The Ergodic Problem

The partition function Z is defined as a sum of Boltzmann factors for discrete stationary states that are available to the canonical ensemble. The partial derivative of Z with respect to the parameter (i.e., at constant V and N) defines 

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Since a celebrated work of Fermi-Pasta-Ulam (FPU), computer experiments have provided powerful tools to attack the ergodic problem of dynamical... 

The establishment of chemical potential equilibrium (with respect to either a setpoint or phase coexistence) is the central component of most Monte Carlo schemes for simulation of the phase behavior and stability of molecular systems. Simulation of the chemical potential (or chemical potential equilibration) in a polymeric system requires more effort than the corresponding calculation for a simple fluid. The reason is that efficient conformational sampling of the polymer is implicitly required for a free-energy calculation and, in fact, the ergodicity problems described in earlier sections are often exacerbated. 

The parameter as defined in the ergodic problem via equation (28-20), has dimensions of reciprocal energy and is given by l/kT. This claim will be justified, and consistency with classical thermodynamics will be demonstrated in Section 28-5. 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

The spontaneous emission in atomic problems and the decay of unstable particles are irreversible processes which manifest the ergodicity of these systems. It is therefore interesting to compare the mechanism of irreversibility which is involved to that in the usual many-body systems such as a classical gas. 

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). 

Equation (10) shows that we can always accomplish our objective if we can measure the full canonical distribution of an appropriate order parameter. By full we mean that the contributions of both phases must be established and calibrated on the same scale. Of course it is the last bit that is the problem. (It is always straightforward to determine the two separately normalized distributions associated with the two phases, by conventional sampling in each phase in turn.) The reason that it is a problem is that the full canonical distribution of the (an) order parameter is typically vanishingly small at values intermediate between those characteristic of the two individual phases. The vanishingly small values provide a real, even quantitative, measure of the ergodic barrier between the phases. If the full -order parameter distribution is to be determined by a direct approach (as distinct from the circuitous approach of Section IV.B, or the off the map approach to be discussed in Section IV.D), these low-probability macrostates must be visited. 

The Nose-Hoover thermostat exhibits non-ergodicity problems for some systems, e.g. the classical harmonic oscillator. These problems can be solved by using a chain... 

A second prominent feature here is the ergodic character (or lack thereof) of the process, depending on the rationality or irrationality of <. This leads inevitably to the fascinating question, Does a real system choose between these values of , and if so, how The boundaries themselves remain neutral with respect to the choice of whenever they are compatible with the flow. Thus, for a slide flow, the walls must be parallel to the slides, whereas for a tube flow, they must be parallel to the tube. In both cases there remains an additional degree of freedom, which is precisely the choice of f. Other examples of indeterminancy arise from the neglect of fluid and particle inertia, as already discussed in Section I (see also the review in Leal, 1980). Whether or not inclusion of nonlinear inertial effects can remove the above indeterminacy, as it often does for the purely hydrodynamic portion of the problem, is a question that lies beyond the scope of the present (linear) Stokesian context. 

A main message in the previous section is that MD simulations of liquids provide a bunch of uncooked materials to be analyzed as ergodic problems of dynamical systems. In particular, anomalous diffusion or slow relaxation discussed so far... 

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. 

The classical-quantum correspondence also results in useful scaling laws for chaotic states. For the stadium problem the classical correlation functions scale as (2m ) l/z, a feature which is solely a consequence of classical ergodicity. As shown in Fig. 17, quantum states labeled chaotic (according to the aperiodic nature of their correlation function) do obey this scaling relation. Specifically, Fig. 17 displays the correlation lengths (A,/2), defined as... 

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