Critical Crossover Functions for Simple Fluids: Towards the Crossover Modelling Uniqueness

Abstract

Based on a single non-universal temperature scaling factor present in a simple fluid case, a detailed analysis of non-universal parameters involved in different critical-to-classical crossover models is given. For the infinite limit of the cutoff wave number, a set of three scaling-parameters is defined for each model such that it shows all the shapes of the theoretical crossover functions overlap on the mean crossover function shapes close to the non-trivial fixed point. The analysis of corresponding links between their fluid-dependent parameters opens a route to define a parametric model of crossover equation-of-state, closely satisfying the universal features calculated from the Ising-like limit in the massive renormalization scheme.

Appendices

Appendix 1: Brief Status of the Crossover Modelling of the Xenon Properties

1.1 Appendix 1.1: Fitting Models and Xenon Data Sources Used in Table 1

All the PAD amplitudes given in Table 1 have been extracted from the fits of the experimental data obtained beyond the xenon PAD, still fixing the exponent values of $\gamma$, $\beta$, and $\Delta$ in the theoretical crossover functions calculated by the different theoretical models W referenced in Sect. 3.1. $\Gamma ^{+}$ is non-available in [26] due to the fit of the effective exponent $\gamma _{e}\left( \Delta \tau ^{*}\right)$ inferred by the susceptibility data, while $a_{\chi }^{+}$ and $a_{M}$ are non-available in [31] due to the numerical form of the calculated crossover functions. However, all these models are consistent with the renormalization group theory since their Ising-like universal features within the PAD are very similar, introducing, a minima, three non-universal parameters to characterize each physical system belonging to the $O\left( 1\right)$ universality class (see Table 2).

More precisely, the main experimental data sources involved in such fittings were obtained in the 1980s. They are the susceptibility data and the turbidity data of Güttinger and Cannell [14], the vapor-liquid coexisting density data measured by Närger and Balzarini [15] in two different samples of xenon, and the specific heat data at constant volume of Edwards et al. [48]. The temperature ranges are typically within $10^{-4}\lesssim \left| \Delta \tau ^{*}\right| \lesssim 10^{-1}$. Accordingly, from 84’s up to now, we have been interested in the following modelings: (i) The MR6 max test-functions calculated in Ref. [16] were used in Ref. [18] to fit susceptibility, turbidity and specific heat data above $T_{c}$; (ii) The CLM crossover function for susceptibility, as detailed for example in Ref. [45], was used in Ref. [26] to fit the normalized susceptibility data above $T_{c}$; (iii) The CPM crossover functions calculated in Ref. [28] were used in Ref. [13] to fit the susceptibility, turbidity, correlation length data above $T_{c}$ and the coexisting density data below $T_{c}$. We note that, in this same Ref. [13], the CMM crossover functions were also compared to CPM and MR ones from the joint fit of the susceptibility and coexisting density data for $T\lessgtr T_{c}$; (iv) The MSR crossover functions calculated in Ref. [22] were used in Ref. [21] to fit the susceptibility data above $T_{c}$; (v) The TGN numerical functions calculated in Ref. [29] were used in Ref. [31] to separately fit the susceptibility data above $T_{c}$ and coexisting density data below $T_{c}$.

We also note that the two-phase domain below $T_{c}$ have been analyzed only from the Närger and Balzarini [15] measurements at finite distance from $T_{c}$. In the following, three additional remarks concerning the fitting results obtained at finite temperature distance above $T_{c}$ and a final comparative comment concerning the fitting results including experimental results obtained within the (one-phase and two-phase) xenon PAD can be formulated.

1.2 Appendix 1.2: Fitting Analyses of the Normalized Susceptibility Above $T_{c}$

The important preliminary remark is that the normalized presentation of the susceptibility data in Ref. [14], avoids de facto the determination of the value of the physical amplitude $\Gamma ^{+}$ in fitting analyses where the critical temperature was also considered as an adjustable parameter. That provides an opportunity for first analyzing the normalized susceptibility data—or the effective exponent $\gamma _{e}\left( \Delta \tau ^{*}\right)$, equivalently—with only a single adjustable parameter in the crossover function where $\Delta$ takes the fixed theoretical value. The clear understanding of such a single parameter fitting can be found in Fig. 1 of Ref. [18] and in Fig. 2 (Xe) of Ref. [26]. Unfortunately, four decades latter, the true crossover behavior to the mean-field value of $\gamma _{e}\left( \Delta \tau ^{*}\right)$ indicated in Fig. 1 of Ref. [18], always remains unexpected by any crossover theory (with a similar remark for $\beta _{\text {eff}}\left( \Delta \tau ^{*}\right)$ in Fig. 3 of Ref. [43]).

