Crossover Master Model of the Equation-of-State for a Simple Fluid: Critical Universality.

Abstract

We present a new extended parametric equation-of-state model for thermodynamic properties and the correlation length for a simple fluid near its liquid–gas critical point. The model involves 16 universal parameters to perfectly match 10 leading universal amplitudes of the asymptotic Ising-like limit of the critical-to-classical crossover functions calculated by Garrabos and Bervillier [Phys. Rev. E 74,021113 (2006)] from the massive renormalization scheme. The universal values of 8 Ising-like amplitude combinations are then matched exactly. The closure of the construction of parameters is determined after a careful analysis of the intrinsic limitation of parametric equations to describe the universal features at the first order of the confluent corrections-to-scaling. In the asymptotic mean-field limit, the crossover master model also reproduces the mean-field amplitude combinations except for the susceptibility case. The new model is compared with the crossover parametric model previously developed by Agayan et. al [Phys. Rev. E 64, 02615 (2001)]. The residuals from comparison with the mean crossover functions of Garrabos and Bervillier are reported to define the application range of the crossover master model to any simple fluid for which the generalized critical coordinates of the liquid–gas critical point are known.

Appendix A: MR-Like Characteristic Parameters of the CMM

The main objective of this Appendix A is to provide an unambiguous analytical determination of the 16 universal parameters of Table 4 involved by the CMM parametric forms of the e.o.s.. Despite the irreducible defaults of the resulting intrinsic CMM, this closed determination then permits to use, for such a similar determination, any theoretical crossover behavior calculated along the renormalized trajectory of the $O\left( 1\right)$ symmetric $\left( \Phi ^{2}\right) ^{2}$ field theory [4]. The first Section of this Appendix A provides the needed normed quantities that are defined in conformity with the use of $\varSigma _{0}$ as a dimensionless energy density reference (see previous section 2.2.). The second Section provides the 76 successive steps of the unequivocal process to determine these 16 normed universal parameters reported in column 5 of Table 4.

1.1 Normed Universal Prefactors for the Universal Thermodynamic and Correlation Properties

The CMM construction only considers normed quantities using as dimensionless references, $\Sigma _{0}$ for the thermodynamics functions, $\left( \Sigma _{0}\right) ^{-\frac{1}{3}}$ for the correlation functions (to be in conformity with the $Q^{+}$ universality), and consequently, $\left( \Sigma _{0}\right) ^{-\frac{5}{3}}$ for the correlation angular functions. The non-universal nature of the physical system is then suppressed using the following four universal parametric equations

$$\begin{aligned} k(\theta )&= 1-b^{2}\theta ^{2}, \end{aligned}$$

(A1)

$$\begin{aligned} l^{*}\left( \theta \right)&= \frac{l(\theta )}{l_{0}}=\frac{\tilde{l}(\theta )}{\tilde{l}_{0}} \nonumber \\&= \theta \left( 1-\theta ^{2}\right) \left( 1+d\,\theta ^{2}+e\,\theta ^{4}+f\,\theta ^{6}\right) \end{aligned}$$

(A2)

$$\begin{aligned} w^{*}\left( \theta \right)&= \frac{w(\theta )}{m_{0}l_{0}\varSigma _{0}}=\frac{\tilde{w}(\theta )}{\tilde{m}_{0}\tilde{l}_{0}\varSigma _{0}} \nonumber \\&= \left( w_{0}^{*}+w_{1}^{*}\theta ^{2}+w_{2}^{*}\theta ^{4}+w_{3}^{*}\theta ^{6}+w_{4}^{*}\theta ^{8}+w_{5}^{*}\theta ^{10}\right) \end{aligned}$$

