On the Mean-Field Spherical Model

Kastner, Michael; Schnetz, Oliver

doi:10.1007/s10955-005-8031-9

Cited by 35 publications

(60 citation statements)

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“…This result is in agreement with the nonanalytic behavior of the exact solution for the density of states of the mean-field spherical model as reported in [2]. In other words, the density of states Ω N becomes "smoother" with increasing number of degrees of freedom, and already for moderate N it supposedly will be impossible to observe such a finite-system nonanalyticity from noisy experimental or numerical data.…”

supporting

confidence: 91%

“…This result, apart from its essential role for further steps of our analysis, is of interest in its own, giving a model-independent and quantitative account of nonanalyticities in the finite-system entropy as observed for the special cases in [2,3,4]. Then, inspired by a calculation in [6], the density of states is split into two terms: The first is a sum of the leading nonanalytic contributions stemming from all the saddle points, whereas the second contains the "harmless" rest.…”

mentioning

confidence: 93%

“…Recently, however, it was observed that the microcanonical entropy, or Boltzmann entropy, of a finite system is not necessarily analytic, and nonanalytic entropy functions of finite systems have been reported for certain classical models [2,3,4].…”

mentioning

confidence: 99%

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Phase Transitions from Saddles of the Potential Energy Landscape

2007

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The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived. For large systems, the order of the nonanalytic term increases unboundedly, leading to an increasing differentiability of the entropy. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, our results provide an explanation of how, and under which circumstances, saddle points of the potential energy landscape may (or may not) be at the origin of a phase transition in the thermodynamic limit. As an application, the puzzling observations by Risau-Gusman et al. on topological signatures of the spherical model are elucidated. Phase transitions, like the boiling and evaporating of water at a certain temperature and pressure, are common phenomena both in everyday life and in almost any branch of physics. Loosely speaking, a phase transition brings about a sudden change of the macroscopic properties of a many-particle system while smoothly varying a parameter (the temperature or the pressure in the above example). The mathematical description of phase transitions is conventionally based on (grand)canonical thermodynamic functions, relating their loss of analyticity to the occurrence of a phase transition. Such a nonanalytic behavior in a (grand)canonical thermodynamic function can occur only in the thermodynamic limit in which the number of degrees of freedom N of the system goes to infinity [1].Many researchers took it for granted that the same were true also for microcanonical thermodynamic functions. Recently, however, it was observed that the microcanonical entropy, or Boltzmann entropy, of a finite system is not necessarily analytic, and nonanalytic entropy functions of finite systems have been reported for certain classical models [2,3,4].In light of their conceptual similarity to the definition of a phase transition, it is tempting to regard finitesystem nonanalyticities of the entropy as phase transitionlike phenomena. This point of view is advocated in [4], and the authors of that reference argue that such nonanalyticities should also be measurable experimentally, at least in very small systems. However, such an interpretation is complicated by the fact that, as discussed in [3], for typical models the number of nonanalytic points of the entropy increases unboundedly with the number of degrees of freedom N .Because of their typically large number, one might assume that nonanalyticities of the finite-system entropy were unrelated to the occurrence of a phase transition in the thermodynamic limit. A theorem by Franzosi and Pettini [5], however, indicates the existence of a relation between finite-system and infinite-system nonanalyticities (and we will come back to that theorem later). The purpose of the present Letter is to clarify and quantify the ...

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Phase Transitions from Saddles of the Potential Energy Landscape

2007

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“…In these models the microscopic PTs separate energetically different dissociation states. Their number and formal order increases with increasing particle number N [57]. In general, our results indicate that, typically, microscopic PTs in the MCE are accompanied by strong qualitative changes of the thermodynamic observables, as e.g.…”

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confidence: 56%

Phase transitions in small systems: Microcanonical vs. canonical ensembles

Dunkel

Hilbert

2006

Physica A: Statistical Mechanics and its Applications

106

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We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (nonanalyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g. temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.

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“…These two levels of large deviations can also be used in general to study the equilibrium properties of mean-field models of particles involving an all-to-all coupling between particles. Examples of such models, for which the general maximum entropy principles mentioned before have been applied successfully, include the mean-field versions of the Curie-Weiss model [82,84,85] and its parent model, the Potts model [48,85,92,225], the Blume-Emery-Griffiths model [11,90,91], the mean-field Hamiltonian model [10], as well as mean-field versions of the spherical model [38,149], and the φ 4 model [36,128,129]. In all of these models, the energy representation function is either a nonlinear function of the empirical vector (Level-2) or a function of properly-chosen Level-1 macrostates, commonly referred to as mean fields or order parameters.…”

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confidence: 99%

The large deviation approach to statistical mechanics

2009

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The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.PACS numbers: 65.40.Gr, Typeset by REVT E X arXiv:0804.0327v2 [cond-mat.stat-mech]

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On the Mean-Field Spherical Model

Cited by 35 publications

References 34 publications

Phase Transitions from Saddles of the Potential Energy Landscape

Phase Transitions from Saddles of the Potential Energy Landscape

Phase transitions in small systems: Microcanonical vs. canonical ensembles

The large deviation approach to statistical mechanics

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