The Spherical Model of a Ferromagnet

Berlin, T. H.; Kac, Mark

doi:10.1103/physrev.86.821

Cited by 1,014 publications

(798 citation statements)

References 3 publications

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“…Such a generalized identification was observed at the third-order (m = 3) discontinuous transition present in the spherical model in three dimensions [7] as well as in the Ising model on planar random graphs if the Hausdorff dimension is used for d [10].…”

Section: Fisher Zerosmentioning

confidence: 89%

“…There are third-order temperature-driven transitions in various ferromagnetic and antiferromagnetic spin models [7,8], as well as spin models coupled to quantum gravity [9,10]. Recent theoretical studies also indicate the presence of third-order transitions in various superconductors [11], DNA under mechanical strain [12], spin glasses [13], lattice and continuum gauge theories [14] and matrix models linked to supersymmetry [15].…”

Section: Introductionmentioning

confidence: 99%

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Properties of higher-order phase transitions

2006

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Experimental evidence for the existence of strictly higher-order phase transitions (of order three or above in the Ehrenfest sense) is tenuous at best. However, there is no known physical reason why such transitions should not exist in nature. Here, higher-order transitions characterized by both discontinuities and divergences are analysed through the medium of partition function zeros. Properties of the distributions of zeros are derived, certain scaling relations are recovered, and new ones are presented.

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Section: Fisher Zerosmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Properties of higher-order phase transitions

2006

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“…We now summarize the exact solution of the spherical model [14] for barotropic flows in the inertial frame and refer the reader to the literature for details [17]. The partition function for the spherical model has the form…”

Section: Solution Of the Spherical Modelmentioning

confidence: 99%

“…The specific implementation of microcanonical enstrophy constraints in this approach leads to (C) -exact solutions of the resulting theories using the Kac-Berlin method [14] of steepest descent for spherical models. The main point discussed below in further detail is that these spherical models fix the low temperature problems of the classical energy-enstrophy theories [4] and yet are solvable in closed form.…”

Section: Introductionmentioning

confidence: 99%

Phase Transitions to Superrotation in a Coupled Fluid—Rotating Sphere System

Lim

IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence

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Summary. A family of spin-lattice models are derived as convergent finite dimensional approximations to the rest frame kinetic energy of a barotropic fluid coupled to a massive rotating sphere. The angular momentum of the fluid component changes under complex torques that are not resolved and the kinetic energy of the fluid is not a conserved Hamiltonian in these models. These models are used in a statistical equilibrium formulation for the energy -relative enstrophy theory of the coupled barotropic fluid -rotating sphere system, known as Kac's spherical model, to study the interactions between the energy cascade to large scales and angular momentum transfer. Exact solution of this model provides critical temperatures and amplitudes of the ground modes -super-rotating solid body flows -in the BECondensed phase.

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“…Relying heavily on the steepest descent methods in [10] and spherical model approach in [11][12][13][14], and using the scalings, β = βN , α = α/N , p = p/N , and H 0 = H 0 /N , we provide only the final formula obtained in view of space constraints:…”

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confidence: 99%

Negative specific heat in a quasi‐2D vortex system

Andersen

Lim

2007

Proc Appl Math and Mech

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Negative specific heat is a dramatic phenomenon where processes decrease in temperature when adding energy. It has been observed in gravo-thermal collapse of globular clusters. We now report finding this phenomenon in bundles of nearly parallel, periodic, single-sign generalized vortex filaments in the electron magnetohydrodynamic (EMH) model for the unbounded plane under strong magnetic confinement. We derive the specific heat using a steepest descent method and a mean field property. Our derivations show that as temperature increases, the overall size of the system increases exponentially and the energy drops. The implication of negative specific heat is a runaway reaction, resulting in a collapsing inner core surrounded by an expanding halo of filaments.While [1] has proven that systems that are not isolated from the environment must have positive specific heat, the specific heat in isolated systems can be negative [2]. Negative specific heat is an unusual phenomenon first discovered in 1968 in microcanonical (isolated system) statistical equilibrium models of gravo-thermal collapse in globular clusters [3]. In gravothermal collapse, a disordered system of stars in isolation under-goes a process of core collapse with the following steps: (1) faster stars are lost to an outer halo where they slow down, (2) the loss of potential (gravitational) energy causes the core of stars to collapse inward some small amount, (3) the resulting collapse causes the stars in the core to speed up. If one considered the "temperature" of the cluster to be the average speed of the stars, this process has negative specific heat because a loss of energy results in an increase in overall temperature.In the intervening four decades, negative specific heat has been observed in few other places (except in cases of small numbers of particles, [4]). We now report finding negative specific heat in a general quasi-2D vorticity model in equilibrium.Our goal is to find the specific heat of this vortex model in statistical equilibrium given an appropriate definition for energy and a microcanonical (isolated) probability distribution. Our approach is to describe the statistical behavior of a large number of discrete, interacting vortex structures and consider the limiting case.We use an approximate model for vortex behavior known as the local-induction approximation (LIA), useful for nearly parallel filaments, combined with a two-dimensional logarithmic interaction:where α is the vortex elasticity in units of energy/length. The generalized angular momentum isΓ i is the vortex circulation. Unlike many systems, here the plane is unbounded, not periodic. This vorticity description as well as others can be found in [5], [6], [7], and, especially, [8,9]. For our isolated, classical system, the energy and angular momentum plus magnetic moment are conserved giving rise to the following probability distribution for the filaments in equilibrium:where H 0 is the total "enthalpy" per vortex per period of the plasma, s is the complete state of the system, ...

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The Spherical Model of a Ferromagnet

Cited by 1,014 publications

References 3 publications

Properties of higher-order phase transitions

Properties of higher-order phase transitions

Phase Transitions to Superrotation in a Coupled Fluid—Rotating Sphere System

Negative specific heat in a quasi‐2D vortex system

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