(A3)

$$\begin{aligned} a_{r}\left( \theta ,Y_{1}\right)&= \frac{a\left( \vartheta ,Y_{1}\right) }{\left( m_{0}\right) ^{-\frac{5}{3}}\left( l_{0}\right) ^{\frac{1}{3}}\left( \varSigma _{0}\right) ^{-\frac{5}{3}}}= \frac{\tilde{a}\left( \vartheta ,Y_{1}\right) }{\left( \tilde{m}_{0}\right) ^{-\frac{5}{3}}\left( \tilde{l}_{0}\right) ^{\frac{1}{3}}\left( \varSigma _{0}\right) ^{-\frac{5}{3}}} \nonumber \\&= \left( a_{r0}+a_{r1}\theta ^{2}+a_{r2}\theta ^{4}\right) Y_{1}+\left( a_{r0}^{\star }+a_{r1}^{\star }\theta ^{2}+a_{r2}^{\star }\theta ^{4}\right) \left( 1-Y_{1}\right) \end{aligned}$$

(A4)

All the 16 unknown parameters can be distinguished in a set $\left\{ b^{2},d,e,f\right\}$ of 4 parameters independent of $\Sigma _{0}$ and a set $\left\{ w_{i}^{*},a_{rj},a_{rj}^{\star }\right\}$ (with $i=\left\{ 0,5\right\}$ and $j=\left\{ 0,2\right\}$), of 12 normed parameters with reference to $\Sigma _{0}$. Accordingly, the 5 normed and 2 reduced quantities of practical interest read as follows

$$\begin{aligned} w_{0}^{*}=\frac{\tilde{w}(0)}{\tilde{m}_{0}\tilde{l}_{0}}\frac{1}{\varSigma _{0}}= & {} \frac{w_{0}}{\varSigma _{0}} \end{aligned}$$

(A5)

$$\begin{aligned} w_{1}^{*}=\frac{1}{2}\frac{\tilde{w}^{,,}(0)}{\tilde{m}_{0}\tilde{l}_{0}}\frac{1}{\varSigma _{0}}= & {} \frac{w_{1}}{\varSigma _{0}} \end{aligned}$$

(A6)

$$\begin{aligned} \varSigma _{1}^{*}=\frac{\varSigma _{1}}{\varSigma _{0}}= & {} 2{\textstyle \underset{j=0}{\overset{j=5}{\sum }}}jw_{j}^{*} \end{aligned}$$

(A7)

$$\begin{aligned} \varSigma _{2}^{*}=\frac{\varSigma _{2}}{\varSigma _{0}}= & {} 2{\textstyle \underset{j=0}{\overset{j=5}{\sum }}}j\left( 2j-1\right) w_{j}^{*} \end{aligned}$$

(A8)

$$\begin{aligned} S_{0}^{*}=\frac{S_{0}}{\varSigma _{0}}= & {} {\textstyle \underset{j=0}{\overset{j=5}{\sum }}}w_{j}^{*} \left( \frac{1}{b^{2}}\right) ^{j} \end{aligned}$$

(A9)

$$\begin{aligned} L_{0}^{*}=-\frac{l'(1)}{2l_{0}}=-\frac{\tilde{l}'(1)}{2\tilde{l}_{0}}= & {} 1+d+e+f \end{aligned}$$

(A10)

$$\begin{aligned} f= & {} L_{0}^{*}-1-d-e \end{aligned}$$

(A11)

When $b^{2}$ is used as entry data in Eq. A1, the knowledge of $w_{0}^{*}$ (Eq. A5) and $w_{1}^{*}$ (Eq. A6) close the normed description of respectively the heat capacity above and below $T_{c}$ and the isothermal susceptibility above $T_{c}$ (see Eqs. 33 to 36 and Eqs. 39, 40). The parameter set $\left\{ w_{0}^{*},w_{1}^{*},\varSigma _{0}^{*}=1,\varSigma _{1}^{*},\varSigma _{2}^{*},S_{0}^{*}\right\}$ of Eqs. A5 to A9 closes the universal parametric form of Eq. A3. In addition, $L_{0}^{*}$ of Eq. A10 provides access to one parameter among $\left\{ d,e,f\right\}$, as for example f of Eq. A11. The parameter set $\left\{ d,e,L_{0}^{*}\right\}$ is then equivalent to the $\left\{ d,e,f\right\}$ set and finally closes the estimation of universal parametric Eq. A1.

