Journal of Statistical Physics

, Volume 164, Issue 3, pp 575–615 | Cite as

Critical Crossover Functions for Simple Fluids: Non-Analytical Scaling Determination of the Ising-Like Crossover Parameter

  • Yves Garrabos
  • Carole Lecoutre
  • Samuel Marre
  • Bernard LeNeindre


A non-analytical scaling determination of the Ising-like crossover parameter is proposed considering the critical isochore of a simple fluid at finite distance from its critical temperature. The mean crossover functions, estimated from the bounded results of the massive renormalization scheme in field theory applied to the \(\left( \Phi ^{2}\right) _{d}^{2}\left( n\right) \) model in three dimensions (\(d=3\)) and scalar order parameter (\(n=1\)), are used to formulate the corresponding scaling equations valid in two well-defined temperature ranges from the critical temperature. The validity range and the Ising-like nature of the corresponding crossover description are discussed in terms of a single Ising-like scale factor characterizing the critical isochore. The asymptotic value of this scale factor can be predicted within the Ising-like preasymptotic domain. Unfortunately, the absence of precise experimental data in such a close vicinity of the critical point leads the direct testing impossible. A contrario, from our scaling equations and the use of precise measurements performed at finite distance from the critical point, its local value can be estimated beyond the Ising-like preasymptotic domain. This non-analytical scaling determination only needs to make reference to the universal features estimated from the mean crossover functions and to introduce a single master dimensionless length common to all the simple fluids. This latter parameter guaranties the uniqueness of the physical length unit used for the theoretical crossover functions and the fluid singular properties when the generalized critical coordinates of the vapor-liquid critical point of each fluid are known. Xenon case along its critical isochore is considered as a typical example to demonstrate the singleness of the Ising-like crossover parameter. With the measurements at finite temperature range of the effective singular behaviors of the isothermal compressibility in the homogeneous domain, and the vapor-liquid coexisting densities in the non homogeneous domain, our scaling equations provide the local estimate of this crossover parameter as a function of the temperature distance from the critical temperature. Our results compare favorably with the single asymptotic value calculated only using four critical coordinates of xenon.


Critical crossover function Crossover parameter Critical isochore Xenon 


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Copyright information

© European Union  2016

Authors and Affiliations

  • Yves Garrabos
    • 1
    • 2
  • Carole Lecoutre
    • 1
    • 2
  • Samuel Marre
    • 1
    • 2
  • Bernard LeNeindre
    • 3
    • 4
  1. 1.CNRS, Institut de Chimie de la Matière Condensée de Bordeaux - UPR 9048Pessac CedexFrance
  2. 2.Université de Bordeaux, ICMCB-UPR 9048Pessac CedexFrance
  3. 3.CNRS, Laboratoire des Sciences des Procédés et des Matériaux- UPR 3407VilletaneuseFrance
  4. 4.Université Paris 13 - Sorbonne Paris CitéVilletaneuseFrance

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