1.3 Appendix 1.3: Fitting Analyses of the Normalized and Dimensional Susceptibility Above $T_{c}$

Fitting of the dimensional susceptibility data shows that $\Gamma ^{+}$ value appears mainly dependent on the theoretical value of the fixed exponent $\gamma$ and the selected value of the reference susceptibility data. Therefore, all the fitting results given in lines 1 to 5 of Table 1 have similar behaviors in the interpolated-PAD in the sense where the level of $\sim$0.4 % accuracy was always revealed in the experimental range $10^{-4}\lesssim \Delta \tau ^{*}\lesssim 5\times 10^{-2}$ (with no significative variation of $T_{c}$ observed) using the (single parameter) rescaled temperature fields in each model. Consequently, such a similarity implies that the single non-universal values of $\vartheta _{\mathcal {L}}^{+}$ or $\Delta \tau _{\text {W}}^{*}$, adjusted far away from $T_{c}$, are directly correlated to thetrue, but unknown, non-universal value of $a_{\chi }^{+}$, the latter amplitude being prominently close to $T_{c}$ in characterizing the Ising-like nature of xenon. Despite such expected links between the adjusted $\Delta \tau _{\text {W}}^{*}$ values and the true $a_{\chi }^{+}$ value, the typical uncertainty of this $a_{\chi }^{+}$ value can be estimated at $\sim$50 % from Table 1 results. In addition, the true uncertainty of the $\Gamma ^{+}$ value is certainly in the range of a few %, a level comparable to the dispersion of the $\Gamma ^{+}$ values reported in the Güttinger and Cannell’s fitting results of Ref. [14] (assuming $1\,\%$ experimental accuracy on $\kappa _{T,\text {expt}}^{*}$ data) and confirmed by the subsequent analyses reported in Table 1 (fixing the exponent values). Such an uncertainty level on $\Gamma ^{+}$ is too large to obtain an indirect accurate determination of $a_{\chi }^{+}$ (or $\vartheta$, equivalently) within the PAD. As an immediate consequence, all the data fittings at finite distance from $T_{c}$ with more than one single adjustable free parameter (as in the fitting cases of lines 1 to 5 in Table 1) are not able to extract the single asymptotic scaling factor $\Delta \tau _{\text {W}}^{*}$.

1.4 Appendix 1.4: Fitting Analyses of Other Data Sources Above $T_{c}$

From the initial results of Ref. [18] using the MR6 crossover functions with $\vartheta _{\mathcal {L}}^{+}$ free, we have shown here the limited interest for supplementary fitting of xenon singular properties (in a finite temperature range above $T_{c}$, along the critical isochore) using CLM, CPM, MSR or TGN to get an accurate estimation of their intrinsic nonuniversal parameters. For instance, in Ref. [18], the fitting of the GC turbidity data above $T_{c}$ where $\vartheta _{\mathcal {L}}^{+}$, $\Gamma ^{+}$ and $\xi _{0}^{+}$ are left free, involved as a result $\vartheta _{\mathcal {L}}^{+}=0.02\pm 0.03$, i.e., a noticeably poor estimation of the scale factor $\vartheta _{\mathcal {L}}^{+}$. Therefore, here we can stress that the Ising-like PAD description of $\kappa _{T}$ and $\xi$ can only be retrieved when are fixed the values of $\vartheta _{\mathcal {L}}^{+}=0.0191$ in the MR6 1984’s fitting and $\vartheta =0.0211752$ in the recent MR7 re-analysis [13] without adjustable parameter.

Similarly, the fitting results of the specific heat data [48] in the temperature range $10^{-4}\lesssim \Delta \tau ^{*}\lesssim 10^{-3}$ were also reported in Ref. [18]. Unfortunately, it has not been possible to obtain a good determination of the scale factor $\vartheta _{\mathcal {L}}^{+}$ as these specific heat measurements are not as accurate as those of the scattered light intensity and turbidity. The fit fixing $\vartheta _{\mathcal {L}}^{+}=0.0191$ with free leading amplitude $A^{+}$ and (here not important) free background constant $B_{exp}$, was without systematic deviation within a supposed experimental error of $2\,\%$. The resulting adjusted value of $A^{+}$ was compatible with $\xi _{0}^{+}=0.186\pm 0.001\,\text {nm}$ by using the universal amplitude combination $R_{\xi }^{+}=\xi ^{+}\left( A^{+}\right) ^{\frac{1}{d}}=0.27$. Obviously, similar results are expected using CLM, CPM, MSR or TGN with similar universal value for $R_{\xi }^{+}$ (see line 11 of Table 2).

1.5 Appendix 1.5: Fitting Analyses Including Experimental Data Within the Xenon PAD

A small number of theoretical analyses have addressed the xenon singular description from experimental data obtained very close to its vapor-liquid critical point, i.e., within the xenon PAD. To our knowledge, only the observations of the Fraunhofer interference patterns due to the Earth’s-gravity-induced density profiles performed in the 1980s very close to $T_{c}$ have provided the indirect access to the susceptibility data and the coexisting density data within the PAD, through the measurements of the optical phase quantity $\tilde{\rho }-\kappa _{T}^{*}\tilde{\mu }$ and the fringe angle [49–52]. However, the analyses of these interference patterns were affected by the complexity of the expressions for $\tilde{\rho }-\kappa _{T}^{*}\tilde{\mu }$, which were, at the date, highly dependent on the selected parametric models of the scaled equation-of-state. Today, it is well-recognized that such parametric models are never in full conformity with the universal features calculated from theoretical renormalization methods [53]. In addition, only two singular properties were not intrinsically able to validate the Ising-like crossover nature of xenon characterized by three non-universal parameters. Nevertheless, it remains interesting to note that some fitting analyses have been performed by using the Wilcox-Estler scaled parametric equation of state [54] (see also detail in Ref. [50]) with only two free exponents ($\beta$ and $\gamma$), i.e., in conformity with the two-scale factor universality. These analyses, only considering the asymptotic contribution of the leading power law terms $\Delta \tilde{\rho }_{LV,\text {expt}}=B\left| \Delta \tau ^{*}\right| ^{\beta }$ and $\kappa _{T,\text {expt}}^{*}=\Gamma ^{+}\left( \Delta \tau ^{*}\right) ^{-\gamma }$, have thus provided access to the exponent-amplitude pairs $\left\{ \beta ;B\right\}$ and $\left\{ \gamma ;\Gamma ^{+}\right\}$ for the free and fixed values of the exponents. The corresponding results were $\left\{ \beta =0.329\pm 0.005;B=1.48\pm 0.06\right\}$, $\left\{ \gamma =1.23\pm 0.003;\Gamma ^{+}=0.062\pm 0.006\right\}$ with free exponents [51] and $\left\{ \beta =0.325;B=1.42\pm 0.03\right\}$, $\big \{ \gamma =1.24;\Gamma ^{+}=0.058\pm 0.002\big \}$ with fixed exponents [52] in agreement with the results reported in Table 1. This good level of agreement was initially underlined by Güttinger and Cannell for the case of their determination of the interpolated $\left\{ \gamma ;\Gamma ^{+}\right\}$ pair because the two experiments rely on completely different effects and there is no region of overlap between the temperature distances from $T_{c}$ of the two data sets.

