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→Introduction 引言
f(x, y, z, \ldots) = x^nf(1, y/x,
f(x, y, z, \ldots) = x^nf(1, y/x,
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots); </math>
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots); </math>
{{NumBlk|2=<math>f(x, y, z, \ldots) = x^nf(1, y/x,
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots);</math>|3={{EquationRef|2}}|：}}
i.e., the <math>n^{th}</math> power of <math>x</math> times some
i.e., the <math>n^{th}</math> power of <math>x</math> times some
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.
</math>
</math>
{{NumBlk|:|<math>x\frac{\partial f}{\partial x}+y\frac{\partial
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.</math>|{{EquationRef|3}}}}
In [[thermodynamics]], if the scale of a system is merely
In [[thermodynamics]], if the scale of a system is merely
and the masses <math>m_1, m_2, \ldots</math> of each of its chemical
and the masses <math>m_1, m_2, \ldots</math> of each of its chemical
constituents are increased by that factor, so the extensive function
constituents are increased by that factor, so the extensive function
<math>S(E, V, m_1, m_2, \ldots)</math> is homogeneous of degree 1 in
<math>S(E, V, m_1, m_2, \ldots)</math> is homogeneous of degree 1 in its extensive arguments:
its extensive arguments:
:<math>\label{eq:4}
:<math>\label{eq:4}
{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).
{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).
</math>
</math>
{{NumBlk|:|<math>S(\lambda E, \lambda V, \lambda
{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).</math>|{{EquationRef|4}}}}
pressure, and <math>\mu_i</math> the chemical potential of the species
pressure, and <math>\mu_i</math> the chemical potential of the species
<math>i\ ,</math> we have the thermodynamic relations <math>\partial
<math>i\ ,</math> we have the thermodynamic relations <math>\partial
S/\partial E = 1/T\ ,</math> <math>\partial S/\partial V = p/T\ ,</math>
S/\partial E = 1/T\ ,</math> <math>\partial S/\partial V = p/T\ ,</math> and <math>\partial S/\partial m_i = - \mu_i/T\ ;</math> so from Euler's theorem,
and <math>\partial S/\partial m_i = - \mu_i/T\ ;</math> so from Euler's
theorem,
:<math>\label{eq:5}
:<math>\label{eq:5}
\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
\cdots) =S, </math>
\cdots) =S, </math>
{{NumBlk|:|<math>\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
\cdots) =S,</math>|{{EquationRef|5}}}}
an important identity. Any extensive function
<math>X(T, p, m_1, m_2, \ldots)\ ,</math> such as the volume V or the
an important identity. Any extensive function <math>X(T, p, m_1, m_2, \ldots)\ ,</math> such as the volume V or the Gibbs free energy <math>E+pV-TS\ ,</math> is homogeneous of the first degree in the <math>m_i</math> at fixed <math>p</math> and <math>T\ ,</math> so
Gibbs free energy <math>E+pV-TS\ ,</math> is homogeneous of the first
degree in the <math>m_i</math> at fixed <math>p</math> and
<math>T\ ,</math> so
:<math>\label{eq:6}
:<math>\label{eq:6}
X = m_1 \frac{\partial X}{\partial
X = m_1 \frac{\partial X}{\partial
m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots , </math>
m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots , </math>
{{NumBlk|:|<math>X = m_1 \frac{\partial X}{\partial
m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots ,</math>|{{EquationRef|6}}}}
an
an
== Scaling laws ==
== Scaling laws ==
The foregoing are scaling relations in classical thermodynamics. In
The foregoing are scaling relations in classical thermodynamics. In more recent times, in statistical mechanics, the expression "scaling laws" has been taken to refer to the homogeneity of form of the thermodynamic and correlation functions near critical points, and to the resulting relations among the exponents that occur in those functions. There are many excellent references for critical phenomena and the associated scaling laws, among them the superb book by Domb [1] and the historic early review by Fisher [2].
more recent times, in statistical mechanics, the expression "scaling
laws" has been taken to refer to the homogeneity of form of the
thermodynamic and correlation functions near critical points, and to
the resulting relations among the exponents that occur in those
functions. There are many excellent references for critical phenomena
and the associated scaling laws, among them the superb book by Domb
[1] and the historic early review by Fisher
[2].