Similarly, Eqs. 30, 31 and 32 can be now rescaled to induce the following normed prefactors

$$\begin{aligned} \frac{P_{0}^{l}}{f_{P,\text {Ising}}^{l}\left( m_{0},l_{0}\right) } \left( \frac{1}{\varSigma _{0}}\right) ^{n_{P}}= & {} \mathfrak {Z}_{P}^{*,l} \end{aligned}$$

(A12)

$$\begin{aligned} \frac{\overline{P_{0}^{l}}}{\overline{f_{P,\text {mf}}^{l} \left( \tilde{m}_{0},\tilde{l}_{0}\right) }}\left( \frac{1}{\varSigma _{0}}\right) ^{n_{P}}= & {} \overline{\mathfrak {Z}_{P}^{*,l}} \end{aligned}$$

(A13)

where $n_{P}=1$ for the thermodynamic properties and $n_{P}=-\frac{1}{3}$ for the correlation lengths. The normed prefactors are labeled with the added superscript $*$. They are related to Eqs. A5 to A11 through the different values of the normed angular functions for $\vartheta =\left\{ 0,1,\left| \frac{1}{b}\right| \right\}$ (also labeled with the added superscript $*$ ). They are obtained from the corresponding angular functions given in Appendix A of Ref. [24]. We note that the $\varSigma _{0}$ dependence in the angular correction-to-scaling functions disappears introducing the normed parameters.The values of the first-order confluent correction prefactors of Eqs. 31 are then unchanged.

Analysis of Ref. [24] is complemented by 26 universal normed prefactors defined in column 2 of Table 6, where, as previously in Table 5, part (a) corresponds to the Ising-like leading prefactors (using Eqs. A(18) to A(23) of Ref. [24]), part (b) corresponds to the first-order confluent prefactors obtained from Eqs. A(45) to A(49) of Ref. [24] and part (c) corresponds to the mean-field leading prefactors (using Eqs. (4.45) to (4.49) of Ref. [24]).

Table 6 Column 2: Parametric definition of the normed universal prefactors (see text) involved in the determination of 26 physical quantities reported in Table 1. (a) 10 Ising-like leading prefactors, (b) 7 first-order confluent prefactors, (c) 9 mean-field-like prefactors. Column 3: Corresponding normed equation (with added label n) given in Ref. [24]. Column 4: line-step number of Table 8 given in Appendix A where is performed the unequivocal determination of each normed prefactor. Column 1: ordering number (added label n refers to a normed prefactor (see Table 5 and text))

The normed equation number (with added label n) corresponds to the equation number given in Ref. [24] (see column 3). In column 2, some universal prefactors are given explicitly when they appear in simple forms of exponent combinations (as in lines 11(*) and 13(e2)), or parameters $b^{2}$ and $w_{0}^{*}$ of Eq. A5 (as in lines 1n(e3(+), 2n(e3(-), and 12). More complex implicit forms express their functional dependence in some auxiliary universal quantities of Eqs (A7) to (A11). In the $\left\{ d,e,f\right\}$-dependent case, the prefactors are given in the functional forms illustrating their $L_{0}^{*}$ dependence in place of f dependence. Therefore, when the universal values of the exponent are fixed to their MR and MF values (see Table 2), the hierarchical calculations of these universal parameters can be performed using the MR-Ising-like and mean-field-like ratios and combinations of Table 3.

A special mention concerns the ordering case of $a_{rj}$ and $a_{rj}^{\star }$ involved in Eq. A4. We recall first Eqs. (5.168) and (5. 169) of the Agayan’s dissertation,

$$\begin{aligned} \frac{\left( \xi _{0}^{+}\right) ^{2}}{\Gamma _{0}^{+}}=a\left( 0\right) \; \qquad \frac{\left( \xi _{0}^{-}\right) ^{2}}{\Gamma _{0}^{-}}=a\left( 1\right) \left| k\left( 1\right) \right| ^{\eta \nu }\;\qquad \frac{\left( \xi _{0}^{c}\right) ^{2}}{\Gamma _{0}^{c}}=a\left( \left| \frac{1}{b}\right| \right) \left| l\left( \left| \frac{1}{b}\right| \right) \right| ^{\frac{\eta \nu }{\beta \delta }} \end{aligned}$$

complemented by the similar equations from the mean-field classical limit,

$$\begin{aligned} \frac{\left( \overline{\xi _{0}^{+}}\right) ^{2}}{\overline{\Gamma _{0}^{+}}} =a^{\star }\left( 0\right) \;\qquad \frac{\left( \overline{\xi _{0}^{-}}\right) ^{2}}{\overline{\Gamma _{0}^{-}}}=a^{\star }\left( 1\right) \; \qquad \frac{\left( \overline{\xi _{0}^{c}}\right) ^{2}}{\overline{\Gamma _{0}^{c}}} =a^{\star }\left( \left| \frac{1}{b}\right| \right) \end{aligned}$$