Appendix 2: Two-Scale Factor Similarity of the Crossover Models with $\Lambda \rightarrow \infty$

1.1 Appendix 2.1: MSR Versus MR

Using the notations and the results of Ref. [22], it is immediate to see that the MSR non-universal quantities needed to construct the universal crossover functions $F_{\chi ,n,\text {MSR}}\left( \frac{\Delta \tau ^{*}}{t_{0}}\right)$ and $F_{M,n,\text {MSR}}\left( \frac{\Delta \tau ^{*}}{t_{0}}\right)$ are $\chi _{n,\text {MSR}}\equiv \chi _{0}\propto \mu ^{-2}$, $M_{n,\text {MSR}}\equiv \phi _{0}\propto \mu ^{\frac{1}{2}}$, and $\Delta \tau _{\text {MSR}}^{*}\equiv t_{0}=\frac{\mu ^{2}}{a}$. De facto, only a (the renormalized temperature-like scaling parameter) and $\mu$ (the arbitrary reference length needed in the estimation of the renormalized coupling parameter u) are involved in such a construction (see Sect. 5.1). Therefore, the implicit role of u remains subtle to understand as the temperature scaling factor $t_{0}=\frac{\mu ^{2}}{a}$ is only defined for the fixed point limit $u=u^{*}$, i.e., without any reference to the mean-field regime. That requires checking the goodness of the fit versus $\left( 1-\frac{u}{u^{*}}\right)$ in order to control the expected power-law dependence of $\mu \propto \left( 1-\frac{u}{u^{*}}\right) ^{\frac{\nu }{\Delta }}$ and $a\propto \left( 1-\frac{u}{u^{*}}\right) ^{\frac{2\nu -1}{\Delta }}$, and the resulting power-law dependence of $t_{0}=\frac{\mu ^{2}}{a}\propto \left( 1-\frac{u}{u^{*}}\right) ^{\frac{1}{\Delta }}$ involved in the expressions of the first Wegner amplitudes where high-order terms were ignored. The main consequence is that only two out of the three-$\left\{ u;\mu ;a\right\}$- non-universal parameters are relevant fitting parameters. As expected, the use of a free ratio $\frac{u}{u^{*}}$ to be adjusted in the fit leaves the fit goodness unchanged for $\left( 1-\frac{u}{u^{*}}\right) \le 10^{-3}$, while the adjusted $\mu$ and a remain the two non-universal parameters characteristic of the physical fluid. The small value of $\left( 1-\frac{u}{u^{*}}\right)$ explains the expected large difference between the magnitude of a and $\mu$. For example $u=0.999u^{*}$, $a=0.333\pm 0.012$ and $\mu =\left( 2.97\pm 0.33\right) \times 10^{-4}$ in the xenon case reported in Ref. [21], while, from the data fitting, the adjusted magnitudes of the asymptotic multiparameter combinations are on a single decade range (see sublines 6 and 7, column 6 of Table 3) and comparable to the corresponding calculated ones in the case of MR (see column 7). Therefore, considering column 5 of Table 4 with a fixed u value and using Eqs. (51) and (52), the two formal parameter links MSR versus MR are as follows (lines 12b and 12c),

$$\begin{aligned} a= & {} k_{a}\vartheta \left( \psi _{\rho }\right) ^{\frac{\alpha -\nu }{\eta \nu }}\end{aligned}$$

(64)

$$\begin{aligned} \mu= & {} k_{\mu }\left( \psi _{\rho }\right) ^{-\frac{2}{\eta }} \end{aligned}$$

(65)

with $t_{0}=\frac{\mu ^{2}}{a}=k_{t_{0}}\vartheta ^{-1}\left( \psi _{\rho }\right) ^{\frac{-2}{\eta \nu }}$ (line 12a). Obviously, $\vartheta =A_{\text {MSR}}^{\vartheta }\left( t_{0}\right) ^{-1}\mu ^{\frac{1}{\nu }}$ and $\psi _{\rho }=A_{\text {MSR}}^{\psi _{\rho }}\mu ^{-\frac{\eta }{2}}$, where the values (lines 5 and 7) of $A_{\text {MSR}}^{\vartheta }$ and $A_{\text {MSR}}^{\psi _{\rho }}$ are calculated from Eqs. (54) and (55). Hence, the prefactors in Eqs. (64) and (65) are

$$\begin{aligned} k_{a}= & {} A_{\text {MSR}}^{\vartheta }\left( A_{\text {MSR}}^{\psi _{\rho }}\right) ^{\frac{\nu -\alpha }{\eta \nu }}\end{aligned}$$

