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Random Processes
Published in X. Rong Li, Probability, Random Signals, and Statistics, 2017
Ergodicity requires that both ensemble average and time average be constants and equal to each other. As a result, for an ergodic random process, its mean (ensemble average) can be obtained from the time average of any single realization of the random process. This is extremely important in practice since it is usually the case that only a single sample function of a random process is available, Fortunately, most random processes involved in real-world problems are ergodic. The usefulness of ergodicity is as follows. In most applications, the ensemble average (mean) of a random process cannot be obtained directly. A typical example is the case in which the random process is a signal whose expression or representation is not known. If by some criterion it is known that the process is ergodic, then its mean can be approximated by the finite time average of any of its single sample functions.
Random Signal Analysis
Published in Peter M. Clarkson, Optimal and Adaptive Signal Processing, 2017
It would seem reasonable that stationarity would be required before such time averaging can be used. In fact, time averaging cannot always be used even if x(n) is stationary. That is, stationarity is a necessary but not a sufficient condition for a time average of the form (2.6.6) to converge to μ. When such a time average can be employed the sequence is said to be ergodic. Various forms of ergodicity can be defined including ergodicity in the mean, ergodicity in mean-square and so on. Each referring to a particular moment. For example, x(n) is ergodic in the mean if the estimator (2.6.6) is consistent: () limM→∞{1M∑n=0M−1x(n)}→μ.
Ensembles
Published in Teunis C. Dorlas, Statistical Mechanics, 2021
where XM=(1/M)∑k=1MX(k)(t) for an arbitrary point t in time ! The idea of ergodicity has come under severe criticism and it is now known that ‘most’mechanical (i.e. Hamiltonian) systems are not ergodic.
Effective temperature scaled dynamics of a flexible polymer in an active bath
Published in Molecular Physics, 2020
Xiuli Cao, Bingjie Zhang, Nanrong Zhao
Figure 2 plots the ensemble-averaged , individual time-averaged and also the mean time-averaged as a function of time interval τ. The system parameters are fixed at , and . As it is shown, first, the individual TA-ACFs reveal a pronounced spread of magnitudes. These trajectory-to-trajectory variations indicate non-ergodic of the underlying relaxation process. Noticeably, such a non-ergodic phenomenon has also been observed in anomalous diffusion dynamics taking place at various heterogeneous systems [48,49]. Second, we find that the ensemble- and mean TA-ACFs are fairly close but do not match perfectly. This discrepancy is certainly the consequence of weak ergodicity breaking. In what follows, our analysis is based on a trajectory-to-trajectory average procedure, where the mean time-averaged are considered.