This special mention refers to the control of $Q^{+}$, $U_{\xi }$, and $Q_{2}$ for the Ising-like critical limit ($Y_{1}\rightarrow 1$), or, $\overline{U_{\xi }}$ (with $\left( \overline{U_{\xi }}\right) ^{2}\equiv \overline{U_{2}}$) and $\overline{Q_{2}}$ for the mean-field-like classical limit ($Y_{1}\rightarrow 0$).

1.1.1 Ising-Like Limit

The parameter $a_{0}$ can be estimated from $a_{0}=\frac{1}{q_{1}\left( 0\right) }\left[ \frac{Q^{+}}{\alpha q_{2}\left( 0\right) }\right] ^{\frac{2}{3}}$ (see Eq. 5.173(n), Agayan thesis), which then leads to the values of $a_{r0}$ and $\mathfrak {Z}_{\xi }^{*,+}$in agreement with $Q^{+}$. Simultaneously, we note that our CMM value $c_{q}^{+}=0.3330078$ (line $\left( 2\right)$ in Table 7), obtained from the MR results, differs from the CPM one $c_{q}^{+}=0.3286$ (see Eq. 5.191, Agayan thesis). Subsequently, in the Agayan dissertation, the parameter $a_{1}$ was calculated to recover only the universal combination $U_{\xi }$. In our present extended Eq. A4, the $a_{r1},a_{r2}$ pair increases the control to the $U_{\xi },Q_{2}$ pair. Indeed, it is now possible to calculate the two quantities $K_{1}=\frac{U_{2}}{U_{\xi }^{2}}\times \frac{1}{\left( b^{2}-1\right) ^{-\eta \nu }}$ and $K_{c}=\left[ \frac{q_{1}^{*}\left( 0\right) }{q_{1}^{*}\left( \frac{1}{b}\right) }\right] ^{\frac{-\eta }{2-\eta }}\left[ Q_{2}\right] ^{\frac{2}{2-\eta }}$, that induce the values of $a_{r1}$, $a_{r2}$, $\mathfrak {Z}_{\xi }^{*,-}$, and $\mathfrak {Z}_{\xi }^{*,c}$. Finally, the complete Ising-like triad $\left\{ a_{ri}\right\}$ induces the unequivocal control of the Ising-like combinations triad $\left\{ Q^{+},U_{\xi },Q_{2}\right\}$. However, as already noted by Agayan, the universal combination $U_{2}$ can be never explicitly controlled through $K_{1}$. Such a result confirms a needed alternative way to account for the MR value of $U_{2}$ (see below).

Table 7 Column 2: Normed functional forms of auxiliary parameters. Column 3: line-step number of Tables 8 in Appendix A where is performed the unequivocal determination of the normed auxiliary parameter

