Non-ergodic

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. 

In a perfect crystal at 0 K all atoms are ordered in a regular uniform way and the translational symmetry is therefore perfect. The entropy is thus zero. In order to become perfectly crystalline at absolute zero, the system in question must be able to explore its entire phase space the system must be in internal thermodynamic equilibrium. Thus the third law of thermodynamics does not apply to substances that are not in internal thermodynamic equilibrium, such as glasses and glassy crystals. Such non-ergodic states do have a finite entropy at the absolute zero, called zero-point entropy or residual entropy at 0 K. 

Here,/Qj ax( ) is a generalized Debye-Waller factor giving account for the decay of the correlations at faster times. In the framework of the MCT it is also called non-ergodicity factor. The characteristic timescale r (T) is the structural relaxation time and (3 is the stretching parameter (0[Pg.73]

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)...

If the non-ergodicity parameter is fixed to a T-independent valne, an increase of from 0.36 to 0.45 with increasing T in the T-range stndied is reported. 

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)...

In conclusion, it is worth reflecting on a classical trajectory study of neutral ethane [335] in which it was found that there were dynamical restrictions to intramolecular energy transfer among C—H motions and between these and C—C motions. It was pointed out [335] that this non-ergodicity might not produce results observable at present levels of experimental resolution. This is probably the situation in mass spectrometry. QET is a respected theory in mass spectrometry because, proceeding from clearly stated assumptions, it is mathematically tractable and is able to explain the currently available experimental data. 

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... 

Crossover Temperature for Various Glass Formers as Reported by the Different Methods From the Temperature Dependence of the Stretching Parameter y(T), Scaling the Time Constant xa — xa(r) [cf. Eq. (42)], Non-ergodicity Parameter 1 —f(T) Obtained from Spectra Analysis, Electron Paramagnetic Resonance (EPR), and from Tests of the Asymptotic Laws of Mode Coupling Theory ... 

Flere (1 - f) and (1 - S) are the losses brought about by the corresponding process and gfastj/3 (t -> oo) = 0 (see Fig. 6). The factor / can be regarded as a generalized non-ergodicity parameter and, hence, it is expected to show a similar anomaly as the Debye-Waller factor /g (see Fig. 5). Such decomposition of the correlation function is useful in spin-lattice relaxation studies, as will be discussed in Section 3.2.4. 

Fig. 5. Non-ergodicity parameter fQ (Debye-Waller factor) of o-terphenyl as obtained from NS experiments for different values of the momentum transfer Q. (Adapted from Ref. 60.)...

In MCT, the non-ergodicity parameter /e(7) is the crucial quantity. It describes the long-time behavior of S(Q,t). In its idealized version, MCT predicts a discontinuous change at a critical temperature T223... 

Whereas the correlation function S(Q, t) decays to zero in the liquid state, this is no longer the case below Tc, where the system becomes non-ergodic. Such scenario results from a generalized oscillator equation with non-linear damping. The slowing down of the molecular dynamics creates an enhanced damping, which in turn slows down the correlation function etc. The density is taken as a control parameter. In Fig. 38a, /(the Q dependence is omitted in the most simple approach) is displayed... 

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