567
个编辑
更改
无编辑摘要
Prof. Benjamin Widom accepted the invitation on 4 February 2009 (self-imposed deadline: 4 August 2009).
[[Category:Statistical Mechanics]]
<strong><nowiki>Scaling laws </nowiki></strong> are the expression of
physical principles in the mathematical language of homogeneous
functions.
==Introduction==
A function <math>f (x, y, z,\ldots)</math> is said to be homogeneous
of degree <math>n</math> in the variables <math>x,y,z,\ldots</math>
if, identically for all <math>\lambda\ ,</math>
:<math>\label{eq:1}
f(\lambda
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
\ldots). </math>
For example, <math>ax^2 + bxy + cy^2</math>
is homogeneous of degree 2 in <math>x</math> and <math>y</math> and of
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
as an alternative expression of homogeneity: <math>f (x, y, z,
\ldots)</math> is homogeneous of degree <math>n</math> in <math>x, y,
z, \ldots</math> if
:<math>\label{eq:2}
f(x, y, z, \ldots) = x^nf(1, y/x,
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots); </math>
i.e., the <math>n^{th}</math> power of <math>x</math> times some
function <math>\phi</math> of the ratios <math>y/x, z/x, \ldots</math>
alone.
If <math>f (x, y, z, \ldots)</math> is homogeneous of degree
<math>n</math> in <math>x, y, z, \ldots</math> it satisfies Euler's
theorem :
:<math>\label{eq:3}
x\frac{\partial f}{\partial x}+y\frac{\partial
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.
</math>
In [[thermodynamics]], if the scale of a system is merely
increased by a factor <math>\lambda</math> with no change in its
intensive properties, then all its extensive properties including its
entropy <math>S\ ,</math> energy <math>E\ ,</math> volume <math>V\ ,</math>
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
<math>S(E, V, m_1, m_2, \ldots)</math> is homogeneous of degree 1 in
its extensive arguments:
:<math>\label{eq:4}
S(\lambda E, \lambda V, \lambda
{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).
</math>
With <math>T</math> the temperature, <math>p</math> the
pressure, and <math>\mu_i</math> the chemical potential of the species
<math>i\ ,</math> we have the thermodynamic relations <math>\partial
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,
:<math>\label{eq:5}
\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
\cdots) =S, </math>
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
:<math>\label{eq:6}
X = m_1 \frac{\partial X}{\partial
m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots , </math>
an
important class of relations.
== Scaling laws ==
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].
Near the Curie point (critical point) of a ferromagnet, which occurs
at <math>T = T_c\ ,</math> the magnetic field <math>H\ ,</math>
magnetization <math>M\ ,</math> and <math>t = T/T_c-1\ ,</math> are
related by
:<math>\label{eq:7}
H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid
^{1/\beta}) </math>
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
^{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|Fig1|Scaling function
<math>j(x)</math>]]
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
\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
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}
\gamma =
\beta(\delta-1). </math>
Equations \eqref{eq:7} and
\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.
A free energy <math>F</math> may be obtained from \eqref{eq: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 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
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}
\alpha +2\beta +\gamma=2. </math>
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.
== 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
=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>
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]
:<math>\label{eq:10}
h(r,t)=r^{-(d-2+\eta)}G(r/\xi). </math>
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
:<math>\label{eq:11}
\xi\sim \mid t\mid ^{-\nu} </math>
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
^{-\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),
:<math>\label{eq:12}
G(x) \sim \left\{
\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow
\infty \\ 1, & x\rightarrow 0 . \end{array} \right. </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
[[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
and <math>\chi</math> the isothermal compressibility, we have as an
exact relation in the Ornstein-Zernike theory
:<math>\label{eq:13}
\rho kT
\chi =1+\rho \int h(r) \rm{d}\tau </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
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]
:<math>\label{eq:14}
(2-\eta)\nu = \gamma . </math>
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}
\mu + \nu = 2-\alpha= \gamma +2\beta, </math>
another
scaling relation [16,17].
== Exponents and space dimension ==
The critical-point exponents depend on the dimensionality
<math>d\ .</math> The theory was found to be illuminated by treating
<math>d</math> as continuously variable and of any magnitude. There
is a class of critical-point exponent relations, often referred to as
hyperscaling, in which <math>d</math> appears explicitly. The
correlation length <math>\xi</math> is the coherence length of density
or magnetization fluctuations. What determines its magnitude is that
the excess free energy associated with the spontaneous fluctuations in
the volume <math>\xi ^d</math> must be of order <math>kT\ ,</math> which
has the finite value <math>kT_c</math> at the critical point. But the
typical fluctuations that occur in such an elemental volume are just
such as to produce the conjugate phase. The free energy
<math>kT</math> is then that for creating an interface of area
<math>\xi^{d-1}\ ,</math> which is <math>\sigma \xi^{d-1}\ .</math> Thus,
as the critical point is approached <math>\sigma \xi^{d-1}</math> has
a finite limit of order <math>kT_c\ .</math> Then from the definitions
of the exponents <math>\mu</math> and <math>\nu\ ,</math>
:<math>\label{eq:16}
\mu = (d-1)\nu, </math>
a hyperscaling relation [16].
