Thermodynamic Limit for Dipolar Media

Banerjee, Shubho; Griffiths, Robert B.; Widom, Michael

doi:10.1023/b:joss.0000026729.83187.79

Cited by 60 publications

(30 citation statements)

References 31 publications

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“…We will see examples of energies of the form (46) in subsection VI A, while in subsection VI B 4, we will work out in detail a specific example corresponding to a realistic system. Using methods similar to those introduced to prove the existence of a thermodynamic limit for short-range forces [278], it can be shown [30,164] that a system of dipolar spins posses a well defined bulk free energy, independent of sample shape, only in the case of zero applied field. The key to the existence of this thermodynamic limit is the reduction in demagnetization energy when uniformly magnetized regions break into ferromagnetically ordered domains [207].…”

Section: E Dipolar Systemsmentioning

confidence: 99%

Statistical mechanics and dynamics of solvable models with long-range interactions

Campa¹,

Dauxois²,

Ruffo³

2009

Physics Reports

876

1,330

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For systems with long-range interactions, the two-body potential decays at large distances as V (r) ∼ 1/r α , with α ≤ d, where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameters space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.

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Section: E Dipolar Systemsmentioning

confidence: 99%

Statistical mechanics and dynamics of solvable models with long-range interactions

Campa¹,

Dauxois²,

Ruffo³

2009

Physics Reports

876

1,330

View full text Add to dashboard Cite

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“…21 These two large categories of studies have at least two features in common: first, the role of chaining 19 in hampering or preventing ordering transitions that at low temperatures would induce macroscopic polarization; second, the role of boundary conditions which may induce demagnetization (depending on the domain's shape) that can prevent the thermodynamic limit from existing. 22 These are indeed the major pitfalls that may undermine a model for dipolar interactions in soft matter. We start by considering boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Phase polarity in a ferrofluid monolayer of shifted-dipole spheres

Piastra

Virga

2012

Soft Matter

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We consider a model for ferrofluids in which the constituent colloidal particles are spheres on a monolayer, with centers freely gliding on a given plane and permanent, point dipoles located along a radius in each sphere, oriented away from its center and freely pointing in all directions in space. Small clusters of these particles, also called shifted-dipole spheres in short, were studied in Kantorovich et al., Soft Matter, 2011, 7, 5217 to describe the effect of the dipole's shift on the ground state arrangement of particles in clusters of different populations at zero temperature. Our approach is somehow complementary to that; in the limit of many particles, we introduce an average Hamiltonian, in terms of which we construct a mean-field theory for a homogeneous ferrofluid monolayer. In an appropriate range of dipole's shifts, we predict a transition from the isotropic, non-polar phase to an ordered, polar phase with macroscopic polarization in the plane of the layer, at a critical reduced temperature, which turns out to be independent of the dipole's shift.

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“…N = 512 Stockmayer particles were simulated in a uniform field with periodic boundary conditions applied. The long-range dipolar interactions were computed using Ewald summations with conducting boundary conditions, which removes depolarisation fields [59]. The equations of motion were integrated using the velocity-Verlet method, with a reduced timestep δt * = 0.01.…”

Section: Molecular Dynamics Simulationsmentioning

confidence: 99%

Thermodynamics of the Stockmayer fluid in an applied field

et al. 2015

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The thermodynamic properties of the Stockmayer fluid in an applied field are studied using theory and computer simulation. Theoretical expressions for the second and third virial coefficients are obtained in terms of the dipolar coupling constant (λ, measuring the strength of dipolar interactions as compared to thermal energy) and dipole-field interaction energy (α, being proportional to the applied field strength). These expressions are tested against numerical results obtained by Mayer sampling calculations. The expression for the second virial coefficient contains terms up to λ 4 , and is found to be accurate over realistic ranges of dipole moment and temperature, and over the entire range of the applied field strength (from zero to infinity). The corresponding expression for the third virial coefficient is truncated at λ 3 , and is not very accurate: higher order terms are very difficult to calculate. The virial coefficients are incorporated in to a thermodynamic theory based on a logarithmic representation of the Helmholtz free energy. This theory is designed to retain the input virial coefficients, and account for some higher order terms in the sense of a resummation. The compressibility factor is obtained from the theory and compared to results from molecular dynamics simulations with a typical value λ = 1. Despite the mathematical approximations of the virial coefficients, the theory captures the effects of the applied field very well. Finally, the vapourliquid critical parameters are determined from the theory, and compared to published simulation results; the agreement between the theory and simulations is good.

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Thermodynamic Limit for Dipolar Media

Cited by 60 publications

References 31 publications

Statistical mechanics and dynamics of solvable models with long-range interactions

Statistical mechanics and dynamics of solvable models with long-range interactions

Phase polarity in a ferrofluid monolayer of shifted-dipole spheres

Thermodynamics of the Stockmayer fluid in an applied field

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