Ergodic properties

MSN.69. A. P. Grecos and I. Prigogine, Kinetic and ergodic properties of quanmm systems— The Friedrichs Model, Physica 59, 77-96 (1972). 

MSN.85. I. Prigogine and A. P. Grecos, Kinetic theory and ergodic properties in quantum mechanics, in 75 Jahre Quantenmechanik, Akademie-Verlag, Berlin, 1977, pp. 57-68. 

First it is clear that the total W decomposes into separate blocks for the separate subshells. Hence we may confine ourselves to a single subshell. According to the ergodic property the remaining block of W is indecomposable. It therefore has a single stationary solution psn. 

Let L and W be random variables whose distributions are the steady-state distributions of the number-in-system and time-in-system for the queue, respectively. Then, in great generality, we have the ergodic properties... 

To see that this is a special case of (49), note that 1 - P(L = 0) is not only the steady-state probability that the server is busy, it is also the steady-state expected number of jobs in service, which in turn, by an ergodic property, is the time-average number of jobs in the system in question. Of course, 1 / /X is the expected time in service and plays the role of W in (49). When Little s law is applied to the waiting area, it tells us that the expected number of jobs waiting is the product of the arrival rate and the average waiting time. 

Because of the ergodic property of a Markov chain, the result at qr=jsds independent of the initial state of seeding, i.e.. the initial values X iq = 0), Xiy(q = 0). This means that the supramolecular approach generating polarity through a process of growth is circumventing the basic difficulty of creating polarity by spontaneous nucleation. 

An even stronger property than the ergodic property is the concept of a mixing system. For a mixing system, the finite time density p(z, t) converges, in the weak sense, to the invariant distribution Poo(z), as f 00. That is, we have, for all test functions (p in some chosen space... 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

The key feature of Brownian dynamics that makes it possible to verify the ergodic property is the fact that each variable of the system is directly in contact with an independent stochastic Wiener process. This ensures that at each point of phase space, all possible directions are sampled and the paths will have freedom to move in any direction. In Langevin dynamics, taking M = / for simplicity,... 

In the case of Nose-Hoover, this problem was known to both of the inventors and has been observed by many authors (see e.g. [251] and [180]). More recently, it has been studied analytically [218], Some confusion arises from the fact that the practical ergodic properties (and other properties) in Nosd-Hoover simulations are sensitive to the particular system under study and the values of parameters. For example it is known that in the case of simple liquids and gases, Nos6-Hoover does (at least under some conditions) enable reliable calculation of thermodynamical quantities [77], whereas in other systems results are poor and cannot be rectified by adjustment of parameters. In general, the control of temperature requires that the thermostat couples tightly to the physical variables, which generally calls for a small thermal mass, but this may be at odds with efficiency reduction (it may lead to a need to use smaller timesteps to resolve the motion of the auxiliary variable). 

The fact that the Hormander condition holds is, as we know from Chap. 6, only part of the story. However based on known results for Langevin dynamics, we conjecture that ergodicity will hold if (i) the Hormander condition holds, (ii) U is sufficiently smooth, and (iii) the configurational phase space is compact, e.g. periodic boundary conditions are employed. However we stress that each possible method will ultimately need to be carefully and systematically checked to verify the ergodic properties. 

H, 1X2 both one for simplicity as their particular values (as long as nonzero) have no bearing on the ergodic property. 

The qualitative theory describes the long-term behaviour of stochastic systems without solving the equations. Recurrence, stationarity and ergodic properties are the most important concepts which characterise the stochastic process and/or the state-space. 

Nevertheless, the pivot algorithm has one distinctive advantage. It exhibits very good ergodicity properties [5], while slithering snake and... 

It is useful to note that Theorem A.3 may be proven by looking at the ergodic properties of the backward recurrence time process A [Asmussen (2003), VII.2]. In particular one sees that the condition mx < oo is precisely the condition for A to be positive recurrent. This implies directly the existence of a unique invariant probability measure. One can write it explicitly ... 

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