(66)

$$\begin{aligned} k_{\mu }= & {} \left( A_{\text {MSR}}^{\psi _{\rho }}\right) ^{\frac{2}{\eta }} \end{aligned}$$

(67)

with $k_{t_{0}}=\frac{\left( k_{\mu }\right) ^{2}}{k_{a}}=\frac{\left( A_{\text {MSR}}^{\psi _{\rho }}\right) ^{\frac{2}{\eta \nu }}}{A_{\text {MSR}}^{\vartheta }}$. The MSR versus MR comparison is closed. In addition the dimensional length scaling factor $l_{0}$ (see last line of Table 2) in MSR is identical to the length unit $\alpha _{c}$ of Eq. (22) in MR. Consequently, introducing the universal ratios $R_{C,\text {W}}^{+}=\frac{\left( \mathbb {Z}_{C}^{+}\right) ^{-1}}{Z_{C,\text {W}}^{+}}$ or $R_{\xi ,\text {W}}^{+}=\frac{\left( \mathbb {Z}_{\xi }^{+}\right) ^{-1}}{Z_{\xi ,\text {W}}^{+}}$, such as $1\equiv R_{C,\text {W}}^{+}\frac{R_{\chi ,\text {W}}}{\left( R_{M,\text {W}}\right) ^{2}}$, $1\equiv R_{C,\text {W}}^{+}\left( R_{\xi ,\text {W}}^{+}\right) ^{d}$ or $1\equiv \left( R_{\chi ,\text {W}}^{+}\right) ^{-d}\frac{R_{\chi ,\text {W}}}{\left( R_{M,\text {W}}\right) ^{2}}$ from our hypothesis on identical universal features whatever W, and noting thus that $A_{\text {MSR}}^{\vartheta }\equiv \left( R_{C,\text {MSR}}^{+}\right) ^{-1}\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{\frac{1}{\nu }}\equiv \left( R_{\xi ,\text {MSR}}^{+}\right) ^{-\frac{1}{\nu }}\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{\frac{1}{\nu }}$, we obtain $t_{0}\vartheta \propto \left( \frac{\mathbb {L}^{\left\{ 1f\right\} }}{\mu }\right) ^{-\frac{1}{\nu }}$. The latter result confirms the $t_{0}$ versus $\vartheta ^{-1}$ asymptotic link, while $\mu$ well appears as the non-classical, metric-like non-universal parameter readily Ising-like similar to $\psi _{\rho }$ for the one-component fluid subclass. We also stress the pivotal scaling role played by the reference lengths, here underlined by the term $\left( \frac{\mathbb {L}^{\left\{ 1f\right\} }}{\mu }\right) ^{-\frac{1}{\nu }}$ in the $t_{0}$ versus $\vartheta ^{-1}$ link. De facto, the hyperscaling-like universal combinations $\left( R_{\xi }^{+}\right) ^{d}=A^{+}\left( \xi ^{+}\right) ^{d}$ and $Q_{c}=\left( \xi ^{+}\right) ^{d}\frac{B^{2}}{\Gamma ^{+}}$ (then $R_{C}=A^{+}\frac{\Gamma ^{+}}{B^{2}}$), are well accounted for since $A^{+}=Z_{C,\text {MSR}}^{+}\mu ^{3}\left( t_{0}\right) ^{\alpha -2}\equiv \mathbb {Z}_{C}^{+}\vartheta ^{2-\alpha }\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{d}$ and $\frac{\xi _{0,\text {MSR}}^{+}}{l_{0}}=Z_{\xi ,\text {MSR}}^{+}\mu ^{-1}\left( t_{0}\right) ^{\nu }\equiv \left( \mathbb {Z}_{\xi }^{+}\right) ^{-1}\vartheta ^{-\nu }\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{-1}=\xi ^{+}$ , with $Z_{C,\text {MSR}}^{+}=\frac{1}{16\pi }$, $Z_{\xi ,\text {MSR}}^{+}\simeq 1.9$, $\left( R_{C}\right) _{\text {MSR}}=0.0580\approx 0.0574=\left( R_{C}\right) _{\text {MR}}$, $\left( R_{\xi }^{+}\right) _{\text {MSR}}^{d}=0.0206\approx 0.0196=\left( R_{\xi }^{+}\right) _{\text {MR}}^{d}$, and $\left( Q_{c}\right) _{\text {MSR}}^{-1}=2.82\approx 2.93=\left( Q_{c}\right) _{\text {MR}}^{-1}$ (see lines 5, 6 & 10 to 12 in corresponding columns 5 & 7 of Table 2).

Finally, choosing a prefixed u value close to $u^{*}$ for consistency with the MSR approximations and $\left\{ \mu ,a\right\}$ as free, independent fitting parameters, induces the quasi-complete similarity between the MSR-$\left\{ u;\mu ;a\right\}$- and the MR-$\left\{ \vartheta ;g_{0};\psi _{\rho }\right\}$- non-universal parameters (with $\vartheta$ and $\psi _{\rho }$ independent). For the remaining slight differences between the estimated universal values of the exponents and amplitude combinations in each model, see lines 7 to 13, columns 5 and 7, of Table 2. The MSR versus MR matching infered from their respective non-universal sets $\left\{ \mu ;a\right\}$ and $\left\{ \vartheta ;\psi _{\rho }\right\}$ extracted at finite distance from $T_{c}$, confirms the Ising-like critical nature of the xenon crossover descriptions based on the renormalization methods with infinite cutoff wave number, without any reference to the parametrization of the mean field regime. Now, the MSR modeling can be applied to any one-component fluid whose generalized critical coordinates of liquid-gas critical point are known, since the unequivocal links $\left\{ \vartheta ;\psi _{\rho }\right\}$ versus $\left\{ Y_{c};Z_{c}\right\}$ are already given by Eqs. (19) and (20) in Sect. 2.

