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Math Tools
Published in Thomas M. Nordlund, Peter M. Hoffmann, Quantitative Understanding of Biosystems, 2019
Thomas M. Nordlund, Peter M. Hoffmann
The previous section would seem to imply that distributions apply when the number of particles is large, perhaps approaching Avogadro’s number, 6.02 × 1023. The idea that a smooth distribution accurately describes the population of particles does require a large number of particles, or else a smaller number, even as small as 1, as long as adequate time is allowed by the experiment or biological process for the smaller number of particles to explore (pass through) the majority of the allowed and accessible states. These two extremes—ensembles of large numbers of particles whose properties are measured during a brief snapshot in time or one particle observed for an infinite length of time—constitute the ergodic hypothesis. A system is ergodic with respect to a measured property p if the distribution function describing that property is the same whether the ensemble average or the time average is measured: (
Selection and Design of the Experiment
Published in Marian (Editor-in-Chief) Muste, Dennis A. Lyn, David M. Admiraal, Robert Ettema, Vladimir Nikora, Marcelo H. Garcia, Experimental Hydraulics: Methods, Instrumentation, Data Processing and Management, 2017
Marian (Editor-in-Chief) Muste, Dennis A. Lyn, David M. Admiraal, Robert Ettema, Vladimir Nikora, Marcelo H. Garcia
For stationary processes, whereby the statistical functions are time-invariant, the signal describing the turbulent quantities can be conceptualized using Reynolds decomposition: i.e., the instantaneous value of a fluctuating quantity is expressed as a mean value and a fluctuation about the mean, as illustrated in Figure 4.5.3a. The Reynolds decomposition for a turbulent fluctuation is mathematically expressed as: f(t)=f¯+f′ If the time-averaged mean value and autocorrelation function of the data from a stationary process are equal to the corresponding ensemble averaged values for any set of samples, the process is ergodic (only stationary random processes can be ergodic). A stationary process is non-ergodic if the above conditions are not fulfilled. Often stationary random processes are ergodic. Therefore, stationary random processes can be fully characterized by a single observed time history record. Analytical procedures for assessing whether or not a flow is stationary are presented in Section 6.12.1, Volume I.
Flight Planning
Published in Yasmina Bestaoui Sebbane, Multi-UAV Planning and Task Allocation, 2020
There are N mobile agents assumed to move by either the first-order or second-order dynamics. An appropriate metric is needed to quantify how well the trajectories are sampling a given probability distribution μ. It is assumed that μ is zero outside a rectangular domain 𝕌 ∈ ℝn and that the agent trajectories are confined to the domain 𝕌. For a dynamical system to be ergodic, the fraction of the time spent by a trajectory must be equal to the measure of the set. Let B(X, R) = {R : ∥Y − X∥ ≤ R} be a spherical set and χ(X, R) be the indicator function corresponding to the set B(X, R). Given trajectory Xj : [0, t] → ℝn, j = 1, ...N, the fraction of the time spent by the agents in the set B(X, R) is given as dt(X,R)=1Nt∑j=1N∫0tχ(X,R)(Xj)(τ)dτ
Adaptive Reweighted Variance Estimation for Monte Carlo Eigenvalue Simulations
Published in Nuclear Science and Engineering, 2020
In this paper, is assumed to be from a discrete-time stationary ergodic Gaussian process. A process is stationary ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process and these properties do not change over time. In a Gaussian process, every finite collection of random variables has a multivariate normal distribution. The autocovariance,
An optical channel modeling of a single mode fiber
Published in Journal of Modern Optics, 2018
Neda Nabavi, Peng Liu, Trevor James Hall
The basic focus of the ergodic theory is the development of conditions under which sample or time averages consisting of arithmetic means of a sequence of measurements on a random process converged to a probabilistic or ensemble average of the measurement. This analysis then can be extended to the channel matrix. Considering: