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Volumn 60, Issue 5, 1999, Pages 5244-5260

Metastable states in spin glasses and disordered ferromagnets

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ARTICLE;

EID: 0013248235     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.60.5244     Document Type: Article
Times cited : (41)

References (79)
  • 1
    • 85036353436 scopus 로고    scopus 로고
    • Recent Progress in Random Magnets, edited by D.H. Ryan, (World Scientific, Singapore, 1992)
    • Recent Progress in Random Magnets, edited by D.H. Ryan, (World Scientific, Singapore, 1992).
  • 19
    • 85036188779 scopus 로고    scopus 로고
    • For recent papers that study metastability as a unifying theme in glasses, spin glasses, neural network models, and other disordered systems, see G. Parisi, in 25 Years of Non-Equilibrium Statistical Mechanics, edited by J.J. Brey, J. Marro, J.M Rubi, and M. San Miguel (Springer-Verlag, Berlin, 1995), pp. 135–142;, K. Binder, ibid., pp. 143–160;, D. Sherrington and A.C.C. Coolen, ibid., pp. 161–176
    • For recent papers that study metastability as a unifying theme in glasses, spin glasses, neural network models, and other disordered systems, see G. Parisi, in 25 Years of Non-Equilibrium Statistical Mechanics, edited by J.J. Brey, J. Marro, J.M Rubi, and M. San Miguel (Springer-Verlag, Berlin, 1995), pp. 135–142;K. Binder, ibid., pp. 143–160;D. Sherrington and A.C.C. Coolen, ibid., pp. 161–176.
  • 24
    • 84884904599 scopus 로고
    • I. Morgenstern, ibid., pp. 305–327;, A.P. Young, ibid., pp. 328–345., J.L. van Hemmen, I. Morgenstern, Springer-Verlag, Berlin
    • See, for example, K. Binder and W. Kinzel, in Heidelberg Colloquium on Spin Glasses, edited by J.L. van Hemmen and I. Morgenstern (Springer-Verlag, Berlin, 1983), pp. 279–304;I. Morgenstern, ibid., pp. 305–327;A.P. Young, ibid., pp. 328–345.
    • (1983) Heidelberg Colloquium on Spin Glasses , pp. 279-304
    • Binder, K.1    Kinzel, W.2
  • 29
    • 36149042419 scopus 로고
    • A.J. Bray and M.A. Moore, J. Phys. C 13, L469 (1980). By studying the number of solutions to the TAP equations 32, the authors were also able to draw some conclusions about numbers of metastable states for (Formula presented)
    • (1980) J. Phys. C , vol.13
    • Bray, A.J.1    Moore, M.A.2
  • 30
    • 0019061437 scopus 로고
    • This paper also contains some results on numbers of metastable states near (Formula presented)
    • C. De Dominicis, M. Gabay, T. Garel, and H. Orland, J. Phys. (Paris) 41, 923 (1980). This paper also contains some results on numbers of metastable states near (Formula presented)
    • (1980) J. Phys. (Paris) , vol.41 , pp. 923
    • De Dominicis, C.1    Gabay, M.2    Garel, T.3    Orland, H.4
  • 40
  • 45
    • 85036401568 scopus 로고    scopus 로고
    • As in previous treatments (see, however, Refs. 29 and 30), our results are confined to energetically stable states. In dynamical treatments, one finds that states that are locally stable with respect to small free energy changes are often invoked. It is not clear how to define such objects, or their companions such as free energy barriers, in a clear and unambiguous way. We do not consider this to be a serious impediment for two reasons: first, at least at low temperatures, the properties of such states are likely to be closely related to the energetically stable states discussed here; and second, we are able to consider the properties of M-spin-flip energetically stable states for arbitrarily large (finite) M. We believe that it is reasonable to expect that these states will play an important dynamical role at positive temperature, presumably with larger M becoming relevant for higher temperatures
    • As in previous treatments (see, however, Refs. 29 and 30), our results are confined to energetically stable states. In dynamical treatments, one finds that states that are locally stable with respect to small free energy changes are often invoked. It is not clear how to define such objects, or their companions such as free energy barriers, in a clear and unambiguous way. We do not consider this to be a serious impediment for two reasons: first, at least at low temperatures, the properties of such states are likely to be closely related to the energetically stable states discussed here; and second, we are able to consider the properties of M-spin-flip energetically stable states for arbitrarily large (finite) M. We believe that it is reasonable to expect that these states will play an important dynamical role at positive temperature, presumably with larger M becoming relevant for higher temperatures.
  • 53
    • 85036147569 scopus 로고    scopus 로고