1.2 Appendix 2.2: CLM and CPM Versus MR

The results of the CLMs or CPM fitting cases are not basically comparable to the MR ones due to a possible, positive non-zero value of the crossover parameter $\bar{u}$ in the construction of the crossover function Y (using the notation of Refs. [25] and [28]). Consequently, from the corresponding four-parameter fittings, the extraction of the true values of the two-scale-factors which are non-universal in the Ising-like asymptotic description close to the non-trivial fixed point remains unclear. Indeed, on a theoretical renormalization group approach where it is assumed that only a single irrelevant field contributes to the first-order confluent corrections to scaling, the first-order amplitudes of the confluent corrections are mandatory linked to the second crossover parameter g or $\bar{u}\Lambda$, which acts as an effective Ginzburg number, while the renormalization of the mean-field-like non-universal parameters needs to only account for two suplementary non-universal bare parameters. For instance, in CPM, the introduction of the free parameters $l_{0}$ and $m_{0}$ to characterize the leading amplitudes is only due to the Ising-like universal features intrinsic to the parametric form of the scaled equation-of-state associated to the Ginzburg-number-like nature of the temperature scaling parameter g. Hence, the two-scale factor universality can correctly be accounted for by using the three-$\left\{ g;\widetilde{l}_{0};\widetilde{m}_{0}\right\}$- parameter description, as well as the two-$\left\{ l_{0};m_{0}\right\}$- parameter description, through the $\widetilde{l}_{0}=l_{0}g^{\beta \delta -\frac{3}{2}}$ and $\widetilde{m}_{0}=m_{0}g^{\beta -\frac{1}{2}}$ parameter rescaling. Similarly, considering the prefactor sets $\left\{ \chi _{0,\text {W}}^{+},M_{0,\text {W}}\right\}$ given in lines 6 and 7 for the model versions in columns 3 to 5 of Table 3, the comparison of the multi-parameter combinations involved in the matching equations, show that the two mean-field-like bare quantities, namely $\left\{ a_{0};u_{0}\right\}$, $\left\{ c_{t};c_{\rho }\right\}$, and $\left\{ \tilde{l}_{0};\tilde{m}_{0}\right\}$, are Ising-like renormalized only using the crossover parameter g (or $\overline{u}\Lambda$), and not the $\left( 1-\bar{u}\right)$ quantity. That is formally noticeable in the asymptotical CLM-II version where are used the renormalized variables (here with subscript II) $t_{\text {II}}=c_{t}\Delta \tau ^{*}$ and $M_{\text {II}}=c_{\rho }\Delta \tilde{\rho }$ (ignoring in xenon case the non-symmetrical behavior of the order-parameter density $\phi$, with $\phi \equiv \Delta \tilde{\rho }$ for simple fluids). In such a specific version, the respective parameter links

$$\begin{aligned} c_{t}= & {} k_{c_{t}}\vartheta \left( \bar{u}\Lambda \right) ^{-\frac{1}{\nu }-2}=\left( A_{\text {CLM-II}}^{\vartheta }\right) ^{-1}\vartheta \left( \bar{u}\Lambda \right) ^{-\frac{1}{\nu }-2}\end{aligned}$$

(68)

$$\begin{aligned} c_{\rho }= & {} k_{c_{\rho }}\left( \psi _{\rho }\right) ^{-1}\left( \bar{u}\Lambda \right) ^{-\frac{\eta }{2}}=A_{\text {CLM-II}}^{\psi _{\rho }}\left( \psi _{\rho }\right) ^{-1}\left( \bar{u}\Lambda \right) ^{-\frac{\eta }{2}} \end{aligned}$$

(69)

(see lines 12b and 12c, column 3 in Table 4) are obvious due to $t=\vartheta \Delta \tau ^{*}$ and $m_{\text {th}}=\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{-d}\left( \psi _{\rho }\right) ^{-1}\Delta \tilde{\rho }$ in MR case. In Eqs. (68) and (69) the implicit scaling role of $\mathbb {L}^{\left\{ 1f\right\} }$ is inserted in the universal prefactors

$$\begin{aligned} k_{c_{t}}= & {} \left( A_{\text {CLM-II}}^{\vartheta }\right) ^{-1} \end{aligned}$$

(70)

$$\begin{aligned} k_{c_{\rho }}= & {} A_{\text {CLM-II}}^{\psi _{\rho }} \end{aligned}$$

(71)

