Exact three-dimensional Casimir force amplitude,Cfunction, and Binder’s cumulant ratio: Spherical model results

Abstract

The three-dimensional mean spherical model on a hypercubic lattice with a film geometry Lϫϱ 2 under periodic boundary conditions is considered in the presence of an external magnetic field H. The universal Casimir amplitude ⌬ and the Binder's cumulant ratio B are calculated exactly and found to be ⌬ϭ Ϫ2(3)/(5)ϷϪ0.153051 and Bϭ2/͕ͱ5ln 3 ͓(1ϩͱ5)/2͔͖. A discussion on the relations between the finite temperature C function, usually defined for quantum systems, and the excess free energy ͑due to the finite-size contributions to the free energy of the system͒ scaling function is presented. It is demonstrated that the C function of the model equals 4/5 at the bulk critical temperature T c . It is analytically shown that the excess free energy is a monotonically increasing function of the temperature T and of the magnetic field ͉H͉ in the vicinity of T c . This property is supposed to hold for any classical d-dimensional O(n),nϾ2, model with a film geometry under periodic boundary conditions when dр3. An analytical evidence is also presented to confirm that the Casimir force in the system is negative both below and in the vicinity of the bulk critical temperature T c . ͓S1063-651X͑98͒04508-5͔

Fluctuation-induced forces in confined ideal and imperfect Bose gases

“…(4.49) remains valid for x < 0. It follows that 55) which is again consistent with the results of [24] and [25]. The scaling function Θ P ∞,3 (x) is plotted in Fig.…”

“…in agreement with [24], [25], and [52]. The residual grand potential ϕ P ∞,res,d can be computed along similar lines.…”

“…This prompted them to conclude that the universality class of the critical behavior at the condensation transition of the imperfect Bose gas is represented by the spherical model. On the other hand, they found that at d = 3 the Casimir amplitude ∆ C associated with the residual free energy at the critical point takes twice the value of its analog for the spherical model [24,25].…”

“…Using a coherent-state functional-integral formulation, we show that the leading asymptotic behavior near its bulk condensation transition is described by the n = ∞ analogs Θ P ∞,d of the above scaling functions Θ P d of the classical O(2n) real-valued φ 4 theory. This means, in particular, that for d = 3 the critical behavior of the imperfect Bose gas with PBCs is described by twice the exactly known scaling function of the mean spherical model [24,25].…”

Critical and near-critical phase behavior and interplay between the thermodynamic Casimir and van der Waals forces in a confined nonpolar fluid medium with competing surface and substrate potentials

“…[47] one has performed measurements of the Casimir force in thin wetting films of binary liquid mixture. One the theoretical side, the effect has been studied via exact calculations in the two-dimensional Ising model [48][49][50][51][52][53][54][55][56][57], the three dimensional spherical model [58][59][60][61][62][63][64][65][66], via conformal-theoretical methods [67][68][69][70][71][72][73], within mean-field type calculations on Ising type [74][75][76][77][78] and XY models [80], through renormalization-group studies via ε-expansion [81][82][83][84][85][86][87][88] and via fixed dimension d techniques [87][88][89] of O(n) models, as well as via MonteCarlo calculations [79,[90][91]…”