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

Journal of Statistical Physics

Volumn 103, Issue 3-4, 2001, Pages 601-618

  Aizenman, M ^a   Goldstein, S ^b   Lebowitz, J L ^b  

^a PRINCETON UNIVERSITY   (United States)

^b RUTGERS UNIVERSITY   (United States)

Author keywords

Bounded variance; One dimension particle systems; Symmetry breaking

Indexed keywords

EID: 0035534757     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1010397401128     Document Type: Article

References (33)

1. 1. 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).
2. 1. 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).
3. 2. 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).
4. 3. 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).
5. 3. 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).
6. 4. M. Aizenman and P. A. Martin, Structure of Gibbs states of one-dimensional Coulomb systems, Comm. Math. Phys. 78:99 (1980).
7. 5. 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).
8. 5. 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).
9. 5. 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).
10. 6. 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).
11. 6. 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).
12. 6. 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).
13. 7. M. Gaudin, Gaz coulombien discret à une dimension, J. Physique 34:511 (1973).
14. 8. H. Kunz, The one-dimensional classical electron gas, Ann. Phys. (N.Y.) 85:303 (1974).
15. 9. 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).
16. 10. Ch. Lugrin and Ph. A. Martin, Functional integration treatment of one-dimensional ionic mixtures, J. Math. Phys. 23:2418 (1982).
17. 11. B. Jancovici, Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46:386 (1981).
18. 12. J. Imbrie, Debye screening for jellium and other Coulomb systems, Commun. Math. Phys. 87:515 (1983).
19. 13. I. Affleck, Exact results on the dimerization transition in SU(n) antiferromagnetic chains. J. Phys. Cond. Matt. 2:405 (1990).
20. 14. M. Aizenman and B. Nachtergaele. Geometric aspects of quantum spin states. Comm. Math. Phys. 164:17 (1994).
21. 15. M. T. Batchelor and M. N. Barber, Spin-s quantum chains and Temperley-Lieb algebras. J. Phys. A 23:L15-L21 (1990).
22. 16. A. Klümper, The spectra of q-state vertex models and related antiferromagnetic quantum spin chains, J. Phys. A 23:809 (1990).
23. 17. M. Aizenman and J. Fröhlich, States of one-dimensional Coulomb systems as simple examples of Θ-vacua and confinement. J. Stat. Phys. 26:247 (1981).
24. 18. Ch. Maes, A. van Moffaert, and F. Redig, Almost Gibbsian versus weakly Gibbsian measures, Stoch. Proc. and Appl. 79:1-15 (1999).
25. 19. M. L. Mehta, Random Matrices (Academic Press, New York, 1990), 2nd ed.
26. 20. J. Beck, Irregularities of distribution. I, Acta Math. 159:1 (1987).
27. 21. 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.
28. 21. 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.
29. 22. 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).
30. 23. H.-O. Georgii, Gibbs Measures and Phase Transitions (W. De Gruyter, Berlin, 1988).
31. 24. K. Schmidt, Cocycles of Ergodic Transformation Groups. Lect, Notes in Math., Vol. 1 (MacMillan Co. of India, 1977).
32. 25. V. Leonov, On the dispersion of time averages of a stationary random process, Theory Probab. Appl. 6:93 (1961).
33. 26. J. Aaronson and B. Weiss, Remark on tightness of cocycles, (1999) preprint.