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the first degree in <math>a, b,</math> and <math>c\ .</math>

the first degree in <math>a, b,</math> and <math>c\ .</math>

By setting <math>\lambda = 1/x</math> in \eqref{eq:1} we have

By setting <math>\lambda = 1/x</math> in ({{EquationNote|1}}) we have

as an alternative expression of homogeneity: <math>f (x, y, z,

as an alternative expression of homogeneity: <math>f (x, y, z,

\ldots)</math> is homogeneous of degree <math>n</math> in <math>x, y,

\ldots)</math> is homogeneous of degree <math>n</math> in <math>x, y,

^{1/\beta})</math>|{{EquationRef|7}}}}

^{1/\beta})</math>|{{EquationRef|7}}}}

where <math>j(x)</math> is the "scaling" function and <math>\beta</math> and <math>\delta</math> are two critical-point exponents [3-7].  Thus, from \eqref{eq:2} and \eqref{eq:7}, as the critical point is approached <math>(H\rightarrow 0</math> and <math>t\rightarrow 0)\ ,</math> <math>\mid H\mid</math> becomes a homogeneous function of <math>t</math> and <math>\mid M\mid

where <math>j(x)</math> is the "scaling" function and <math>\beta</math> and <math>\delta</math> are two critical-point exponents [3-7].  Thus, from ({{EquationNote|2}}) and ({{EquationNote|7}}), as the critical point is approached <math>(H\rightarrow 0</math> and <math>t\rightarrow 0)\ ,</math> <math>\mid H\mid</math> becomes a homogeneous function of <math>t</math> and <math>\mid M\mid

^{1/\beta}</math> of degree <math>\beta \delta\ .</math>  The scaling function <math>j(x)</math> vanishes proportionally to <math>x+b</math> as <math>x</math> approaches <math>-b\ ,</math> with <math>b</math> a positive constant; it diverges proportionally to <math>x^{\beta(\delta-1)}</math> as <math>x\rightarrow \infty\ ;</math> and <math>j(0) = c\ ,</math> another positive constant (Fig. 1). Although \eqref{eq:7} is confined to the immediate neighborhood of the critical point <math>(t, M, H</math> all near 0), the scaling variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless traverses the infinite range <math>-b < x < \infty\ .</math>

^{1/\beta}</math> of degree <math>\beta \delta\ .</math>  The scaling function <math>j(x)</math> vanishes proportionally to <math>x+b</math> as <math>x</math> approaches <math>-b\ ,</math> with <math>b</math> a positive constant; it diverges proportionally to <math>x^{\beta(\delta-1)}</math> as <math>x\rightarrow \infty\ ;</math> and <math>j(0) = c\ ,</math> another positive constant (Fig. 1). Although ({{EquationNote|7}}) is confined to the immediate neighborhood of the critical point <math>(t, M, H</math> all near 0), the scaling variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless traverses the infinite range <math>-b < x < \infty\ .</math>

[[Image:scaling_laws_widom_nocaption_Fig1.png|thumb|300px|right|Scaling function  

[[Image:scaling_laws_widom_nocaption_Fig1.png|thumb|300px|right|Scaling function  

<math>j(x)</math>|链接=Special:FilePath/Scaling_laws_widom_nocaption_Fig1.png]]

<math>j(x)</math>|链接=Special:FilePath/Scaling_laws_widom_nocaption_Fig1.png]]

When <math>\mid H\mid = 0+</math> and <math>t<0\ ,</math> so that <math>M</math> is then the spontaneous magnetization, we have from

When <math>\mid H\mid = 0+</math> and <math>t<0\ ,</math> so that <math>M</math> is then the spontaneous magnetization, we have from ({{EquationNote|7}}), <math>\mid M\mid = (-\frac{t}{b})^\beta\ ,</math> where <math>\beta</math> is the conventional symbol for this critical-point exponent.  When <math>M\rightarrow 0</math> on the critical isotherm <math>(t=0)\ ,</math> we have <math>H \sim cM\mid

