Abstract
We examine critically the issue of phase transitions in one-dimensional systems with short range interactions. We begin by reviewing in detail the most famous non-existence result, namely van Hove's theorem, emphasizing its hypothesis and subsequently its limited range of applicability. To further underscore this point, we present several examples of one-dimensional short ranged models that exhibit true, thermodynamic phase transitions, with increasing level of complexity and closeness to reality. Thus having made clear the necessity for a result broader than van Hove's theorem, we set out to prove such a general non-existence theorem, widening largely the class of models known to be free of phase transitions. The theorem is presented from a rigorous mathematical point of view although examples of the framework corresponding to usual physical systems are given along the way. We close the paper with a discussion in more physical terms of the implications of this non-existence theorem.
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REFERENCES
E. H. Lieb and D. C. Mattis, Mathematical Physics in One Dimension (Academic Press, London, 1966).
J. Bernasconi and T. Schneider, Physics in One Dimension (Springer, Berlin, 1981).
D. H. Dunlap, H. L. Wu, and P. Phillips, Phys. Rev. Lett. 65:88(1990).
F. A. B. F. de Moura and M. L. Lyra, Phys. Rev. Lett. 81:3735(1998).
P. Carpena, P. Bernaola-Galvá;n, P. C. Ivanov, and H. E. Stanley, Nature 418:955(2002).
P. Carpena, P. Bernaola-Galvá;n, P. C. Ivanov, and H. E. Stanley, Nature 421:764(2003).
M. R. Evans, Brazilian J. Phys. 30:42(2000).
L. van Hove, Physica 16:137(1950) (reprinted in ref. 1, p. 28).
D. Ruelle, Statistical Mechanics: Rigorous Results (Addison-Wesley, Reading, 1989).
L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, New York, 1980).
C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, 2000).
R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).
G. Forgacs, J. M. Luck, T. M. Nieuwenhuizen, and H. Orland, Phys. Rev. Lett. 57:2184(1986); B. Derrida, V. Hakim, and J. Vannimenus, J. Stat. Phys. 66:1189(1992); G. Giguliarelli and A. L. Stella, Phys. Rev. E 53:5035(1996); P. S. Swain and A. O. Parry, J. Phys. A 30:4597(1997); T. W. Burkhardt, J. Phys. A 31:L549(1998).
D. Ruelle, Comm. Math. Phys. 9:267(1968).
G. Rushbrooke and H. Ursell, Proc. Cambridge Phil. Soc. 44:263(1948).
F. J. Dyson, Comm. Math. Phys. 12:91(1969).
J. Fröhlich and T. Spencer, Comm. Math. Phys. 81:87(1982).
C. Kittel, Am. J. Phys. 37:917(1969).
J. F. Nagle, Am. J. Phys. 36:1114(1968).
H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1971).
K. Huang, Statistical Mechanics (Wiley, Singapore, 1987).
M. Plischke and B. Bergersen, Equilibrium Statistical Physics (World Scientific, Singapore, 1994).
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1995).
S. T. Chui and J. D. Weeks, Phys. Rev. B 23:2438(1981).
J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford University Press, Oxford, 1992).
T. W. Burkhardt, J. Phys. A 14:L63(1981).
T. Dauxois, M. Peyrard, and A. R. Bishop, Phys. Rev. E 47:R44(1993); T. Dauxois and M. Peyrard, Phys. Rev. E 51:4027(1995).
N. Theodorakopoulos, T. Dauxois, and M. Peyrard, Phys. Rev. Lett. 85:6(2000); T. Dauxois, N. Theodorakopoulos, and M. Peyrard, J. Stat. Phys. 107:869(2002).
N. Theodorakopoulos, http://arxiv.org/abs/cond-mat/0210188 (2002).
A. Campa and A. Giansanti, Phys. Rev. E 68:3585(1998).
P. Meyer-Nieberg, Banach Lattices (Springer, Berlin, 1991).
A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces (Springer, Berlin, 1997).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Dover, New York, 1993).
J. A. Cuesta and A. Sá;nchez, J. Phys. A 35:2373(2002).
T. Tsuzuki and K. Sasaki, Progr. Theoret. Phys. Supp. 94:73(1988).
S. Ares, J. A. Cuesta, A. Sá;nchez, and R. Toral, Phys. Rev. E 67:046108(2003).
M. V. Simkin and V. P. Roychowdhury, Complex Sys. 14:269(2003).
C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems: An Introduction (Cambridge University, Cambridge, 1993), Chapter 21.
D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, 1978).
V. Baladi, Positive Transfer Operators and Decay of Correlations (World Scientific, Singapore, 2000).
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An erratum to this article is available at http://dx.doi.org/10.1007/s10955-009-9862-6.
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Cuesta, J.A., Sánchez, A. General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions. Journal of Statistical Physics 115, 869–893 (2004). https://doi.org/10.1023/B:JOSS.0000022373.63640.4e
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DOI: https://doi.org/10.1023/B:JOSS.0000022373.63640.4e