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The history of the ergodic hypothesis in statistical mechanics is marked by several changes in the content of the assumptions and is not a closed subject. For our purposes, we use the term ``ergodic'' only to imply the equivalence of ensemble and time averages. See A. S. Wightman, in Statistical Mechanics at the Turn of the Decade, edited by E. G. D. Cohen (Decker, New York, 1971), pp. 1-32, for a discussion of ergodic theory.
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5
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33847005283
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The ergodic hypothesis is also central in the computer simulation technique of molecular dynamics which is a well-established method for studying condensed matter systems. The essential aspects of this technique were described by
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(1964)
Phys. Rev.
, vol.136
, pp. A405
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Rahman, A.1
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14
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84927308503
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For a lucid discussion of the connection between rigidity and broken symmetry, see P. W. Anderson, Basic Notions of Condensed Matter Physics (Benjamin, Menlo Park, CA, 1984); see also D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, Reading, MA, 1975).
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16
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84927308502
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See also D. R. Nelson, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1983), Vol. 7.
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24
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0001663751
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The Transition from Analytic Dynamics to Statistical Mechanics
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For reviews describing the KAM theorem and its applications for dynamical systems see (a), in Fundamental Problems in Statistical Mechanics, edited by E. , North-Holland
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(1973)
Amsterdam, 1975) and Adv. Chem. Phys.
, vol.24
, pp. 155
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Ford, J.1
Cohen, G.D.2
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26
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84927308501
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(c) M. V. Berry, in Topics in Nonlinear Dynamics (La Jolla Institute), Proceeding of the Workshop on Topics in Nonlinear Dynamics, AIP Conf. Proc. No. 46, edited by S. Jorna (AIP, New York, 1978), p. 16;
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27
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(d) Y. M. Treve, ibid., p. 147.
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29
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84927308499
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The invariant measure refers to the set of configurations belonging to a specified component. These configurations are the set of phase points that lead to a given local free-energy minimum in the quench process.
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31
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84927308498
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The long-time limit of curlepaj(t) and curlepbj(t) are well-defined quantities. There will be 1/N corrections to the long-time limit of the quantities which have no consequence in the thermodynamic limit. These 1/N corrections are not relevant in the definition of the energy metric. Thus d(t) can be unambiguously defined even when N is finite.
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32
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0042091834
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(a) It has been suggested recently [T. R. Kirkpatrick and D. Thirumalai, J. Phys. A (to be published)] that these metastable states can be viewed as saddle point solutions which minimize the free energy calculated using a coarse-grained density-functional Hamiltonian. Such a precise characterization of the metastable state is only possible because the scenario suggested above is based on a mean-field theory. (b) For a heuristic argument suggesting that the number of metastable free-energy minima should grow exponentially with N see
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(1984)
Science
, vol.225
, pp. 983
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Stillinger, F.H.1
Weber, T.A.2
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38
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84927308497
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These ideas can be made precise using certain mean-field random Potts models.
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43
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84927308496
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J. Non-Cryst. Solids (to be published).
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Angell, C.A.1
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44
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See, for example, H.-U. Künzi, in Glassy Metals II, edited by H. Beck and H.-J. Güntherodt (Springer-Verlag, New York, 1983).
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50
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84927308494
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In this study the change of elastic modulus was found to be a few parts per million over a period of days.
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84927308493
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The averaging time should be larger than any transient time in the problem.
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55
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84927308492
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J.-B. Suck and H. Rudin, in Ref. 32.
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60
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84927308491
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The distribution function P( curlep ;t) can be roughly thought of as the density of single-particle states for the glass. It can be shown that variance of the distribution function is related to the Fourier transform of the dynamic energy correlation function < curlep (t) curlep (0) > . Using the Gaussian distribution P( curlep , τ ) approx exp (- curlep2/2 < curlep2> ), and by assuming that transport in the glassy phase proceeds when the energy of a single particle exceeds a typical barrier height beta, an average relaxation time can be computed. This would be given by τ ( β ) approx tintcurlepstarapprox kT inf exp (- curlep2/2 < curlep2> ) exp ( β curlep )d curlep. Evaluating this integral and taking appropriate limits, τ ( β ) can be expressed in terms of the variance.
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61
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If this variance is taken to be proportional to a characteristic temperature T0 [, ], then τ ( β ) approx exp [(T0/T)2]. The variance can be related to entropy changes and this would lead to τ ( β ) approx exp [A/(T-TK)2], where TK is the temperature where the excess configurational entropy vanishes.
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(1987)
H. Bassler, Phys. Rev. Lett.
, vol.58
, pp. 767
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63
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36549101292
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However, it should be pointed out that one expects structural relaxation in glasses to proceed via a correlated or cooperative mechanism involving more than one particle. A recent study [, ] has explicitly examined such motions in the glassy state. Thus a proper account of relaxation times has to consider the distribution of energy describing such events which is more difficult to compute. Using such distributions may lead naturally to the more accepted Vogel-Fulcher form for the temperature dependence of relaxation times.
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(1988)
J. Chem. Phys.
, vol.88
, pp. 3879
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Miyagawa, H.1
Hiwatari, Y.2
Bernu, B.3
Hansen, J.P.4
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For a numerical study of Arnold diffusion see B. B. Chirokov, J. Ford, and F. Vivaldi, in Nonlinear Dynamics and the Beam-Beam Interaction (Brookhaven National Laboratory, 1979), Proceedings of the Symposium on Nonlinear Dynamics and Beam-Beam Interaction, AIP Conf. Proc. No. 57, edited by M. Month and J. C. Herrea (AIP, New York, 1979), p. 323.
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