1.1.2 Mean-Field-Like Limit

The estimation of the mean-field-like triad $a_{ri}^{\star }$ follows in a similar manner. The parameter $a_{r0}^{*}$ can be estimated from the normed Eq. 5.186(n) of the Agayan’s thesis, using now $\overline{c_{q}^{+}}=1$, and noting that $\mathfrak {Z}_{B_{cr}}^{*}=-2w_{0}^{*}$. Subsequent estimation of $\overline{\mathfrak {Z}_{\xi }^{*,-}}$ occurs also in similar agreement. From the Agayan’s CPM, the non-zero value of $a_{r1}^{\star }$ only provides the control of $\overline{U_{\xi }}$ and gives thus a measure of the non-exact mean-field value of $\overline{U_{2}}$ since $a_{r1}^{\star }$ only vanishes when $\left( \overline{U_{\xi }}\right) ^{2}\equiv \overline{U_{2}}$, identically. However, we note that the mean-field like universal combination $\overline{Q_{2}}$ (see line 20(e) in Table (3)) plays a similar role to $Q_{2}$ when the $\left\{ a_{r1}^{\star },a_{r2}^{\star }\right\}$ pair is accounted for in the classical limit ($Y_{1}\rightarrow 0$) of Eq. 12. Consequently, the estimations of $\overline{D_{0}^{c}}=\frac{\tilde{l}\left( \frac{1}{b}\right) }{\left[ \overline{m_{1}}\left( \frac{1}{b}\right) \right] ^{3}}$ and $\overline{\Gamma _{0}^{c}}=\tilde{q}_{1}\left( \frac{1}{b},0\right) \times \left[ \tilde{l}\left( \frac{1}{b},0\right) \right] ^{1-\frac{1}{3}}$ to validate the relation $\overline{\Gamma _{0}^{c}}=\frac{1}{3\left( \overline{D_{0}^{c}}\right) ^{\frac{1}{3}}}$, show that the two quantities$\overline{K_{1}}=\left( \overline{U_{\xi }}\right) ^{-2}\times \frac{\overline{\Gamma _{0}^{+}}}{\overline{\Gamma _{0}^{-}}}$ and $\overline{K_{c}}=\left( \frac{\overline{\xi _{0}^{c}}}{\overline{\xi _{0}^{+}}}\right) ^{2}\frac{\overline{\Gamma _{0}^{+}}}{\overline{\Gamma _{0}^{c}}}$ (equal to unity for the ideal mean-field behavior), can be used to obtain the two equations $a_{r1}^{*}=a_{r0}^{*}\times \frac{\left( \overline{K_{c}}-1\right) b^{4}-\left( \overline{K_{1}}-1\right) }{b^{2}-1}$ and $a_{r2}^{*}=a_{r0}^{*}\times \frac{\left( \overline{K_{c}}-1\right) b^{4}-\left( \overline{K_{1}}-1\right) b^{2}}{1-b^{2}}$ (which replace Eq. 5.187(n) of Agayan thesis). Adding the practical simplification where $\overline{Q_{2}}=1\equiv \overline{K_{c}}$ finally provides the values of $a_{r1}^{\star }$, $a_{r2}^{\star }$, $\overline{\mathfrak {Z}_{\xi }^{*,-}}$, and $\overline{\mathfrak {Z}_{\xi }^{*,c}}$.

Therefore, the classical triad $\left\{ a_{ri}^{\star }\right\}$ induces the unequivocal control of the Ising-like triad $\left\{ c_{q}^{+},\overline{U_{\xi }},\overline{Q_{2}}\right\}$. Nevertheless, the non-zero values of $a_{r1}^{*}$ and $a_{r2}^{*}$ are measuring the non-exact mean-field value of $\overline{U_{2}}$ since $\overline{K_{1}}$ differs from the unity only if $\overline{U_{2}}=\frac{\overline{\Gamma _{0}^{+}}}{\overline{\Gamma _{0}^{-}}}\ne 2$ (that is precisely the case in intrinsic CMM as in the Agayan’s CPM). We also note that the classical $\overline{U_{2}}$-value is never controlled from the classical triad $\left\{ a_{ri}^{\star }\right\}$, a similar situation to the unchecked $U_{2}$-value from the Ising-like triad $\left\{ a_{ri}\right\}$.

1.2 Closed Estimation of the Normed Universal Prefactors

Ordering estimation of the quantities involved in our unequivocal process are listed in columns 6 and 7 from successive line-steps 1 to 77 of Tables 8. Columns 1, 2, and 3 give the ordering account for the normed prefactors (see Table 6), their corresponding universal ratios or combinations (from Table 3), and the resulting universal normed parameters (from Table 5), respectively. Column 4 reports the references used to support the estimation or the functional condition controlled at this step.

