Abstract
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.
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REFERENCES
D. C. Brydges and P. A. Martin, Coulomb systems at low density: A review, J. Stat. Phys. 96:1163 (1999). See also Ph. Martin, Sum rules in charged fluids, Rev. Mod. Phys. 60:1075-1127 (1988).
A. Lenard, Exact statistical mechanics of a one-dimensional system with Coulomb forces. III. Statistics of the electric field, J. Math. Phys. 4:533 (1963).
J. L. Lebowitz and E. H. Lieb, The existence of thermodynamics for real matter with Coulomb forces, Phys. Rev. Lett. 22:631 (1969). E. H. Lieb and J. L. Lebowitz, The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei, Advances in Math. 9:316 (1972).
M. Aizenman and P. A. Martin, Structure of Gibbs states of one-dimensional Coulomb systems, Comm. Math. Phys. 78:99 (1980).
H. van Beijeren and B. U. Felderhof, Mol. Phys. 38:1179 (1979); Ch. Gruber, Ch. Lugrin, and Ph. A. Martin, J. Stat. Phys. 22:193 (1980); Ph. A. Martin and T. Yalcin, J. Stat. Phys. 22:435 (1980).
J. L. Lebowitz, Phys. Rev. A 72:773 (1983); J. L. Lebowitz in Strongly Coupled Coulomb Systems, G. J. Kalman, J. M. Rommel, and K. Blagoev, eds. (Plenum Press, New York and London, 1998); D. Levesque, J.-J. Weis, and J. L. Lebowitz, Charge fluctuations in the two-dimensional one-component plasma, J. Stat. Phys. 100:209-222 (2000).
M. Gaudin, Gaz coulombien discret à une dimension, J. Physique 34:511 (1973).
H. Kunz, The one-dimensional classical electron gas, Ann. Phys. (N.Y.) 85:303 (1974).
H. J. Brascamp and E. H. Lieb, Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma, in Functional Integration and Its Applications, A. M. Arthurs, ed. (Oxford Univ. Press, 1975).
Ch. Lugrin and Ph. A. Martin, Functional integration treatment of one-dimensional ionic mixtures, J. Math. Phys. 23:2418 (1982).
B. Jancovici, Exact results for the two-dimensional one-component plasma, Phys. Rev. Lett. 46:386 (1981).
J. Imbrie, Debye screening for jellium and other Coulomb systems, Commun. Math. Phys. 87:515 (1983).
I. Affleck, Exact results on the dimerization transition in SU(n) antiferromagnetic chains, J. Phys. Cond. Matt. 2:405 (1990).
M. Aizenman and B. Nachtergaele, Geometric aspects of quantum spin states, Comm. Math. Phys. 164:17 (1994).
M. T. Batchelor and M. N. Barber, Spin-s quantum chains and Temperley-Lieb algebras, J. Phys. A 23:L15-L21 (1990).
A. Klümper, The spectra of q-state vertex models and related antiferromagnetic quantum spin chains, J. Phys. A 23:809 (1990).
M. Aizenman and J. Fröhlich, States of one-dimensional Coulomb systems as simple examples of 3-vacua and confinement, J. Stat. Phys. 26:247 (1981).
Ch. Maes, A. van Moffaert, and F. Redig, Almost Gibbsian versus weakly Gibbsian measures, Stoch. Proc. and Appl. 79:1-15 (1999).
M. L. Mehta, Random Matrices (Academic Press, New York, 1990), 2nd ed.
J. Beck, Irregularities of distribution. I, Acta Math. 159:1 (1987).
P. Choquard, P. J. Forrester, and E. R. Smith, The two dimensional one component plasma at T=2; The semiperiodic strip, J. Stat. Phys. 33:13 (1983); B. Jancovici and J. L. Lebowitz, Bounded fluctuations and translation symmetry breaking: A solvable model, J. Stat. Phys., this issue.
A. C. D. van Enter, R. Fernandez, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory, J. Stat. Phys. 72:879-1167 (1993).
H.-O. Georgii, Gibbs Measures and Phase Transitions (W. De Gruyter, Berlin, 1988).
K. Schmidt, Cocycles of Ergodic Transformation Groups. Lect. Notes in Math., Vol. 1 (MacMillan Co. of India, 1977).
V. Leonov, On the dispersion of time averages of a stationary random process, Theory Probab. Appl. 6:93 (1961).
J. Aaronson and B. Weiss, Remark on tightness of cocycles, (1999) preprint.
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Aizenman, M., Goldstein, S. & Lebowitz, J.L. Bounded Fluctuations and Translation Symmetry Breaking in One-Dimensional Particle Systems. Journal of Statistical Physics 103, 601–618 (2001). https://doi.org/10.1023/A:1010397401128
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DOI: https://doi.org/10.1023/A:1010397401128