\eqref{eq:7}, <math>\mid M\mid = (-\frac{t}{b})^\beta\ ,</math> where <math>\beta</math> is the conventional symbol for this

critical-point exponent.  When <math>M\rightarrow 0</math> on the critical isotherm <math>(t=0)\ ,</math> we have <math>H \sim cM\mid

M\mid ^{\delta-1}\ ,</math> where <math>\delta</math> is the conventional symbol for this exponent.  From the first of the two

M\mid ^{\delta-1}\ ,</math> where <math>\delta</math> is the conventional symbol for this exponent.  From the first of the two

properties of <math>j(x)</math> noted above, and Eq.\eqref{eq:7}, one may calculate the magnetic susceptibility <math>(\partial

properties of <math>j(x)</math> noted above, and Eq.({{EquationNote|7}}), one may calculate the magnetic susceptibility <math>(\partial

M/\partial H)_T\ ,</math> which is then seen to diverge proportionally to <math>\mid t\mid ^{-\beta(\delta-1)}\ ,</math> both at <math>\mid

M/\partial H)_T\ ,</math> which is then seen to diverge proportionally to <math>\mid t\mid ^{-\beta(\delta-1)}\ ,</math> both at <math>\mid

H\mid = 0+</math> with <math>t<0</math> and at <math>H=0</math> with <math>t>0</math> (although with different coefficients).  The

H\mid = 0+</math> with <math>t<0</math> and at <math>H=0</math> with <math>t>0</math> (although with different coefficients).  The

A free energy <math>F</math> may be obtained from \eqref{eq:7} by integrating at fixed temperature, since <math>M = -(\partial

 

A free energy <math>F</math> may be obtained from ({{EquationNote|7}}) by integrating at fixed temperature, since <math>M = -(\partial

F/\partial H)_T\ ,</math> and the corresponding heat capacity <math>C_H</math> then follows from <math>C_H = -(\partial ^2

F/\partial H)_T\ ,</math> and the corresponding heat capacity <math>C_H</math> then follows from <math>C_H = -(\partial ^2

F/\partial T^2)_H\ .</math>  One then finds from \eqref{eq:7} that <math>C_H</math> at <math>H=0</math> diverges at the critical point

F/\partial T^2)_H\ .</math>  One then finds from ({{EquationNote|7}}) that <math>C_H</math> at <math>H=0</math> diverges at the critical point

proportionally to <math>\mid t\mid ^{-\alpha}</math> (with different coefficients for <math>t\rightarrow 0-</math> and <math>t\rightarrow

proportionally to <math>\mid t\mid ^{-\alpha}</math> (with different coefficients for <math>t\rightarrow 0-</math> and <math>t\rightarrow

0+)\ ,</math> with the critical-point exponent <math>\alpha</math> related to <math>\beta</math> and <math>\gamma</math> by the scaling law [9]

0+)\ ,</math> with the critical-point exponent <math>\alpha</math> related to <math>\beta</math> and <math>\gamma</math> by the scaling law [9]

== Critical exponents ==

== Critical exponents ==

What were probably the historically earliest versions of critical-point exponent relations like \eqref{eq:8} and \eqref{eq:9} are due to Rice [10] and to Scott [11].  It was before Domb and Sykes [12] and Fisher [13] had noted that the exponent <math>\gamma</math> was in reality greater than its mean-field value <math>\gamma =1</math> but when it was already clear from Guggenheim's corresponding-states analysis [14] that <math>\beta</math> had a value much closer to 1/3 than to its mean-field value of 1/2. Then, under the assumption <math>\gamma

What were probably the historically earliest versions of critical-point exponent relations like ({{EquationNote|8}}) and ({{EquationNote|9}}) are due to Rice [10] and to Scott [11].  It was before Domb and Sykes [12] and Fisher [13] had noted that the exponent <math>\gamma</math> was in reality greater than its mean-field value <math>\gamma =1</math> but when it was already clear from Guggenheim's corresponding-states analysis [14] that <math>\beta</math> had a value much closer to 1/3 than to its mean-field value of 1/2. Then, under the assumption <math>\gamma

