Ergodicity making

It is very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

Viewed from the perspectives of configuration space provided by the caricature in Fig. 2, the most direct approach to the phase-coexistence problem calls for a full frontal assault on the ergodic barrier that separates the two phases. The extended sampling strategies discussed in Section III.C make that possible. The framework we need is a synthesis of Eqs. (10) and (32). We will refer to it generically as Extended Sampling Interface Traverse (ESIT). 

To characterize these invariant structures and the changes of reaction coordinates, the concept of finite-time Lyapunov exponents can be useful [44]. The original definition of the Lyapunov exponents needs ergodicity (see, e.g.. Ref. 45) to make sure that the time average of the exponents converges. However, for chaotic itinerancy, the exponents would not converge. Moreover, the finite-time Lyapunov exponents can be more useful to detect whether... 

Here we also include the contribution of Okushima, in which the concept of the Lyapunov exponents is extended to orbits of finite duration. The mathematical definition of the Lyapunov exponents requires ergodicity to ensure convergence of the definition. On the other hand, various attempts have been made to extend this concept to finite time and space, to make it applicable to nonergodic systems. Okushima s idea is one of them, and it will find applications in nonstationary reaction processes. 

Although we have assumed in Eq. [209] that the velocity profile in the confined fluid is linear, it is not immediately obvious that this is technically possible in the absence of moving boundary conditions. A parallel to this situation is the comparison between Nose-Hoover (NH) thermostats and Nose-Hoover chain (NHC) thermostats. Although the Nose-Hoover equations of motion can be shown to generate the canonical phase space distribution function, for a pedagogical problem like the simple harmonic oscillator (SHO), the trajectory obtained from the NH equations of motion has been found not to fill up the phase space, whereas the NHC ones do. The SHO is a stiff system and thus to make it ergodic, one needs additional degrees of freedom in the form of an NHC.2 ... 

Here we make use of the density of states, as numerically obtained through Eqs. (37) and (41), to formulate the ergodicity in the present dynamics. 

If a system is uniformly hyperbolic, every point in phase space has both stable and unstable directions, and the maximum Lyapunov exponent with respect the maximum entropy measure is positive. The system has the mixing property and is therefore ergodic. The correlation function of observables also shows exponential decay. Uniformly hyperbolicity, which is sometimes rephrased as strong chaos in physical literature, is a well-established class of systems and is controllable by means of many mathematical tools [15]. In hyperbolic systems, there are no sources to make the relaxation process slow. 

For turbulence that is both homogeneous and stationary (statistically not changing over time), the time, space and ensemble averages should all be equal. This is called the ergodic condition, which is sometimes assumed to make the turbulence problem more tractable. 

Does the concept of chemical equilibrium make sense in this context It is not clear that the concept of equilibrium makes sense, even for a closed thermodynamic system. Matter might flow persistently across the reaction graph for the lifetime of the universe non-ergodically, without ever reaching an equilibrium distribution. 

A construction makes use of only an insignificant fraction of the Gibbs canonical ensemble and hence is essentially out of equilibrium. This is different from thermodynamic nonequilibrium—it arises because the system is being investigated at time scales much shorter than those required for true statistical equilbrium. Such systems exhibit broken ergodicity [68], as epitomized by a cup of coffee in a closed room to which cream is added and then stirred. The cream and coffee equilibrate within a few seconds (during which vast amounts of microinformation are generated within the whorled patterns) the cup attains room temperature within tens of minutes and days may be required for the water in the cup to saturate the air in the room. 

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