The series of equations start from our initial choices of the three first-order confluent amplitudes $A_{1}^{\pm }$ and $\Gamma _{1}^{+}$ to define previous Eqs. 68 to 72. The prefactors $\mathfrak {Z}_{C}^{1,+}$, $\mathfrak {Z}_{\chi }^{1,+}$ and their ratio, are only dependent on the $\alpha ,\gamma ,\Delta$ exponent values and definitively departs from their MR value. Selecting then the MR condition $\frac{A_{1}^{+}}{A_{1}^{-}}=1.20386$ (see line 9(e)-(e1) in Table 3) leads to $b^{2}=2.44726$. Here is eliminated an alternative way where another selected MR ratio value $\frac{A_{1}^{-}}{\Gamma _{1}^{+}}=0.782374$ (see line $\frac{10}{9}$(i) in Table 3) can produce a different value $b_{C\chi }^{2}=2.36$. However the impact of this latter value remains on the same order of magnitude ($\simeq 3\%$) for intrinsic versus ideal CMM. Moreover, the uncertainties in the determination of the first-order confluent amplitudes along the critical isochore remain larger than 3% (see the $\Gamma _{1}^{+}\left( \equiv a_{\chi }^{1,+}\right)$ case as typical example reported in Table 4 of Ref. [18]).

Table 8 Hierarchical determination step by step (column 1) of the normed, universal prefactors, universal parameters, and universal auxiliary quantities involved in the CMM correlation functions

The most essential point remains that only one among the 7 first-order confluent amplitudes is independent in intrinsic CMM accordingly to the MR results, while 5 among the 7 take same ideal CMM value. De facto, when $b^{2}=2.44726$ (line-step 4), all the quantities of line-steps 5 to 17 (with $\varSigma _{0}^{*}=1$), which are either $\alpha ,\gamma ,\Delta$-dependent or $\left\{ d,e,f\right\}$ non-dependent, can be estimated. In particular, that includes $\mathfrak {Z}_{B_{cr}}^{*}$ (line-step 13) and $\overline{\mathfrak {Z}_{C}^{*,-}}$ (line-step 15).

At the line step 17, we note that the unequivocal determination of the three (with 2 normed) parameters $b^{2},w_{0}^{*},\varSigma _{1}^{*}$ of Table 4 complies with the MR values of the three universal ratios $\frac{A_{1}^{+}}{A_{1}^{-}},U_{0},\frac{B_{1}}{\varGamma _{1}^{+}}$ At the opposite, the values of three universal ratios $\frac{A_{1}^{+}}{\Gamma _{1}^{+}},\frac{A_{1}^{+}}{B_{1}},\frac{A_{1}^{-}}{B_{1}}$ depart intrinsically from their corresponding MR value.

The line-steps from 18 to 32 provide the additional estimations of $L_{0}^{*}$ and $w_{1}^{*}$, starting from the selected condition $\tilde{w}^{,,}(0)_{\text {Ising}}=\tilde{w}^{,,}(0)_{\text {MF}}$ between the auxiliary functional forms reported in lines 2 and 3 of Table 7. Accounting in addition for the MR value of $Q^{+}$ and for fixed value $\overline{c_{q}^{+}}=1$ (as in CPM), all the prefactors defined in line-steps 26 to 32 are then back obtained.