=1</math> and <math>\beta \simeq 1/3\ ,</math> Rice had concluded from the equivalent of \eqref{eq:8} that <math>\delta = 1+1/\beta

=1</math> and <math>\beta \simeq 1/3\ ,</math> Rice had concluded from the equivalent of ({{EquationNote|8}}) that <math>\delta = 1+1/\beta

\simeq 4</math> (the correct value is now known to be closer to 5) and Scott had concluded from the equivalent of \eqref{eq:9} that <math>\alpha =1-2\beta \simeq 1/3</math> (the correct value is now known to be closer to 1/10).  The mean-field values are <math>\delta

\simeq 4</math> (the correct value is now known to be closer to 5) and Scott had concluded from the equivalent of ({{EquationNote|9}}) that <math>\alpha =1-2\beta \simeq 1/3</math> (the correct value is now known to be closer to 1/10).  The mean-field values are <math>\delta

=3</math> and (as noted above) <math>\alpha =0\ .</math>

=3</math> and (as noted above) <math>\alpha =0\ .</math>

Thus, as <math>r\rightarrow \infty</math> in any fixed thermodynamic state (fixed t) near the critical point, <math>h</math> decays with increasing <math>r</math> proportionally to <math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ ,</math> as in the [https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation?oldformat=true '''Ornstein-Zernike theory'''].  If, instead, the critical point is approached <math>(\xi \rightarrow \infty)</math> with a fixed, large <math>r\ ,</math> we have <math>h(r)</math> decaying with <math>r</math> only as an inverse power, <math>r^{-(d-2+\eta)}\ ,</math> which corrects the <math>r^{-(d-2)}</math> that appears in the Ornstein-Zernike theory in that limit. The scaling law \eqref{eq:10} with scaling function <math>G(x)</math> interpolates between these extremes.

Thus, as <math>r\rightarrow \infty</math> in any fixed thermodynamic state (fixed t) near the critical point, <math>h</math> decays with increasing <math>r</math> proportionally to <math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ ,</math> as in the [https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation?oldformat=true '''Ornstein-Zernike theory'''].  If, instead, the critical point is approached <math>(\xi \rightarrow \infty)</math> with a fixed, large <math>r\ ,</math> we have <math>h(r)</math> decaying with <math>r</math> only as an inverse power, <math>r^{-(d-2+\eta)}\ ,</math> which corrects the <math>r^{-(d-2)}</math> that appears in the Ornstein-Zernike theory in that limit. The scaling law({{EquationNote|1=10}}) with scaling function <math>G(x)</math> interpolates between these extremes.

In the language of fluids, with <math>\rho</math> the number density and <math>\chi</math> the isothermal compressibility, we have as an exact relation in the Ornstein-Zernike theory  

In the language of fluids, with <math>\rho</math> the number density and <math>\chi</math> the isothermal compressibility, we have as an exact relation in the Ornstein-Zernike theory  

with <math>k</math> Boltzmann's constant and where the integral is over all space with <math>\rm{d} \tau</math> the element of volume. The same relation holds in the ferromagnets with <math>\chi</math> then the magnetic susceptibility and with the deviation of <math>\rho</math> from the critical density <math>\rho_c</math> then the magnetization <math>M\ .</math>  At the critical point <math>\chi</math> is infinite and correspondingly the integral diverges because the decay length <math>\xi</math> is then also infinite.  The density <math>\rho</math> is there just the finite positive constant <math>\rho_c</math> and <math>T</math> the finite <math>T_c\ .</math>  Then from the scaling law \eqref{eq:10}, because of the homogeneity of <math>h(r,t)</math> and because the main contribution to the diverging integral comes from large <math>r\ ,</math> where \eqref{eq:10} holds, it follows that <math>\chi</math> diverges proportionally to <math>\xi^{2-\eta} \int