through Eqs. (54) and (55) [see also below Eqs. (72) and (73)]. The fact that the temperature scaling factor $\Delta \tau _{\text {CLM-II}}^{*}$ explicitly contains the product $\bar{u}\Lambda$, leads to the corresponding power-law terms $\left( \overline{u}\Lambda \right) ^{\frac{1}{\nu }-2}$ and $\left( \overline{u}\Lambda \right) ^{-\frac{\eta }{2}}$ in Eqs. (68) and (69). The Ising-like renormalization of the mean-field-like non-universal parameters $c_{t}$ and $c_{\rho }$ is only governed by $\overline{u}\Lambda$ (as the Ising-like critical limit for the crossover function Y, only needs the knowledge of the product $\bar{u}\Lambda$, not separately $\bar{u}$). Such a pivotal role of $\overline{u}\Lambda$ already mentionned in 2000 [42], can be evidenced here by extending the previous estimations of the prefactor sets $\left\{ \chi _{n,\text {CLM-II}}^{+},M_{n,\text {CLM-II}}\right\}$ [see Eqs. (58) and (61)] to the cases of $\left\{ C_{n,\text {CLM-II}}^{+},\xi _{n,\text {CLM-II}}^{+}\right\}$ for the specific heat and correlation length above $T_{c}$, where $C_{n,\text {CLM-II}}^{+}=\left( \overline{u}\Lambda \right) ^{d}\left( 1-\bar{u}\right) ^{\frac{\alpha -2}{\Delta }}$, and $\xi _{n,\text {CLM-II}}^{+}=\left( \overline{u}\Lambda \right) ^{-1}\left( 1-\bar{u}\right) ^{\frac{\nu }{\Delta }}$. As a remarkable result, these two correlated (since $1\equiv C_{n,\text {CLM-II}}^{+}\left( \xi _{n,\text {CLM-II}}^{+}\right) ^{d}$) asymptotic prefactors are non-dependent on the non-universal parameters $c_{t}$ and $c_{\rho }$. The respective comparisons with $C_{r}^{+}=\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{d}$ and $\xi _{r}^{+}=\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{-1}$, non-dependent of $\vartheta$ and $\psi _{\rho }$ in the case of MR, in order to match $A^{+}$ and $\xi ^{+}$, implie $\frac{\mathbb {L}^{\left\{ 1f\right\} }}{\overline{u}\Lambda }\left( 1-\bar{u}\right) ^{\frac{\nu }{\Delta }}=R_{\xi ,\text {CLM-II}}^{+}=\left( R_{C,\text {CLM-II}}^{+}\right) ^{-\frac{1}{d}}$. $R_{\xi ,\text {CLM-II}}^{+}$ and $R_{C,\text {CLM-II}}^{+}$ are the CLM-II versions of the universal ratios $R_{C,\text {M}}^{+}$ and $R_{\xi ,\text {M}}^{+}$ previously introduced in Appendix 2.1. As for $\left( \frac{\mathbb {L}^{\left\{ 1f\right\} }}{\mu }\right) ^{-\frac{1}{\nu }}$ in the case of MSR, the concomitant pivotal roles of $\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{-\frac{1}{\nu }}$ and $\left( \overline{u}\Lambda \right) ^{-\frac{1}{\nu }}$ in the case of CLM-II, infer (anticipating the discussion below) the scaling length nature of the constraint reported in line 9, column 3 in Table 4, since $\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{-\frac{1}{\nu }}=L_{\text {CLM-II}}^{\vartheta }\left( R_{\xi ,\text {CLM-II}}^{+}\right) ^{-\frac{1}{\nu }}=L_{\text {CLM-II}}^{\vartheta }\left( R_{C,\text {CLM-II}}^{+}\right) ^{\frac{1}{2-\alpha }} \left( \mathrm{with } 1\equiv R_{C,\text {CLM-II}}^{+}\left( R_{\xi ,\text {CLM-II}}^{+}\right) ^{d}\right)$.

Such a Ising-like renormalization of the bare parameters only using $\overline{u}\Lambda$ or g is implicitly contained in all the crossover model versions [25], including CPM [27, 28], based on simplest, similar, three-term mean-field-like Hamiltonians (including a square-gradient term), but the related data fitting validation needs to mandatorily be performed within the PAD. Unfortunately, as already mentioned, the absence of experimental data within the PAD prevents de facto any accurate independent estimation of the free sets $\left\{ a_{0};u_{0}\right\}$, $\left\{ c_{t};c_{\rho }\right\}$, or $\left\{ l_{0};m_{0}\right\}$. When the data fitting is performed beyond the PAD, if a single value of $\Delta \tau _{\text {W}}^{*}$ exists, then the correlated prefactor set which can be extracted from the fit is not $\left\{ \chi _{0,\text {W}}^{+},M_{0,\text {W}}\right\}$, but $\left\{ \chi _{n,\text {W}}^{+},M_{n,\text {W}}\right\}$ where we have already noted that each readily independent prefactor shows a remaining $\left( 1-\bar{u}\right)$ dependence. As an unavoidable result, the two non-universal parameters that characterize the asymptotic two-scale-factor universality of the fluid are always dependent on the two crossover parameters in such data fittings using four adjustable parameters at a finite distance from $T_{c}$.

Consequently, CLMs or CPM joint fits of the susceptibility and coexisting density data always contain four non-universal parameters for only three resulting independent values of the free multi-parameter combinations involved in the PAD description conforms to the one calculated along the renormalized trajectory. Fitting of any supplementary singular property measured at finite distance from $T_{c}$ along the critical isochore, do not change the situation since the universal features of the $O\left( 1\right)$ universality class are intrinsically accounted for in the construction of these parametric models. That makes it necessary to introduce an arbitrary character in the practical use of these parametric models, as for instance, when was used by the authors [28] the fixed value $\frac{\Lambda }{\left( c_{t}\right) ^{\frac{1}{2}}}=\pi$ in their fits of the singular behaviors of $^{3}\text {He}$ properties. The same arbitrary value was also assumed in Ref. [13] for the CPM fitting data (with comparable values for the asymptotic multiparameter combinations given in lines 6 and 7 of Table 3) and for the estimation of the first-order confluent corrections in the case of CMM to obtain ideal match with MR without adjustable parameter. Such arbitrariness cannot be suppressed due to the basic construction of the mean-field limit $Y\rightarrow 1$ for the crossover function Y, using separately $\bar{u}$ and $\bar{u}\Lambda$.