Near the Curie point (critical point) of a ferromagnet, which occurs
Near the Curie point (critical point) of a ferromagnet, which occurs
H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid
H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid
^{1/\beta}) </math>
^{1/\beta}) </math>
{{NumBlk|:|<math>H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid
^{1/\beta})</math>|{{EquationRef|7}}}}
where <math>j(x)</math> is the "scaling" function
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
and <math>\beta</math> and <math>\delta</math> are two critical-point
<math>t\rightarrow 0)\ ,</math> <math>\mid H\mid</math> becomes a homogeneous function of <math>t</math> and <math>\mid M\mid
exponents [3-7]. Thus, from \eqref{eq:2} and \eqref{eq:7},
^{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
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
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
variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless
<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
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
<math>M</math> is then the spontaneous magnetization, we have from
\eqref{eq:7}, <math>\mid M\mid = (-\frac{t}{b})^\beta\ ,</math> where <math>\beta</math> is the conventional symbol for this
\eqref{eq:7}, <math>\mid M\mid = (-\frac{t}{b})^\beta\ ,</math>
critical-point exponent. When <math>M\rightarrow 0</math> on the critical isotherm <math>(t=0)\ ,</math> we have <math>H \sim cM\mid
where <math>\beta</math> is the conventional symbol for this
M\mid ^{\delta-1}\ ,</math> where <math>\delta</math> is the conventional symbol for this exponent. From the first of the two
critical-point exponent. When <math>M\rightarrow 0</math> on the
properties of <math>j(x)</math> noted above, and Eq.\eqref{eq:7}, one may calculate the magnetic susceptibility <math>(\partial
critical isotherm <math>(t=0)\ ,</math> we have <math>H \sim cM\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
M\mid ^{\delta-1}\ ,</math> where <math>\delta</math> is 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
conventional symbol for this exponent. From the first of the two
conventional symbol for the susceptibility exponent is <math>\gamma\ ,</math> so we have [8]
properties of <math>j(x)</math> noted above, and Eq.\eqref{eq: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
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
conventional symbol for the susceptibility exponent is
<math>\gamma\ ,</math> so we have [8]
:<math>\label{eq:8}
:<math>\label{eq:8}
\gamma =
\gamma =
\beta(\delta-1). </math>
\beta(\delta-1). </math>
{{NumBlk|:|<math>\gamma =
\beta(\delta-1). </math>|{{EquationRef|8}}}}
Equations \eqref{eq:7} and
Equations \eqref{eq:7} and
\eqref{eq:8} are examples of scaling laws, Eq.\eqref{eq:7}
\eqref{eq:8} are examples of scaling laws, Eq.\eqref{eq:7} being a statement of homogeneity and the exponent relation \eqref{eq:8} a consequence of that homogeneity.
being a statement of homogeneity and the exponent relation
\eqref{eq:8} a consequence of that homogeneity.
A free energy <math>F</math> may be obtained from \eqref{eq:7} by
A free energy <math>F</math> may be obtained from \eqref{eq:7} by integrating at fixed temperature, since <math>M = -(\partial
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
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
<math>C_H</math> then follows from <math>C_H = -(\partial ^2
proportionally to <math>\mid t\mid ^{-\alpha}</math> (with different coefficients for <math>t\rightarrow 0-</math> and <math>t\rightarrow
F/\partial T^2)_H\ .</math> One then finds from \eqref{eq:7} that
0+)\ ,</math> with the critical-point exponent <math>\alpha</math> related to <math>\beta</math> and <math>\gamma</math> by the scaling law [9]
<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
0+)\ ,</math> with the critical-point exponent <math>\alpha</math>
related to <math>\beta</math> and <math>\gamma</math> by the scaling
law [9]
:<math>\label{eq:9}
:<math>\label{eq:9}
\alpha +2\beta +\gamma=2. </math>
\alpha +2\beta +\gamma=2. </math>
{{NumBlk|:|<math>\alpha +2\beta +\gamma=2. </math>|{{EquationRef|9}}}}When <math>2\beta+\gamma=2</math> the resulting <math>\alpha =0</math> means, generally, a logarithmic rather than power-law divergence together with a superimposed finite discontinuity occurring between <math>t=0+</math> and <math>t=0-</math> [4]. In the 2-dimensional Ising model the discontinuity is absent and only the logarithm remains, while in mean-field (van der Waals, Curie-Weiss, Bragg-Williams) approximation the logarithm is absent but the
When <math>2\beta+\gamma=2</math> the resulting <math>\alpha =0</math>
means, generally, a logarithmic rather than power-law divergence
together with a superimposed finite discontinuity occurring between
<math>t=0+</math> and <math>t=0-</math> [4]. In the
2-dimensional Ising model the discontinuity is absent and only the
logarithm remains, while in mean-field (van der Waals, Curie-Weiss,
Bragg-Williams) approximation the logarithm is absent but the
discontinuity is still present.
discontinuity is still present.
== Critical exponents ==
== Critical exponents ==
What were probably the historically earliest versions of
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
critical-point exponent relations like \eqref{eq:8} and
=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
\eqref{eq:9} are due to Rice [10] and to Scott [11].
\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
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
\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
=3</math> and (as noted above) <math>\alpha =0\ .</math>
=3</math> and (as noted above) <math>\alpha =0\ .</math>
The long-range spatial correlation functions in ferromagnets and
The long-range spatial correlation functions in ferromagnets and fluids also exhibit a homogeneity of form near the critical point. At magnetic field <math>H=0</math> (assumed for simplicity) the correlation function <math>h(r,t)</math> as a function of the spatial separation <math>r</math> (assumed very large) and temperature near the critical point (t assumed very small), is of the form [5,15]
fluids also exhibit a homogeneity of form near the critical point. At
magnetic field <math>H=0</math> (assumed for simplicity) the
correlation function <math>h(r,t)</math> as a function of the spatial
separation <math>r</math> (assumed very large) and temperature near
the critical point (t assumed very small), is of the form [5,15]
:<math>\label{eq:10}
:<math>\label{eq:10}
h(r,t)=r^{-(d-2+\eta)}G(r/\xi). </math>
h(r,t)=r^{-(d-2+\eta)}G(r/\xi). </math>
{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi). </math>|{{EquationRef|10}}}}
Here <math>d</math> is the
Here <math>d</math> is the