with <math>k</math> Boltzmann's constant and where the integral is over all space with <math>\rm{d} \tau</math> the element of volume. The same relation holds in the ferromagnets with <math>\chi</math> then the magnetic susceptibility and with the deviation of <math>\rho</math> from the critical density <math>\rho_c</math> then the magnetization <math>M\ .</math>  At the critical point <math>\chi</math> is infinite and correspondingly the integral diverges because the decay length <math>\xi</math> is then also infinite.  The density <math>\rho</math> is there just the finite positive constant <math>\rho_c</math> and <math>T</math> the finite <math>T_c\ .</math>  Then from the scaling law ({{EquationNote|1=10}}), because of the homogeneity of <math>h(r,t)</math> and because the main contribution to the diverging integral comes from large <math>r\ ,</math> where ({{EquationNote|1=10}}) holds, it follows that <math>\chi</math> diverges proportionally to <math>\xi^{2-\eta} \int

G(x)x^{d-1}\rm{d}</math><math>x\ .</math>  But the integral is now finite because, by \eqref{eq:12}, <math>G(x)</math> vanishes

G(x)x^{d-1}\rm{d}</math><math>x\ .</math>  But the integral is now finite because, by ({{EquationNote|1=12}}), <math>G(x)</math> vanishes

exponentially rapidly as <math>x\rightarrow \infty\ .</math> Thus, from \eqref{eq:11} and from the earlier <math>\chi \sim \mid

exponentially rapidly as <math>x\rightarrow \infty\ .</math> Thus, from ({{EquationNote|1=11}}) and from the earlier <math>\chi \sim \mid

t\mid^{-\gamma}</math> we have the scaling law [15]

t\mid^{-\gamma}</math> we have the scaling law [15]

:<math>\label{eq:14}

:<math>\label{eq:14}

(2-\eta)\nu = \gamma .  </math>

(2-\eta)\nu = \gamma .  </math>

{{NumBlk|:|<math>(2-\eta)\nu = \gamma . </math>|{{EquationRef|14}}}}The surface tension <math>\sigma</math> in liquid-vapor equilibrium, or the analogous excess free energy per unit area of the interface between coexisting, oppositely magnetized domains, vanishes at the critical point (Curie point) proportionally to <math>(-t)^\mu</math> with <math>\mu</math> another critical-point exponent. The interfacial region has a thickness of the order of the correlation

{{NumBlk|:|<math>(2-\eta)\nu = \gamma . </math>|{{EquationRef|14}}}}The surface tension <math>\sigma</math> in liquid-vapor equilibrium, or the analogous excess free energy per unit area of the interface between coexisting, oppositely magnetized domains, vanishes at the critical point (Curie point) proportionally to <math>(-t)^\mu</math> with <math>\mu</math> another critical-point exponent. The interfacial region has a thickness of the order of the correlation

length <math>\xi</math> so <math>\sigma/\xi</math> is the free energy per unit volume associated with the interfacial region. That is in its magnitude and in its singular critical-point behavior the same free energy per unit volume as in the bulk phases, from which the heat capacity follows by two differentiations with respect to the temperature. Thus, <math>\sigma/\xi</math> vanishes proportionally to <math>(-t)^{2-\alpha}\ ;</math> so, together with \eqref{eq:9},

length <math>\xi</math> so <math>\sigma/\xi</math> is the free energy per unit volume associated with the interfacial region. That is in its magnitude and in its singular critical-point behavior the same free energy per unit volume as in the bulk phases, from which the heat capacity follows by two differentiations with respect to the temperature. Thus, <math>\sigma/\xi</math> vanishes proportionally to <math>(-t)^{2-\alpha}\ ;</math> so, together with ({{EquationNote|1=9}}),

:<math>\label{eq:15}

:<math>\label{eq:15}

== Exponents and space dimension ==

== Exponents and space dimension ==

The critical-point exponents depend on the dimensionality

The critical-point exponents depend on the dimensionality <math>d\ .</math>  The theory was found to be illuminated by treating

