Non-ergodic Case Research Articles

Equipartition and ergodicity in closed one-dimensional systems of hard spheres with different masses.

We show by computer simulation that a one-dimensional closed system of hard spheres with different masses exhibits equipartition. This is true even when the system contains as few as two particles or is nonergodic. Use of periodic boundary conditions gives very different results from the fixed boundaries. For more than two particles, it is shown that, for most mass ratios, the probability of an exact return to the initial state is vanishingly small. The density of states in momentum space accessible by a particular particle corresponds to a uniform density of states in the region allowed by conservation laws. There are special mass ratios for which ergodicity fails and recurrence occurs. Even for these nonergodic cases, equipartition is obtained

Thermodynamic limit and central limit theorem for point random fields in non-ergodic case

The central limit theorem for a class of point random fields, including also non- ergodic measures, is proven. The result, mathematically easy, shows the differences in quality between point random fields (continuous case) and random fields (discrete case).

The non-ergodic Jackson network

We generalize Jackson's theorem to the non-ergodic case. Here, despite the fact that the entire Jackson network will not achieve steady state, it is still possible to determine the maximal subnetwork that does. We do so by formulating and algorithmically solving a new non-linear throughput equation. These results, together with the ergodic results and the ones for closed networks, completely characterize the large-time behavior of any Jackson network.

The non-ergodic Jackson network

We generalize Jackson's theorem to the non-ergodic case. Here, despite the fact that the entire Jackson network will not achieve steady state, it is still possible to determine the maximal subnetwork that does. We do so by formulating and algorithmically solving a new non-linear throughput equation. These results, together with the ergodic results and the ones for closed networks, completely characterize the large-time behavior of any Jackson network.

Asymptotically minimax tests of composite hypotheses for nonergodic type processes

Asymptotically efficient tests satisfying a minimax type criterion are derived for testing composite hypotheses involving several parameters in nonergodic type stochastic processes. It is shown, in particular, that the analogue of the usual Neyman's C (α) type test (i.e., the score test) is not efficient for the nonergodic case. Moreover, the likelihood-ratio statistic is not fully efficient for the model discussed in the paper. The efficient statistic derived here is a modified version of the score-statistic discussed previously by Basawa and Koul (1979).

Convergence of non-ergodic dynamical systems

Let {Tt} be a flow on a probability space (S,L,μ}) which describes the time evolution of a dynamical system with state space S, and interpret μ as the initial distribution of the system. Then the distribution of the system at time t is given by μTt−1. Our aim is to study the asymptotic behavior of μTt−1both in general and in the particular cases of random rate and almost periodic systems. The results seem to indicate that convergence or mean convergence is the normal behavior in the non-ergodic case.

Block and sliding-block source coding

A new form of source coding subject to a fidelity criterion, called sliding-block coding, has recently been introduced by Gray, Neuhoff, and Ornstein. A method of conversion from block coding to sliding-block coding, or vice versa, with an arbitrarily small increase in rate is shown for ergodic sources. The ergodic theorem is used to convert from sliding-block codes to block codes. A theorem similar to one of Rohlin is used to convert from block codes to sliding-block codes. A proof of this theorem and some discussion of the nonergodic case appear in the appendices.

Validity of approximate methods in linear response theory for an interacting spin-phonon system

In order to gain insight into their degree of validity, certain commonly used approximate methods for finding dynamic susceptibilities are compared with the exact Kubo susceptibility for a linearly interacting spin-phonon system. Approximate methods investigated include: truncation or decoupling of equations of motion for the commutator and anticommutator Green's functions, and a memory-function method suggested by G\otze and W\olfle. It is shown that approximate methods, while adequate for weak dipole-phonon coupling, can lead to conspicuous deviations from the exact susceptibility in certain frequency ranges. Results are also sensitive to the shape of the phonon spectrum. Ergodic and nonergodic cases are discussed.