Bounded fluctuations and translation symmetry breaking in one-dimensional particle systems

M. Aizenman, S. Goldstein, J. L. Lebowitz

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19 Scopus citations


We present general results for one-dimensional systems of point charges (signed point measures) on the line with a translation invariant distribution μ for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the charges are restricted to multiples of a common unit, and their average charge density does not vanish, then the boundedness of the variance implies translation-symmetry breaking - in the sense that there exists a function of the charge configuration that is nontrivially periodic under translations and hence that μ is not "mixing." Analogous results are formulated also for one dimensional lattice systems under some constraints on the values of the charges at the lattice sites and their averages. The general results apply to one-dimensional Coulomb systems, and to certain spin chains, putting on common grounds different instances of symmetry breaking encountered there.

Original languageEnglish (US)
Pages (from-to)601-618
Number of pages18
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - May 2001

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Bounded variance
  • One dimension particle systems
  • Symmetry breaking


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