<math>d\ .</math>  The theory was found to be illuminated by treating

<math>d</math> as continuously variable and of any magnitude. There

<math>d</math> as continuously variable and of any magnitude. There

is a class of critical-point exponent relations, often referred to as

is a class of critical-point exponent relations, often referred to as

hyperscaling, in which <math>d</math> appears explicitly. The

hyperscaling, in which <math>d</math> appears explicitly. The

correlation length <math>\xi</math> is the coherence length of density

correlation length <math>\xi</math> is the coherence length of density

or magnetization fluctuations.  What determines its magnitude is that

or magnetization fluctuations.  What determines its magnitude is that

a hyperscaling relation [16]. With \eqref{eq:15} we then have also [16]

a hyperscaling relation [16]. With ({{EquationNote|1=15}}) we then have also [16]

:<math>\label{eq:17}

:<math>\label{eq:17}

which, with \eqref{eq:8} and \eqref{eq:14}, yields also [18]

which, with ({{EquationNote|1=8}}) and ({{EquationNote|1=14}}), yields also [18]

:<math>\label{eq:18}

:<math>\label{eq:18}

{{NumBlk|:|<math>2-\eta = \frac{\delta -1}{\delta +1} d. </math>|{{EquationRef|18}}}}

{{NumBlk|:|<math>2-\eta = \frac{\delta -1}{\delta +1} d. </math>|{{EquationRef|18}}}}

 

Unlike the scaling laws ({{EquationNote|1=8}}), ({{EquationNote|1=9}}), ({{EquationNote|1=14}}), and ({{EquationNote|1=15}}), which make no explicit reference to the dimensionality, the <math>d</math>-dependent exponent relations ({{EquationNote|1=16}})-({{EquationNote|1=18}}) hold only for <math>d<4\ .</math>  At <math>d=4</math> the exponents assume the values they have in the mean-field theories but logarithmic factors are then appended to the simple power laws. Then for <math>d>4\ ,</math> the terms in the thermodynamic functions and correlation-function parameters that have as their exponents those given by the mean-field theories are the leading terms. The terms with the original <math>d</math>-dependent exponents, which for <math>d<4</math> were the leading terms, have been overtaken, and, while still present, are now sub-dominant.

Unlike the scaling laws \eqref{eq:8}, \eqref{eq:9}, \eqref{eq:14}, and \eqref{eq:15}, which make no explicit reference to the dimensionality, the <math>d</math>-dependent exponent relations \eqref{eq:16}-\eqref{eq:18} hold only for <math>d<4\ .</math>  At <math>d=4</math> the exponents assume the values they have in the mean-field theories but logarithmic factors are then appended to the simple power laws. Then for <math>d>4\ ,</math> the terms in the thermodynamic functions and correlation-function parameters that have as their exponents those given by the mean-field theories are the leading terms. The terms with the original <math>d</math>-dependent exponents, which for <math>d<4</math> were the leading terms, have been overtaken, and, while still present, are now sub-dominant.

This progression in critical-point properties from <math>d<4</math> to <math>d=4</math> to <math>d>4</math> is seen clearly in the phase transition that occurs in the analytically soluble model of the ideal Bose gas. There is no phase transition or critical point in it for <math>d \le 2\ .</math>  When <math>d>2</math> the chemical potential

This progression in critical-point properties from <math>d<4</math> to <math>d=4</math> to <math>d>4</math> is seen clearly in the phase transition that occurs in the analytically soluble model of the ideal Bose gas. There is no phase transition or critical point in it for <math>d \le 2\ .</math>  When <math>d>2</math> the chemical potential

\Lambda^d \sim \left\{ \begin{array} {lc }(-\mu)^{d/2-1}, & 2<d<4 \\

\Lambda^d \sim \left\{ \begin{array} {lc }(-\mu)^{d/2-1}, & 2<d<4 \\

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right. </math>|3={{EquationRef|19}}}}

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right. </math>|3={{EquationRef|19}}}}

When <math>2<d<4</math> the mean-field <math>-\mu</math> is still present but is dominated by <math>(-\mu)^{d/2-1}\ ;</math> when

