Ergodic Hypothesis

We have now presented two averages for macroscopic properties 1. The trajectory average over consecutive time points  

The ensemble average over distributed points in phase space  

A fundamental hypothesis in statistical thermodynamics is that the two averages are equal. This is the ergodic hypothesis. The physical implication of the ergodic hypothesis is that any system afforded with infinite time will visit all the points of phase space with a frequency proportional to their probability density. In other words, in a single trajectory the system spends an amount of time at each microstate that is proportional to its probability. The ergodic hypothesis is still a hypothesis because there has been no formal proof of its truth. 

Nonetheless, the ergodic hypothesis has logical, physical underpinnings. If one does not accept the ergodic hypothesis, one is faced with logical inconsistencies in the physical description of matter. Importantly, the adoption of the ergodic hypothesis results in a correct prediction of thermodynamic quantities of matter. This is the ultimate test for any hypothesis or theory. 

It is the ergodic hypothesis that allowed Gibbs to shift attention from trajectories to probabihties in phase space. Instead of considering orbits of microstate chains crossing the phase space in time, one can envision the phase space as a continuum with a position-dependent density. Because the latter can be determined more readily than the former, statistical thermodynamics can be employed to connect microscopic to macroscopic states. 

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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. 

The ergodic hypothesis essentially states that no matter where a system is started, it is possible to get to any other point in phase space. For U and A this leads to the following expressions. 

With a computer program that evaluates r as a function of time for a given U(r) we can use the ergodic hypothesis (which states that the time average over a long time is equal to the configuration average) and write... 

In order to evaluate the autocorrelation function we again exploit the ergodic hypothesis and replace the average over phase space ( ) by a time average writing,... 

The second fundamental assumption is the ergodic hypothesis. Accord-... 

In its extreme form the ergodic hypothesis is clearly untenable. Only probability statements can be made in statistical mechanics and these have nothing to do with sequences in time [117]. Not surprisingly, a totally convincing proof of the ergodic theorem in its many guises has not been formulated. The current concensus still is that an axiomatic basis, completely independent of ergodic theory should be chosen [115] for the construction of statistical mechanics. 

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 ... 

In this section we present a derivation of the FT based on stochastic dynamics. In contrast to deterministic systems, stochastic dynamics naturally incorporates crucial assumptions needed for the derivation, such as the ergodicity hypothesis. The derivation we present here follows the approach introduced by Crooks-Kurchan-Lebowitz-Spohn [38, 39] and includes some results recently obtained by Seifert [40] using Langevin systems. 

The ergodic hypothesis assumes Eq. (3.9) to be valid and independent of choice of to. It has been proven for a hard-sphere gas that Eqs. (3.5) and (3.9) are indeed equivalent (Ford 1973). No such proof is available for more realistic systems, but a large body of empirical evidence suggests that the ergodic hypothesis is valid in most molecular simulations. 

Here, the angular brackets denote an ensemble average, which is the same as time average from the ergodic hypothesis. 8ap and 8,- are Kronecker delta, and 8(f — t ) is Dirac s delta function. 

Molecular dynamics is frequently portrayed as a method based on the ergodicity hypothesis which states that the trajectory of a system propagating in time through the phase space following the Newtonian laws of motion given by the equations ... 

An alternative to MD, not relying upon the ergodicity hypothesis, is the Monte-Carlo procedure which yields the required thermodynamical average of observable A by performing the numerical estimate of the following integral (note that A does not depend on momenta) ... 

Perhaps the point to emphasise in discussing theories of translational energy release is that the quasiequilibrium theory (QET) neither predicts nor seeks to describe energy release [576, 720], Neither does the Rice— Ramspergei Kassel—Marcus (RRKM) theory, which for the purposes of this discussion is equivalent to QET. Additional assumptions are necessary before QET can provide a basis for prediction of energy release (see Sect. 8.1.1) and the nature of these assumptions is as fundamental as the assumption of energy randomisation (ergodic hypothesis) or that of separability of the transition state reaction coordinate (Sect. 2.1). The only exception arises, in a sense by definition, with the case of the loose transition state [Sect. 8.1.1(a)]. 

This special choice is generally taken over in the later literature, even though the ergodic hypothesis is then not even mentioned.108... 

It is advisable therefore to call the density distributions (30) and (31) ergodic in order to remind ourselves that so far we have no other justification for their choice than the invocation of the ergodic hypothesis. 

It is at this point that the hypothesis about the gas model being ergodic enters. Because of the doubts about the internal consistency of the ergodic hypothesis, this investigation cannot be considered free of objection (Section 10a).111... 

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