite value of the current no matter how small the former may be. Since 1/λ cannot be zero physically, this is a seeming contradiction of the second law of thermodynamics, which originates from the Langevin equation. We discuss this point from an alternative master equation without assuming the Langevin equation.
KEYWORDS: Signal to noise ratio, LCDs, Signal detection, Interference (communication), Sensors, Oscillators, Numerical simulations, Complex systems, Stochastic processes, Systems modeling
Noise-assisted propagation of periodic signals is investigated for one-dimensional arrays composed of one-way coupled level-crossing detectors (LCD). Analytical expressions are obtained for the signal decay length through chains and the signal decay time through rings, where noise is uncorrelated so that the signal transmission from a LCD to the neighboring one is Markovian. Recent numerical simulations for one-dimensional arrays of one-way coupled bistable oscillators are discussed in comparison to the present analytical results.
Conference Committee Involvement (2)
Noise in Complex Systems and Stochastic Dynamics II
26 May 2004 | Maspalomas, Gran Canaria Island, Spain
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