When <math>2<d<4</math> the mean-field <math>-\mu</math> is still present but is dominated by <math>(-\mu)^{d/2-1}\ ;</math> when

==Origin of homogeneity; block spins==

==Origin of homogeneity; block spins==

A physical explanation for the homogeneity in \eqref{eq:7} and

A physical explanation for the homogeneity in ({{EquationNote|1=7}}) and ({{EquationNote|1=10}}) and for the exponent relations that are

\eqref{eq:10} and for the exponent relations that are

consequences of them is provided by the Kadanoff [[block-spin]] picture

consequences of them is provided by the Kadanoff [[block-spin]] picture

[5], which was itself one of the inspirations for the

[5], which was itself one of the inspirations for the

Therefore, by \eqref{eq:2}, <math>f(t,H)=t^{d/y}

 

Therefore, by ({{EquationNote|1=2}}), <math>f(t,H)=t^{d/y}

\phi(H^{y/x}/t)=H^{d/x}\psi(t/H^{y/x})</math> where <math>\phi</math> and <math>\psi</math> are functions only of the ratio <math>H^{y/x}/t\ .</math>  At <math>H=0</math> the first of these gives <math>f(t,0)=\phi(0)t^{d/y}\ .</math> But two temperature derivatives of <math>f(t,0)</math> gives the contribution to the heat capacity per spin, diverging as <math>t^{-\alpha}\ ;</math> so <math>d/y=2-\alpha\ .</math>  Also, on the critical isotherm <math>(t=0)\ ,</math> the second relation above gives <math>f(0,H)=\psi(0)H^{d/x}\ .</math>  But the magnetization per spin is <math>-(\partial f/\partial H)_T\ ,</math> vanishing as <math>H^{d/x-1}\ ,</math> so <math>d/x-1=1/\delta\ .</math>  The exponents <math>d/x</math> and <math>d/y</math> have thus been identified in terms of the thermodynamic exponents: the heat-capacity exponent <math>\alpha</math> and the critical-isotherm exponent <math>\delta\ .</math>  In the meantime, again with <math>-(\partial

\phi(H^{y/x}/t)=H^{d/x}\psi(t/H^{y/x})</math> where <math>\phi</math> and <math>\psi</math> are functions only of the ratio <math>H^{y/x}/t\ .</math>  At <math>H=0</math> the first of these gives <math>f(t,0)=\phi(0)t^{d/y}\ .</math> But two temperature derivatives of <math>f(t,0)</math> gives the contribution to the heat capacity per spin, diverging as <math>t^{-\alpha}\ ;</math> so <math>d/y=2-\alpha\ .</math>  Also, on the critical isotherm <math>(t=0)\ ,</math> the second relation above gives <math>f(0,H)=\psi(0)H^{d/x}\ .</math>  But the magnetization per spin is <math>-(\partial f/\partial H)_T\ ,</math> vanishing as <math>H^{d/x-1}\ ,</math> so <math>d/x-1=1/\delta\ .</math>  The exponents <math>d/x</math> and <math>d/y</math> have thus been identified in terms of the thermodynamic exponents: the heat-capacity exponent <math>\alpha</math> and the critical-isotherm exponent <math>\delta\ .</math>  In the meantime, again with <math>-(\partial

f/\partial H)_T</math> the magnetization per spin, the homogeneity of form of <math>f(t,H)</math> in \eqref{eq:20} is equivalent to that of <math>H(t,M)</math> in \eqref{eq:7}, from which the scaling laws <math>\gamma=\beta(\delta-1)</math> and <math>\alpha +

f/\partial H)_T</math> the magnetization per spin, the homogeneity of form of <math>f(t,H)</math> in ({{EquationNote|1=20}}) is equivalent to that of <math>H(t,M)</math> in ({{EquationNote|1=7}}), from which the scaling laws <math>\gamma=\beta(\delta-1)</math> and <math>\alpha +

2\beta + \gamma =2</math> are known to follow.

