Canonical solution of a system of long-range interacting rotators on a lattice

Campa, Alessandro; Giansanti, Andrea; Moroni, Daniele

doi:10.1103/physreve.62.303

Cited by 74 publications

(97 citation statements)

References 6 publications

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“…Besides confirming the scaling properties with α, they have also shown that the number of states is of order exp(N ). The exact solution of these models in the canonical ensemble has been obtained by Campa et al [43] and Vollmayr-Lee and Luijten [44]. The scaling of the magnetization and of the energy curve with the α-exponent has been exhibited in full detail.…”

Section: The Ising Model With 1/r α Interactionsmentioning

confidence: 99%

“…In all these cases the interaction is singular at shortrange. Interesting toy models (which generalize the HMF model) without a short distance singularity have been recently proposed [41,42,43,44]. In these latter models, XY spins are put on a lattice and interact through a slowly decreasing non integrable 1/r α law (α < d).…”

Section: The Ising Model With 1/r α Interactionsmentioning

confidence: 99%

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Large Deviation Techniques Applied to Systems with Long-Range Interactions

et al. 2005

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We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by R.S. Ellis, Physica D 133, 106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibrium effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the α-Ising model in one-dimension with 0 ≤ α < 1. Keywords: Long-range interactions, Large deviation techniques, Mean-field limit.PACS numbers: 05.20.-y Classical statistical mechanics A system with long-range interactions is characterized by an interparticle potential V (r) which decreases at large distances r slower than a power r −α with α < d, d being the dimension of the embedding space [1]. Classical examples are self-gravitating [2] and Coulomb [3] systems, vortices in two-dimensional fluid mechanics [4], waveparticles interaction [5,6] and trapped charged particles [7]. The behaviour of such systems is interesting both from the dynamical point of view, because they display peculiar quasi-stationary states that are related to the underlying Vlasov equations [8], and from the static point of view, because equilibrium statistical mechanics shows new types of phase transitions and cases of ensemble inequivalence [9]. In this paper we will restrict ourselves to the second aspect.In long-range interacting systems, essentially all the particles contribute to the local field: the fluctuations around the mean value are small because of the law of large numbers. This explains qualitatively why the mean-field scaling, which amounts to let the number of particles go to infinity at fixed volume [10,11], is usually extremely good. However, let us remind that for long-range interacting systems, microcanonical and canonical ensembles are not necessarily equivalent in the mean-field limit [12,13,14]. Moreover, because of the non additivity of the energy, the usual construction of the canonical ensemble cannot be applied. This is the reason why the microcanonical ensemble is considered by some authors [2,15] as the only "physically motivated" one. Hence, it is extremely important to develop rigorous techniques to solve non trivial physical models in the microcanonical ensemble. One finds in books the solution for the perfect gas, but generalizations to interacting particle systems are difficult.The goal of this paper is to advocate the use of large deviation techniques as a tool to explicitly derive microcanonical and canonical equilibrium solutions for a wide class of models. As a first step in this direction, we will discuss here a general solution method to treat in full detail mean-field models without short distance singu...

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Section: The Ising Model With 1/r α Interactionsmentioning

confidence: 99%

Section: The Ising Model With 1/r α Interactionsmentioning

confidence: 99%

Large Deviation Techniques Applied to Systems with Long-Range Interactions

et al. 2005

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“…Next, we consider the system at K < 1=2. This case has been studied in the context of discrete XY models [36][37][38] and was shown to have a phase transition in the limit that the coupling vanishes as a power of the system size, which in our case is ¼ 1=T; i.e., there is a critical value for ð Þ 1À2K . Hence, at T ¼ 0 the system is fully ordered and hcosX i ¼ 1.…”

mentioning

confidence: 99%

Phase Transitions for a Collective Coordinate Coupled to Luttinger Liquids

2013

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We study various realizations of collective coordinates, e.g., the position of a particle, the charge of a Coulomb box, or the phase of a Bose or a superconducting condensate, coupled to Luttinger liquids with N flavors. We find that for a Luttinger parameter ð1=2Þ < K < 1 there is a phase transition from a delocalized phase into a phase with a periodic potential at strong coupling. In the delocalized phase the dynamics is dominated by an effective mass, i.e., diffusive in imaginary time, while on the transition line it becomes dissipative. At K ¼ ð1=2Þ there is an additional transition into a localized phase with no diffusion at zero temperature.

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“…Within the class of long-range interacting systems, classical spin models, widely investigated during the last years [7], are the most easy-to-handle both from the analytical and the numerical point of view. Within such class of systems, (to be more precise, a class of anisotropic Heisenberg models) the existence of a threshold of disconnection in the energy surface has been demonstrated [1] for an interparticle interaction with infinite range.…”

Section: Introductionmentioning

confidence: 99%

Topological nonconnectivity threshold in long-range spin systems

et al. 2006

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We demonstrate the existence of a topological disconnection threshold, recently found in Ref.[1], for generic 1 − d anisotropic Heisenberg models interacting with an inter-particle potential R −α when 0 < α < 1 (here R is the distance among spins). We also show that if α is greater than the embedding dimension d then the ratio between the disconnected energy region and the total energy region goes to zero when the number of spins becomes very large. On the other hand, numerical simulations in d = 2, 3 for the long-range case α < d support the conclusion that such a ratio remains finite for large N values. The disconnection threshold can thus be thought as a distinctive property of anisotropic long-range interacting systems.

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Canonical solution of a system of long-range interacting rotators on a lattice

Cited by 74 publications

References 6 publications

Large Deviation Techniques Applied to Systems with Long-Range Interactions

Large Deviation Techniques Applied to Systems with Long-Range Interactions

Phase Transitions for a Collective Coordinate Coupled to Luttinger Liquids

Topological nonconnectivity threshold in long-range spin systems

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