Trigonometric models for scaling behavior near criticality

Abstract

Parametric scaling representations of the asymptotic equation of state, correlation lengths, and singular part of the Helmholtz free energy in the critical region of a system with a scalar order parameter are considered with the aim of fitting all 15 available independent universal amplitude ratios and describing naturally van der Waals loops. Defects in previous linear and cubic models and their extensions are described: they cannot represent analytically connected van der Waals loops or approximate satisfactorily all the known amplitude ratios. However, novel trigonometric parametric models for the free energy ͑and correlations͒ meet all the desiderata as demonstrated by the explicit numerical data presented.

Anomalies of the Physical Properties of the Mixture Methane–penthane in the Critical Point Vicinities of the System Liquid-Vapour

“…Здесь α = 0.11, β = 0.325 и γ = 2-α -2β -универсальные критические показатели трехмерной модели Изинга [21]. Скейлинговские плотности φ 1 и φ 2 , сопряженные скейлинговским полям h 1 и h 2 могут быть определены из соотношения…”

Interfacial tensions near critical endpoints: experimental checks of EdGF theory

“…The universal, scaled equation of state m = Q ± (h) [for t > < 0] may be used in the form of the extended sine model of [11] to implement the conversion toh. The corresponding h follows via h = Bh|t| ∆ /C + where, for the present discussion we use (4.8) for C + as in Fig.…”

Off-critical Casimir effect in Ising slabs with antisymmetric boundary conditions ind=3

“…In this case, complications arise in the application of local-functional methods for two main reasons: (i) the approximation GðxÞ ¼ x 2 no longer holds and one needs to use the far more complicated form of GðxÞ as introduced in [4]; (ii) one needs to extend the bulk scaling functions Y À ðÁÞ, Z À ðÁÞ into the two-phase region, a somewhat ad hoc procedure although possible if one uses trigonometric parametric models (instead of the linear model) [17,21] giving rise to ''nonclassical van der Waals loops.'' However, this more complicated calculation is possible and forms the subject of ongoing research.…”

“…For our purposes, pertaining to the present physical problem situated in a one-phase region, the original ''linear'' parametric model [12,20] was found to suffice [22]. At d ¼ 3, we take ¼ 0:328 and ¼ 0:632 (all other exponents follow from the scaling relations) and a satisfactory fit to the bulk amplitude ratios, being properties of bulk scaling functions, is provided by taking b 2 ¼ 1:30 and a 2 ¼ 0:28 in the notation of [21], in the linear model. The universal scaling function W þþ ðyÞ that follows from our calculations in d ¼ 3 is presented in Fig.…”

“…We represent bulk scaling functions using parametric models introduced by Schofield [20]. These have been developed further [17,21] and are believed to give the best available fits to bulk data and, by their very construction, to give scaling functions satisfying required analyticity properties. For our purposes, pertaining to the present physical problem situated in a one-phase region, the original ''linear'' parametric model [12,20] was found to suffice [22].…”