The final approach consists in introducing in line-step 33 a function $f\left( x,y\right)$ of the two variables x and y, which replace the two parameters d and e, respectively, and to then write all the remaining unknown quantities in terms of x and y. The needed first closure equation $C1\left( x,y\right)$ is thus provided considering in line-step 36 the identity $S_{0,\text {Ising}}^{*,c}=S_{0,\text {MF}}^{*,c}$ between Ising and mean-field auxiliary functional forms (see lines 5 and 6 of Table 7), which then account for two additional normed amplitude combinations $R_{\chi }$ and $\overline{U^{c}}=\overline{R_{\chi }}$ (see lines 4(e) and 17(e) of Table 3). This analytical process provides two respective values of the same parameter $S_{0}^{*}$, whose equality involves a single solution $y\left( x\right) =y\left( S_{0}^{*}\right)$ (see line-step 36 in Table 8). In a similar manner, the needed second closure equation is based on the identity between two respective values of the same functional (Ising-like, confluent, and mean-field like) parameters $\varSigma _{2}^{*}$, only dependent on x, y. This second closure equation is not unique since the three different functional forms of $\varSigma _{2}^{*}$ (see lines 7, 8, and 9 of Table 7) account for three universal normed combinations $U_{2}$, $\frac{\Gamma _{1}^{+}}{\Gamma _{1}^{-}}$, and $\overline{U_{2}}$ (see lines 2(e), 11(e), and 15(i) in Table 3), respectively. The corresponding results are not compatible with a single common solution for $y\left( \Sigma _{2}^{*}\right)$, as illustrated by the equations reported in column 6 line-steps 40 to 42, and labelled $C2a\left( x,y\right) ,C2b\left( x,y\right) ,C2c\left( x,y\right)$ in column 7 of Table 8. To maintain asymptotic Ising-like coherence in the construction process the selected second equation is then Eq. $C2a\left( x,y\right)$, which accounts for $U_{2}$ and $\frac{\Gamma _{1}^{+}}{\Gamma _{1}^{-}}$, simultaneously. Such a choice provides the needed alternative way mentioned just above (section “Ising-Like Limit”) to account for $U_{2}$. However the alternative choice that maintains the control of $\overline{U_{2}}$ in place of $\frac{\Gamma _{1}^{+}}{\Gamma _{1}^{-}}$ will be estimated in a forthcoming work for comparison on the crossover effect. The common single solution $x_{0}$ of the condition $C1\left( x,y\right) =C2a\left( x,y\right)$ (see line-step 43) provides the estimation of $d=x_{0}$, $e=y\left( x_{0}\right)$ and then in return $f=L_{0}^{*}-1-x_{0}-y_{0}$, $\Sigma _{2}^{*}=\Sigma _{2,\text {Ising}or\text{CC}}^{*}\left( x_{0},y_{0}\right)$ and $S_{{0,\text{Ising}or\text{MF}}}^{{*,c}} \left( {x_{0} ,y_{0} } \right)$. We mention the resulting difference $\Sigma _{2,\text {MF}}^{*}\left( d,e\right) \ne \Sigma _{2}^{*}$ of $\simeq 5.6\%$ (line step 48). The parameter set $\left\{ w_{2}^{*},w_{3}^{*},w_{4}^{*},w_{5}^{*}\right\}$ (lines steps 49 to 52) involved in Eq. 15 results from the parameter set $\left\{ w_{1}^{*},\Sigma _{1}^{*},\Sigma _{2}^{*},S_{0}^{*}\right\}$, leading to the successive knowledge of the complete remaining line-steps until 77, except the ones where parametric formulation from CPM, or related MR values, are here not available. We note that the construction of Eq. A4 is now fully achieved.

Obviously, the complete identification between the MR crossover functions and intrinsic CMM needs to reintroduce the reference sum $\varSigma _{0}$ throughout, either $w_{0}^{*}=\frac{w_{0}}{\varSigma _{0}}=0.266804$ combined to Eq. 70, or $b^{2}=2.44726$ combined to Eq. 71. As mentioned previously, $w_{0}^{*}$ and $b^{2}$ are both universal independent parameters in Table 8, as well as $w_{0}$ and $b^{2}$ are both universal independent parameters in Table 4, indicated by the label . As previously expected, $w_{0}$, or $\varSigma _{0}$, defines the single energy density reference of dimensionless CMM, above or below $T_{c}$.

The final important remark concerning the above unequivocal process is its possible use whatever the exact numerical results of the universal exponents and universal combinations provided by any theoretical crossover functions calculated for the $N=1$-vector model of three-dimensional (3D) Ising like systems and/or the $O\left( 1\right)$ symmetric $\left( \Phi ^{2}\right) ^{2}$ Field Theory (FT) framework [4]. We also note a recent work by Dohm [44] suggesting that the Ising universality class includes not only isotropic fluids but also weakly anisotropic $\Phi ^{4}$ model. This equation-of-state modeling effort could also apply to such weakly anisotropic magnetic systems provided that the isotropic correlation length is replaced by the mean correlation length of the anisotropic system. In such a case, the model could be tested with numerical simulations of isotropic Ising models in an external magnetic field.