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^{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 \eqref{eq:2} and \eqref{eq:7}, as the critical point is approached <math>(H\rightarrow 0</math> and

^{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>

<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>

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

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

{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi).  </math>|{{EquationRef|10}}}}

{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi).  </math>|{{EquationRef|10}}}}

 

Here <math>d</math> is the dimensionality of space, <math>\eta</math> is another critical-point exponent, and <math>\xi</math> is the correlation length (exponential

Here <math>d</math> is the

dimensionality of space, <math>\eta</math> is another critical-point

exponent, and <math>\xi</math> is the correlation length (exponential

decay length of the correlations), which diverges as  

decay length of the correlations), which diverges as  

:<math>\label{eq:11}

:<math>\label{eq:11}

\xi\sim \mid t\mid ^{-\nu} </math>

\xi\sim \mid t\mid ^{-\nu} </math>

 

as the critical point is

as the critical point is approached, with <math>\nu</math> still another critical-point exponent. Thus, <math>h(r,t)</math> (with <math>H=0)</math> is a homogeneous function of <math>r</math> and <math>\mid t\mid

approached, with <math>\nu</math> still another critical-point

^{-\nu}</math> of degree <math>-(d-2+\eta)\ .</math>  The scaling function <math>G(x)</math> has the properties (to within constant

exponent. Thus, <math>h(r,t)</math> (with <math>H=0)</math> is a

homogeneous function of <math>r</math> and <math>\mid t\mid

^{-\nu}</math> of degree <math>-(d-2+\eta)\ .</math>  The scaling

function <math>G(x)</math> has the properties (to within constant

factors of proportionality),  

factors of proportionality),  

\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow

\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow

\infty \\ 1, & x\rightarrow 0 . \end{array} \right.  </math>

\infty \\ 1, & x\rightarrow 0 . \end{array} \right.  </math>

 

Thus, as

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.

<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

[[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.

In the language of fluids, with <math>\rho</math> the number density

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  

and <math>\chi</math> the isothermal compressibility, we have as an

exact relation in the Ornstein-Zernike theory  

:<math>\label{eq:13}

:<math>\label{eq:13}

\rho kT

\rho kT

\chi =1+\rho \int h(r) \rm{d}\tau </math>

\chi =1+\rho \int h(r) \rm{d}\tau </math>

 

with <math>k</math>

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

Boltzmann's constant and where the integral is over all space with

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

<math>\rm{d} \tau</math> the element of volume. The same relation holds in

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

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

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

exponentially rapidly as <math>x\rightarrow \infty\ .</math>  Thus, from

\eqref{eq: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

 

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},

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},

:<math>\label{eq:15}

:<math>\label{eq:15}

\mu + \nu = 2-\alpha= \gamma +2\beta, </math>

\mu + \nu = 2-\alpha= \gamma +2\beta, </math>

 

another

another scaling relation [16,17].

scaling relation [16,17].

== Exponents and space dimension ==

== Exponents and space dimension ==

:<math>\label{eq:16}

:<math>\label{eq:16}

\mu = (d-1)\nu, </math>

\mu = (d-1)\nu, </math>

 

a hyperscaling relation [16].

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

With \eqref{eq:15} we then have also [16]

:<math>\label{eq:17}

:<math>\label{eq:17}

d\nu = 2-\alpha = \gamma+2\beta, </math>

d\nu = 2-\alpha = \gamma+2\beta, </math>

which, with

 

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

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

:<math>\label{eq:18}

:<math>\label{eq:18}

2-\eta = \frac{\delta -1}{\delta +1} d.  </math>

2-\eta = \frac{\delta -1}{\delta +1} d.  </math>

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

<math>\mu</math> (not to be confused with the surface-tension exponent <math>\mu</math>) vanishes identically for all <math>\rho \Lambda ^d

\ge \zeta (d/2)\ ,</math> where <math>\rho</math> is the density,   <math>\Lambda</math> is the thermal de Broglie wavelength <math>h/\sqrt {2\pi mkT}</math> with <math>h</math> Planck's constant and <math>m</math> the mass of the atom, and <math>\zeta (s)</math> is the Riemann zeta function.  As <math>\rho \Lambda^d \rightarrow

\zeta(d/2)</math> from below, <math>\mu</math> vanishes through a range of negative values. As <math>\mu \rightarrow 0-\ ,</math> the difference <math>\zeta(d/2)-\rho \Lambda^d</math> vanishes (to within

 

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

<math>\mu</math> (not to be confused with the surface-tension exponent

<math>\mu</math>) vanishes identically for all <math>\rho \Lambda ^d

\ge \zeta (d/2)\ ,</math> where <math>\rho</math> is the density,

<math>\Lambda</math> is the thermal de Broglie wavelength

<math>h/\sqrt {2\pi mkT}</math> with <math>h</math> Planck's constant

and <math>m</math> the mass of the atom, and <math>\zeta (s)</math> is

the Riemann zeta function.  As <math>\rho \Lambda^d \rightarrow

\zeta(d/2)</math> from below, <math>\mu</math> vanishes through a

range of negative values. As <math>\mu \rightarrow 0-\ ,</math> the

difference <math>\zeta(d/2)-\rho \Lambda^d</math> vanishes (to within

positive proportionality factors) as  

positive proportionality factors) as  

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right.

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right.

</math>

</math>

 

When <math>2<d<4</math> the mean-field <math>-\mu</math> is

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

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

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

<math>d>4</math> the singular <math>(-\mu)^{d/2-1}</math> is still

present but is dominated by the mean-field <math>-\mu\ .</math>

This behavior is reflected again in the [[renormalization-group theory]]

This behavior is reflected again in the [[renormalization-group theory]] [19-21].  In the simplest cases there are two competing fixed points for the renormalization-group flows, one of them associated with <math>d</math>-dependent  

[19-21].  In the simplest cases there are two competing fixed points for  

the renormalization-group flows, one of them associated with <math>d</math>-dependent  

exponents that satisfy both the <math>d</math>-independent scaling relations and

exponents that satisfy both the <math>d</math>-independent scaling relations and

the hyperscaling relations, the other with the <math>d</math>-independent  

the hyperscaling relations, the other with the <math>d</math>-independent