    • Almost every (whether it’s referring to (Formula presented) or some function of them like the metastable states that are the prime subject of this paper) means a set of probability 1, where the probability measure (i.e., ensemble) in question should be clear from the context. What is left out of a set of probability 1 is insignificant in the sense that it does not contribute to any quantitites (such as overlap distributions) defined in terms of an average over the relevant measure. The distinction between almost every and every is significant and forgetting it can easily lead to confusion and seeming contradiction. So, for example, most (but not all) of the results in this paper about metastable states are only valid for almost every one and these are precisely the ones that are relevant for deep quench dynamics
    • Almost every (whether it’s referring to (Formula presented) or some function of them like the metastable states that are the prime subject of this paper) means a set of probability 1, where the probability measure (i.e., ensemble) in question should be clear from the context. What is left out of a set of probability 1 is insignificant in the sense that it does not contribute to any quantitites (such as overlap distributions) defined in terms of an average over the relevant measure. The distinction between almost every and every is significant and forgetting it can easily lead to confusion and seeming contradiction. So, for example, most (but not all) of the results in this paper about metastable states are only valid for almost every one and these are precisely the ones that are relevant for deep quench dynamics.
  • 60
    • 85036370319 scopus 로고    scopus 로고
    • Random Struct. Algorithms (to be published)
    • S. Nanda and C.M. Newman, Random Struct. Algorithms (to be published).
    • Nanda, S.1    Newman, C.M.2
  • 65
    • 0000265842 scopus 로고    scopus 로고
    • For recent numerical work on one such model, a 2D bond-diluted ferromagnet, see S. Jain, Phys. Rev. E 59, 2493R (1999).
    • (1999) Phys. Rev. E , vol.59
    • Jain, S.1
  • 66
    • 85036424713 scopus 로고    scopus 로고
    • The proof shows more than uncountability of all the M-spin-flip-stable states: the probability measure on the (Formula presented)’s (inherited from its dependence on (Formula presented) and (Formula presented) and their probability measures) is continuous—i.e., no single spin configuration has strictly positive weight
    • The proof shows more than uncountability of all the M-spin-flip-stable states: the probability measure on the (Formula presented)’s (inherited from its dependence on (Formula presented) and (Formula presented) and their probability measures) is continuous—i.e., no single spin configuration has strictly positive weight.
  • 70
    • 85036326904 scopus 로고    scopus 로고
    • For example, one might consider the following situation. Allow single-spin-flip dynamics to run until convergence to a specific (Formula presented) Using this as the starting state, raise the temperature to a small nonzero value T and allow positive-temperature Glauber dynamics to run for a specified time (Formula presented) after which the temperature is reduced back to zero. It is not unreasonable to speculate that the Hamming distance of the final state (Formula presented) reached in this way from the initial (Formula presented) increases with (Formula presented) for (Formula presented) small. Such considerations may be relevant to some aging experiments 4, but our analysis makes clear that any possible correlations of this kind cannot be extended to provide information on the properties of the larger set of metastable states. This example illustrates that one needs to be careful to specify what is meant, for example, by correlations between barriers and overlaps; the meanings may be determined by the experimental situation under investigation
    • For example, one might consider the following situation. Allow single-spin-flip dynamics to run until convergence to a specific (Formula presented) Using this as the starting state, raise the temperature to a small nonzero value T and allow positive-temperature Glauber dynamics to run for a specified time (Formula presented) after which the temperature is reduced back to zero. It is not unreasonable to speculate that the Hamming distance of the final state (Formula presented) reached in this way from the initial (Formula presented) increases with (Formula presented) for (Formula presented) small. Such considerations may be relevant to some aging experiments 4, but our analysis makes clear that any possible correlations of this kind cannot be extended to provide information on the properties of the larger set of metastable states. This example illustrates that one needs to be careful to specify what is meant, for example, by correlations between barriers and overlaps; the meanings may be determined by the experimental situation under investigation.
  • 77
    • 85036407979 scopus 로고    scopus 로고
    • The energy density results are for strongly disordered coupling distributions with finite mean
    • The energy density results are for strongly disordered coupling distributions with finite mean.


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