2\beta + \gamma =2</math> are known to follow.

A related argument yields the scaling law \eqref{eq:10} for the correlation function <math>h(r,t)\ ,</math> with <math>H=0</math> again for simplicity.  In the re-scaled model, <math>t</math> becomes <math>L^yt\ ,</math> as before, while <math>r</math> becomes <math>r/L\ .</math>  There may also be a factor, say <math>L^p</math> with some exponent <math>p\ ,</math> relating the magnitudes of the original and rescaled functions; thus,  

A related argument yields the scaling law ({{EquationNote|1=10}}) for the correlation function <math>h(r,t)\ ,</math> with <math>H=0</math> again for simplicity.  In the re-scaled model, <math>t</math> becomes <math>L^yt\ ,</math> as before, while <math>r</math> becomes <math>r/L\ .</math>  There may also be a factor, say <math>L^p</math> with some exponent <math>p\ ,</math> relating the magnitudes of the original and rescaled functions; thus,  

:<math>\label{eq:21}

:<math>\label{eq:21}

L^{p}h(r/L,L^yt);</math>|3={{EquationRef|21}}}}

L^{p}h(r/L,L^yt);</math>|3={{EquationRef|21}}}}

 

i.e., <math>h(r,t)</math> is homogeneous of degree <math>p</math> in <math>r</math> and <math>t^{-1/y}\ .</math> Then from the alternative form ({{EquationNote|1=2}}) of the property of homogeneity,  

i.e., <math>h(r,t)</math> is homogeneous of degree <math>p</math> in <math>r</math> and <math>t^{-1/y}\ .</math> Then from the alternative form \eqref{eq:2} of the property of homogeneity,  

:<math>\label{eq:22}

:<math>\label{eq:22}

{{NumBlk|1=:|2=<math>h(r,t)h(r,t)\equiv r^p G(r/t^{-1/y})</math>|3={{EquationRef|22}}}}

{{NumBlk|1=:|2=<math>h(r,t)h(r,t)\equiv r^p G(r/t^{-1/y})</math>|3={{EquationRef|22}}}}

 

with a scaling function <math>G\ .</math>  Comparing this with ({{EquationNote|1=10}}), and recalling that the correlation length <math>\xi</math> diverges at the critical point as <math>t^{-\nu}</math> with exponent <math>\nu\ ,</math> we identify <math>p=-(d-2+\eta)</math> and <math>1/y=\nu\ .</math>  The scaling law <math>(2-\eta)\nu=\gamma\ ,</math> which was a consequence of the homogeneity of form of <math>h(r,t)\ ,</math> again holds, while from <math>1/y=\nu</math> and the earlier <math>d/y=2-\alpha</math> we now have the hyperscaling law ({{EquationNote|1=17}}), <math>d\nu=2-\alpha\ .</math>

with a scaling function <math>G\ .</math>  Comparing this with \eqref{eq:10}, and recalling that the correlation length <math>\xi</math> diverges at the critical point as <math>t^{-\nu}</math> with exponent <math>\nu\ ,</math> we identify <math>p=-(d-2+\eta)</math> and <math>1/y=\nu\ .</math>  The scaling law <math>(2-\eta)\nu=\gamma\ ,</math> which was a consequence of the homogeneity of form of <math>h(r,t)\ ,</math> again holds, while from <math>1/y=\nu</math> and the earlier <math>d/y=2-\alpha</math> we now have the hyperscaling law \eqref{eq:17}, <math>d\nu=2-\alpha\ .</math>

The block-spin picture thus yields the critical-point scaling of the thermodynamic and correlation functions, and both the <math>d</math>-independent and <math>d</math>-dependent relations among the scaling exponents. The essence of this picture is confirmed in the renormalization-group theory [19,20].

The block-spin picture thus yields the critical-point scaling of the thermodynamic and correlation functions, and both the <math>d</math>-independent and <math>d</math>-dependent relations among the scaling exponents. The essence of this picture is confirmed in the renormalization-group theory [19,20].