With \eqref{eq:15} we then have also [16]
:<math>\label{eq:17}
d\nu = 2-\alpha = \gamma+2\beta, </math>
which, with
\eqref{eq:8} and \eqref{eq:14}, yields also [18]
:<math>\label{eq:18}
2-\eta = \frac{\delta -1}{\delta +1} d. </math>
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
<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
:<math>\label{eq:19}
\zeta(d/2)-\rho
\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>
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
<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]]
[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
the hyperscaling relations, the other with the <math>d</math>-independent
exponents of the mean-field theories [21]. The first determines the leading
critical-point behavior when <math>d<4\ .</math> At <math>d=4</math> the two fixed
points coincide and the exponents are now those of the mean-field
theories but with logarithmic factors appended to the mean-field power
laws. For <math>d>4</math> the two fixed points separate again and
the leading critical-point behavior now comes from the one whose
exponents are those of the mean-field theories. The effects of both
fixed points are present at all <math>d\ ,</math> but the dominant
critical-point behavior comes from only the one or the other,
depending on <math>d\ .</math>
==Origin of homogeneity; block spins==
A physical explanation for the homogeneity in \eqref{eq:7} and
\eqref{eq:10} and for the exponent relations that are
consequences of them is provided by the Kadanoff [[block-spin]] picture
[5], which was itself one of the inspirations for the
renormalization-group theory [19,20].
In a lattice spin model (Ising model), one considers blocks of spins,
each of linear size <math>L\ ,</math> thus containing <math>L^d</math>
spins, with <math>L</math> much less than the diverging correlation
length <math>\xi</math> (Fig. 2).
[[Image:scaling_laws_widom_nocaption_Fig2.png|thumb|300px|right|Fig2|Block spins]]
Each block interacts with its neighbors through their common boundary
as though it were a single spin in a re-scaled model. Each block is
of finite size so the spins in its interior contribute only analytic
terms to the free energy of the system. The part of the free-energy
density (free energy per spin) that carries the critical-point
singularities and their exponents comes from the interactions between
blocks. Let this free-energy density be <math>f(t,H)\ ,</math> a
function of temperature through <math>t=T/T_c-1</math> and of the
magnetic field <math>H\ .</math> The correlation length is the same in
the re-scaled picture as in the original, but measured as a number of
lattice spacings it is smaller in the former by the factor
<math>L\ .</math> Thus, the re-scaled model is effectively further from
its critical point than the original was from its; so with
<math>H</math> and <math>t</math> both going to 0 as the critical
point is approached, the effective <math>H</math> and <math>t</math>
in the re-scaled model are <math>L^xH</math> and <math>L^yt</math>
with positive exponents <math>x</math> and <math>y\ ,</math> so
increasing with <math>L\ .</math> From the point of view of the
original model the contribution to the singular part of the free
energy made by the spins in each block is <math>L^df(t,H)\ ,</math>
while that same quantity, from the point of the view of the re-scaled
model, is <math>f(L^yt, L^xH)\ .</math> Thus,
:<math>\label{eq:20}
f(L^yt,
L^xH) \equiv L^df(t,H); </math>
i.e., by \eqref{eq:1},
<math>f(t,H)</math> is a homogeneous function of <math>t</math> and
<math>H^{y/x}</math> of degree <math>d/y\ .</math>
Therefore, by \eqref{eq: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
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 +
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,
:<math>\label{eq:21}
h(r,t) \equiv
L^{p}h(r/L,L^yt); </math>
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}
h(r,t)\equiv r^p G(r/t^{-1/y}) </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].
== References ==
[1] C. Domb, ''The Critical Point'' (Taylor & Francis, 1996).
[2] M.E. Fisher, Repts. Prog. Phys. '''30''', part 2
(1967) 615.
[3] C. Domb and D.L. Hunter, Proc. Phys. Soc. '''86'''
(1965) 1147.
[4] B. Widom, J. Chem. Phys. '''43''' (1965) 3898.
[5] L.P. Kadanoff, Physics '''2''' (1966) 263.
[6] A.Z. Patashinskii and V.L. Pokrovskii,
Soviet Physics JETP '''23''' (1966) 292.
[7] R.B. Griffiths, Phys. Rev. '''158''' (1967)
176.
[8] B. Widom, J. Chem. Phys. '''41''' (1964) 1633.
[9] J.W. Essam and M.E. Fisher, J. Chem. Phys. ''' 38''' (1963) 802.
[10] O.K. Rice, J. Chem. Phys. '''23''' (1955) 169.
[11] R.L. Scott, J. Chem. Phys. '''21''' (1953) 209.
[12] C. Domb and M.F. Sykes, Proc. Roy. Soc. A '''240''' (1957) 214.
[13] M.E. Fisher, Physica '''25''' (1959) 521.
[14] E.A. Guggenheim, J. Chem. Phys. '''13''' (1945) 253.
[15] M.E. Fisher, J. Math. Phys. '''5''' (1964) 944.
[16] B. Widom, J. Chem. Phys. '''43''' (1965) 3892.
[17] P.G. Watson, J. Phys. C1 (1968) 268.
[18] G. Stell, Phys. Rev. Lett. '''20''' (1968) 533.
[19] K.G. Wilson, Phys. Rev. B '''4''' (1971) 3174.
[20] K.G. Wilson, Phys. Rev. B '''4''' (1971) 3184.
[21] K.G. Wilson and M.E. Fisher, Phys. Rev. Lett. '''28''' (1972) 240.
<!-- Authors, please check this list and remove any references that are irrelevant. This list is generated automatically to reflect the links from your article to other accepted articles in Scholarpedia. -->
<b>Internal references</b>
* Tomasz Downarowicz (2007) [[Entropy]]. Scholarpedia, 2(11):3901.
* Eugene M. Izhikevich (2007) [[Equilibrium]]. Scholarpedia, 2(10):2014.
* Giovanni Gallavotti (2008) [[Fluctuations]]. Scholarpedia, 3(6):5893.
* Cesar A. Hidalgo R. and Albert-Laszlo Barabasi (2008) [[Scale-free networks]]. Scholarpedia, 3(1):1716.
==See also==