According to the above situation, we must now account for a single common $\vartheta$ value that governs the joint matching of the leading and the confluent amplitudes of the PAD description, i.e., $\left( \vartheta \right) _{1}=\left( \vartheta \right) _{0}\equiv \vartheta$ in Eqs. (53) and (56). The resulting constrained multiparameter combinations $L_{\text {M}}^{\vartheta }=\frac{A_{\text {M}}^{\vartheta }}{R_{\text {M}}^{1}}$ (see line 9 of Table 4) never limit a possible continuous change of the multiplicative term $\left( 1-\bar{u}\right) ^{\frac{1}{\Delta }}$ with $\overline{u}\ge 0$. This change can thus be compensated by the correlated continuous change of the product $\bar{u}\Lambda$, which limits the capability of proof that a single value for the temperature scaling factor $\Delta \tau _{\text {M}}^{*}$ exists. Such implicit correlations between these three quantities are also easy to understand with the CLM-II parameter set $\left\{ \Lambda ;\overline{u};c_{t};c_{\rho }\right\}$, where the ratio $\frac{1-\bar{u}}{L_{\text {M}}^{\vartheta }}=\bar{u}\Lambda$ remains free after the constraint $\left( \vartheta \right) _{1}=\left( \vartheta \right) _{0}\equiv \vartheta$ (see line 10, column 4 in Table 4). An accurate illustration was already shown in Ref. [42, 43] using the explicit correlations between $Y_{c}$ and $Z_{c}$ and the CLM-II fitting results $\left\{ c_{t};c_{\rho }\right\}$ obtained for six differents simple fluids. A similar view can be given on the large field of the theoretical studies focussed on the modeling of different critical-to-classical crossovers. Indeed, it should also be noted the case of an asymptotic version of CLM [41] used through an explicit comparison with the numerical critical-to-classical crossover calculations in Ising lattice models of different interaction ranges, via a tunable range R of interactions [30]. Such a particular asymptotic CLM modeling via a tunable $\overline{u}$ [45] leads to the recovery of common mean-field limits for $\overline{u}\rightarrow 0$ and $R\rightarrow \infty$, while introducing phenomenological arguments to define the practical continuous adjustements of the $\overline{u}\left( R\right) \sim \frac{1}{R^{4}}$ dependence at small R to account for finite-sized effects up to the limit $R=1$. Therefore, in the absence of similar additional theoretical arguments able to link (at least) one crossover parameter to the well-known short-ranged interactions between xenon particles, such four free parameters of the CLMs or CPM crossover fittings remain not easy to compare with the three non-universal parameters of the MR crossover functions. Nevertheless, the matching of the CLM-I&II and CPM PAD description to the MR one can partly be formulated through the universale multiparameter combinations given respectively in lines 9 and 10, columns 3 to 5 of Table 4. For example, from a generalization of Eqs (68) and (69) to the three models, the different prefactors $k_{i}$, where the subscript i refers to the selected non-universal parameter of the model, are universal quantities still depending on $\mathbb {L}^{\left\{ 1f\right\} }$ and the universal ratios of lines 1 to 3. As mentioned by the quantity noted $\left\{ \right\} ^{\star }$ in subline 9, column 5, only the addition of an arbitrary constraint, such as $\frac{\Lambda }{\left( c_{t}\right) ^{\frac{1}{2}}}=\pi$, closes the unequivocal link between the respective non-universal (reduced) parameter set $\left\{ \bar{u};l_{0};m_{0}\right\}$ of CPM versus the one $\left\{ \vartheta ;g_{0};\psi _{\rho }\right\}$ of MR (see also Table 3).

1.3 Appendix 2.3: CLM0 and CPM0 Versus MR via MSR

On the basis of the present work, it is noticeable that the PAD matching can be formally constructed by fixing $\overline{u}=0$ in the above model versions. CLM0-I&II and CPM0 thus appear to be conform to the infinite cutoff wavenumber $\Delta \rightarrow \infty$ and a finite value of $\overline{u}\Lambda$. From lines 8 or 9 with $\overline{u}=0$, the constrained subparameters $g\propto \vartheta ^{-1}$ or $\overline{u}\Lambda$(fixed) given in line 11 of Table 4 can be derived in a straightforward manner. Now, the closed unequivocal link between the respective non-universal parameter sets of the W versus MR comparison is obtained through the equations given respectively in lines 12a, 12b and 12c, columns 3 to 5 of Table 4, where line 12a replaces previous line 9. Here is mentioned the $\bar{u}\Lambda$ quantity noted $\left\{ \right\} ^{\star }$ in line 12, column 4, which is fixed in the case of CLM0-II. The universal prefactors are defined in terms of the universal ratios given in Table 4, as follows

$$\begin{aligned} k_{a_{0}}= & {} \frac{\left( R_{\text {CLM-I}}^{1,+}\right) ^{\gamma -1}}{R_{\chi ,\text {CLM-I}}^{+}}\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{d}; k_{c_{t}}=\left( A_{\text {CLM-II}}^{\vartheta }\right) ^{-1}; k_{l_{0}}=\frac{R_{M,\text {CPM}}}{R_{\chi ,\text {CPM}}^{+}}\end{aligned}$$

