A theorem on the absence of phase transitions in one-dimensional growth models with on-site periodic potentials

Cuesta, José A.; Sánchez, Ángel

doi:10.1088/0305-4470/35/10/303

Cited by 15 publications

(25 citation statements)

References 20 publications

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“…Note that this does not imply that it must exhibit a phase transition: indeed, in [34] it was shown that a suitable change of variables casts the operator in a form compatible with the theorem, thus establishing the impossibility of phase transitions in this model (and in fact in a much wider class). Interestingly, the same problem arises in van Hove's theorem [8]; van Hove writes first his general transfer operator in a non-compact form, but he is able to rewrite it as a compact operator and to prove his theorem.…”

Section: Previous Examples Of Phase Transitions In the Context Of Thementioning

confidence: 99%

General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions

Cuesta

Sánchez

2004

Journal of Statistical Physics

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105

100

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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.Running head: Phase transitions in short-ranged 1D systems

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Section: Previous Examples Of Phase Transitions In the Context Of Thementioning

confidence: 99%

General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions

Cuesta

Sánchez

2004

Journal of Statistical Physics

Self Cite

105

100

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“…Thus, in the combined high-temperature and high-density limit the PS model is amenable to an exact analytical treatment for any number of dimensions [26]. An interesting consequence of the boundedness of the PS potential in the 1D case is the plausible existence of a fluid-crystal phase transition [26], thus providing one of the rare examples of phase transitions in 1D systems [32]. In the complementary low-density domain, the cavity function y(r) has been recently determined to second order in density at any temperature [30].…”

Section: Introductionmentioning

confidence: 99%

Low-temperature and high-temperature approximations for penetrable-sphere fluids: Comparison with Monte Carlo simulations and integral equation theories

2007

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The two-body interaction in dilute solutions of polymer chains in good solvents can be modeled by means of effective bounded potentials, the simplest of which being that of penetrable spheres (PSs). In this paper we construct two simple analytical theories for the structural properties of PS fluids: a low-temperature (LT) approximation, that can be seen as an extension to PSs of the well-known solution of the Percus-Yevick (PY) equation for hard spheres, and a high-temperature (HT) approximation based on the exact asymptotic behavior in the limit of infinite temperature. Monte Carlo simulations for a wide range of temperatures and densities are performed to assess the validity of both theories. It is found that, despite their simplicity, the HT and LT approximations exhibit a fair agreement with the simulation data within their respective domains of applicability, so that they complement each other. A comparison with numerical solutions of the PY and the hypernetted-chain approximations is also carried out, the latter showing a very good performance, except inside the core at low temperatures.

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“…A surprising consequence of the boundedness of the PS potential in the 1D case is the plausible existence of a fluid-crystal phase transition, 18 thus providing one of the rare examples of phase transitions in 1D systems. 21 Statistical mechanics has a long tradition of studying 1D systems, especially in those cases where an exact solution to the many-body problem has been found. 22 Of course, the 1D PS model does not intend to describe all the properties of real polymers in solution, for which spatial dimensionality is known to be important.…”

Section: Introductionmentioning

confidence: 99%

Structure of penetrable-rod fluids: Exact properties and comparison between Monte Carlo simulations and two analytic theories

Malijevský

Santos

2006

The Journal of Chemical Physics

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Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as the dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper the structural properties of one-dimensional penetrable rods are studied. We first derive the exact correlation functions of the penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. It is seen that, in contrast to what is generally believed, the Percus-Yevick equation does not yield the exact cavity function in the hard-rod limit. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit T--> infinity and a low-temperature approximation inspired by the exact result in the opposite limit T--> 0. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A comparison with numerical solutions of the Percus-Yevick and the hypernetted-chain approximations is also carried out. Finally, a perspective on the extension of our two heuristic theories to the more realistic three-dimensional case is provided.

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A theorem on the absence of phase transitions in one-dimensional growth models with on-site periodic potentials

Cited by 15 publications

References 20 publications

General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions

General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions

Low-temperature and high-temperature approximations for penetrable-sphere fluids: Comparison with Monte Carlo simulations and integral equation theories

Structure of penetrable-rod fluids: Exact properties and comparison between Monte Carlo simulations and two analytic theories

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