Measuring Ergodicity

As discussed in the example from the work of Hodel et al. [3], one of the most efficient ways to improve the accuracy of free energy calculations with a given force field is to enhance the conformational sampling. Thus, it is important to assess the extent to which phase space is covered. 

U for the jth particle along the a trajectory after n moves exploring space according to a known equilibrium distribution to be 

w(xfc) is the weighting factor for any property at a given position on the fcth step xfc. For example, for a constant-temperature molecular dynamics or a Metropolis MC run, the weighting factor is unity. However, we wish to leave some flexibility in case we want to use non-Boltzmann distributions then, the weighting factor will be given by a more complicated function of the coordinates. The ergodic measure is then defined as a sum over N particles 

For an ergodic system, if the simulation length n — oo, then du(n) — 0. By analogy with molecular dynamics, for large n we expect the form of the convergence to be [5] 

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Thus we have an alternative route to the experimentally observable property A it is the statistical average of the results of measurement on very many identical systems. The ergodic hypothesis tells us that this interpretation and the time-dependent interpretation are equivalent. 

Xo is Irrational Using the same argument as given above, orbits for irrational Xq must be nonperiodic, with the attractor in this case being the entire interval. Because any finite sequence of digits appears infinitely many times within the binary decimal representation of almost all irrational numbers in [0,1] (all except for a set of measure zero), the orbit of almost all irrationals approaches any x G [0,1] to within a distance e << 1 an infinite number of times i.e., the Bernoulli shift is ergodic. 

Intensity at a Point of Superposition (1.17) The measurable physical parameter of an optical wave is its energy density or intensity. If two fields are superimposed, the measured intensity is given by the sum of the individual intensities plus aterm which describes the long term correlation of the field amplifudes. Long ferm means time scales which are large compared to the inverse of the mean frequency uj/2Tt (about 10 s) the time scale is set by the time resolution of the detector. This is why the held product term is expressed in the form of an ergodic mean ). An interferometer produces superimposed helds, the correlation of which carries the desired information about the astronomical source. We will discuss exactly how this happens in the following sections. 

Mountain, R.D. Thirumalai, D., Measures of effective ergodic convergence in liquids, J. Phys. Chem. 1989, 93, 6975-6979... 

In view of the ergodic hypothesis the average value of an observable property may be regarded as the quantity measured under specified conditions. In this way the internal energy of a system corresponds to the average energy of the canonical distribution ... 

Because the velocity u contains the random component u, the concentration c is a stochastic function since, by virtue of Eq. (2.2), c is a function of u. The mean value of c, as expressed in Eq. (2.5), is an ensemble mean formed by averaging c over the entire ensemble of identical experiments. Temporal and spatial mean values, by contrast, are obtained by averaging v ues from a single member of the ensemble over a period or area, respectively. The ensemble mean, which we have denoted by the angle brackets ( ), is the easiest to deal with mathematically. Unfortunately, ensemble means are not measurable quantities, although under the conditions of the ergodic theorem they can be related to observable temporal or spatial averages. In Eq. (2.7) the mean concentration (c) represents a true ensemble mean, whereas if we decompose c as... 

Fig. 4.3 Scaling representation of the spin-echo data at the first static structure factor peak Qmax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve for 1,4-polybutadiene at Qmax=l-48 A L The scale r(T) is taken from a macroscopic viscosity measurement [130]. Inset Temperature dependence of the non-ergodicity parameter/(Q) near the lines through the points correspond to the MCT predictions (Eq. 4.37) (Reprinted with permission from [124]. Copyright 1988 The American Physical Society)...

Fig. 4.35 Right-hand side Monomeric friction coefficients derived from the viscosity measurements on PB [205]. The open and solid symbols denote results obtained from different molecular weights. Solid line is the result of a power-law fit. Dashed line is the Vogel-Fulcher parametrization following [205]. Left hand side Temperature dependence of the non-ergodicity parameter. The three symbols display results from three different independent experimental runs. Solid line is the result of a fit with (Eq. 4.37) (Reprinted with permission from [204]. Copyright 1990 The American Physical Society)...

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. 

Here K denotes the Lebesgue measure of K. Of crucial importance is the Birkhoff ergodic theorem which states that for / // (fi), a > 1,... 

As mentioned in the introduction, blinking NCs exhibit a nonergodic behavior. In particular the ensemble average intensity (I) is not equal to the time average 7. Of course in the ergodic phase—that is, when both the mean on and off times are finite—we have (I) = 7, in the limit of long measurement time. More generally... 

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