(72)

$$\begin{aligned} k_{u_{0}}= & {} \frac{\left( R_{\text {CLM-I}}^{1,+}\right) ^{\gamma -2\beta }}{R_{\chi ,\text {CLM-I}}^{+}\left( R_{M,\text {CLM-I}}\right) ^{2}}\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{-3d}; k_{c_{\rho }}=A_{\text {CLM-II}}^{\psi _{\rho }}; k_{m_{0}}=R_{M,\text {CPM}}\left( \mathbb {L}^{\left\{ 1f\right\} }\right) ^{d}\nonumber \\ \end{aligned}$$

(73)

The above model matching now enlarges the physical understanding of the non-universal parameters. For instance, from the MSR versus CPM0 comparison, $t_{0}=\frac{\mu ^{2}}{a}$ and $g=\frac{\left( \overline{u}\Lambda \right) ^{2}}{c_{t}}$ take similar nature of temperature scaling factors, which confers a strict functional equivalence between the dimensionless length scaling parameters $\mu$ and $\overline{u}\Lambda$ on one hand, and the dimensionless temperature scaling parameters a and $c_{t}$ on the other. Therefore, in a first step, we can assume $g_{\chi ,\text {MSR}}^{1,+}=g_{\chi ,\text {CPM}}^{1,+}$, taking advantage of the small differences between the universal values $g_{\chi ,\text {MSR}}^{1,+}=0.545$ of SRM and $g_{\chi ,\text {CPM}}^{1,+}=0.590$ of CPM (see line 1, columns 4 and 7, in Table 2). Hence, the confluent amplitude matching between $a_{\chi }^{+}$ of both models implies $g=t_{0}\left( 1-\frac{u}{u^{*}}\right) ^{-\frac{1}{\Delta }}$. Introducing thus $\mu =\mu _{\text {MSR}}^{*}\left( 1-\frac{u}{u^{*}}\right) ^{\frac{\nu }{\Delta }}$, $a=a_{\text {MSR}}^{*}\left( 1-\frac{u}{u^{*}}\right) ^{\frac{2\nu -1}{\Delta }}$, with resulting $t_{0}=\frac{\left( \mu _{\text {MSR}}^{*}\right) ^{2}}{a_{\text {MSR}}^{*}}\left( 1-\frac{u}{u^{*}}\right) ^{\frac{1}{\Delta }}$ for the case of MSR, finally leads to $g=\frac{\left( \mu _{\text {MSR}}^{*}\right) ^{2}}{a_{\text {MSR}}^{*}}$ for the case of CPM0. The bare parameters $\mu _{\text {MSR}}^{*}$ and $a_{\text {MSR}}^{*}$ are only defined for the fixed point limit $u=u^{*}$, i.e., without any reference to the mean-field regime. Similarly, the asymptotical Ising-like matching between B and $\Gamma ^{+}$ values estimated on CPM0 and MSR provides $\widetilde{m}_{0}=\frac{Z_{M,\text {SRM}}}{Z_{M,\text {CPM}}}\left( \frac{a_{\text {MSR}}^{*}}{\mu _{\text {MSR}}^{*}}\right) ^{\frac{1}{2}}\left( 1-\frac{u}{u^{*}}\right) ^{\frac{\nu -2\beta }{2\Delta }}$ and $\widetilde{l}_{0}=\frac{Z_{\chi ,\text {CPM}}}{Z_{\chi ,\text {SRM}}}\frac{Z_{M,\text {SRM}}}{Z_{M,\text {CPM}}}\left( \frac{a_{\text {MSR}}^{*}}{\mu _{\text {MSR}}^{*}}\right) ^{\frac{1}{2}}a_{0}\left( 1-\frac{u}{u^{*}}\right) ^{\frac{2\beta -\nu }{2\Delta }}$, which closes the expected non-analytic links between the MSR-$\left\{ u;\mu ;a\right\}$ non-universal parameters and the CPM0-$\left\{ g;\widetilde{l}_{0};\widetilde{m}_{0}\right\}$ ones and simultaneously reduces by one the number of independent non-universal parameters. The results of the MSR data fittings with a prefixed u value close to $u^{*}$, with $\mu$ and a as free fitting independent parameters, are then strictly similar to the ones of the CPM0 data fittings with a prefixed $\overline{u}=0$ value and g, $\widetilde{m}_{0}$, and $\widetilde{l}_{0}$ as free fitting parameters. Subsequently, $m_{0}=\widetilde{m}_{0}g^{\frac{1}{2}-\beta }$ and $l_{0}=\widetilde{l}_{0}g^{\frac{3}{2}-\beta \delta }$ come out as being the two Ising-like independent, non-universal parameters of CPM0, similar to the two Ising-like independent, non-universal parameters $\mu$ and a of SRM. Our previous comparison between (mean and/or master) MR and SRM fitting results extends the similarity (in number and physical nature) to the three MR-$\left\{ \vartheta ;g_{0};\psi _{\rho }\right\}$, and/or master -$\left\{ Y_{c};\alpha _{c};Z_{c}\right\}$ parameter sets (with $\mathbb {L}^{\left\{ 1f\right\} }=g_{0}\alpha _{c}$), where the two Ising-like independent, non-universal scale factors are $\vartheta$ and $\psi _{\rho }$ (and/or $Y_{c}$ and $Z_{c}$). The next step, not examined here, will focus on the development of better matching methods in order to account for the small differences between $g_{\chi ,\text {MSR}}^{1,+}$ and $g_{\chi ,\text {CPM}}^{1,+}$. In such a future work, it will also become important to integrate the reduction of diffences between the universal values of the critical exponents and the amplitude combinations, using an upgraded version of CMM which matches as well as possible the universal estimates from the MR scheme.