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第78行： + − <strong><nowiki>Scaling laws </nowiki></strong> are the expression of physical principles in the mathematical language of homogeneous functions. − 标度律是物理原理在齐次函数数学语言中的表达。 − − 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> 第97行：
第87行： − 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 ({{EquationNote|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> 第116行：
第97行： − − 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 [https://wiki.swarma.org/index.php/%E7%83%AD%E5%8A%9B%E5%AD%A6 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. − 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 ({{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 ({{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> 第198行：
第133行： − 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 − 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.({{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 − 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 ({{EquationNote|7}}) and − ({{EquationNote|8}}) are examples of scaling laws, Eq.({{EquationNote|7}}) being a statement of homogeneity and the exponent relation ({{EquationNote|8}}) a consequence of that homogeneity. 第219行：
第142行： − 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 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 − 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> − + − 当 <math>2\beta+\gamma=2</math>，则有 <math>\alpha =0</math>，这通常意味着对数发散而不是幂律发散，并且在 <math>t=0+</math> 和 <math>t=0-</math> 之间存在叠加有限不连续。在二维伊辛模型中，仅有对数关系而这种不连续是不存在的；而在平均场近似中情形相反。 − === Critical exponents 临界指数 === − 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 ({{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 ({{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 − − + + + + − 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] 第251行：
第164行： − 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> 第271行：
第173行： − 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 − :<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 ({{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 ({{EquationNote|1=12}}), <math>G(x)</math> vanishes − 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] − :<math>\label{eq:14} − (2-\eta)\nu = \gamma . </math> − + − another scaling relation [16,17]. − === Exponents and space dimension 临界点指数与空间维度 ===+ − 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 ({{EquationNote|1=15}}) we then have also [16] − :<math>\label{eq:17}+ − d\nu = 2-\alpha = \gamma+2\beta, </math> − − which, with ({{EquationNote|1=8}}) and ({{EquationNote|1=14}}), yields also [18] − :<math>\label{eq:18}+ − 2-\eta = \frac{\delta -1}{\delta +1} d. </math> − − 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. − 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> 第353行：
第209行： − 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 R'''enormalization-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> − 这一行为也反映在重整化群理论中。最简单的情形是，重整化群流中有两个相互竞争的不动点，一点与依赖 <math>d</math> 的指数相关，同时满足与 <math>d</math> 无关的标度关系和超标度关系，另一点则与平均场理论的 <math>d</math> 无关指数相关。前者决定了当 <math>d<4\ </math> 时的主导临界点行为。<math>d=4</math> 时，这两个不动点重合，指数现在是平均场理论的指数，但在平均场幂律中增加了对数因子。对于 <math>d>4</math>，两固定点再次分开，此时主导临界点行为源自平均场理论的指数。综上所述，两固定点产生的影响覆盖所有 <math>d\ </math>的取值范围，但是随着 <math>d\ </math>取值的变化，主导临界点行为会在二者之间切换。+ − ===Origin of homogeneity; block spins 齐次性的成因与块自旋=== − A physical explanation for the homogeneity in ({{EquationNote|1=7}}) and ({{EquationNote|1=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]. − 来自于重整化群理论的卡丹诺夫块自旋（图2）为({{EquationNote|1=7}})和({{EquationNote|1=10}})中的齐次性以及由它们推导出的指数关系提供了物理解释。+ − 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). − + − 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 ({{EquationNote|1=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 ({{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 − 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. 第424行：
第242行： − 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} − 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 ({{EquationNote|1=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 ({{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> − 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]. − + + +
→物理学中的标度律
== 物理学中的标度律 ==
== 物理学中的标度律 ==
'''标度律 Scaling laws'''是物理原理在齐次函数数学语言中的表达。
如果对所有 <math>\lambda\ </math> 都满足关系{{NumBlk|2=<math>f(\lambda
如果对所有 <math>\lambda\ </math> 都满足关系{{NumBlk|2=<math>f(\lambda
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
则称函数 <math>f (x, y, z,\ldots)</math> 是变量 <math>x,y,z,\ldots</math> 的 <math>n</math> 次齐次函数。
则称函数 <math>f (x, y, z,\ldots)</math> 是变量 <math>x,y,z,\ldots</math> 的 <math>n</math> 次齐次函数。
例如，<math>ax^2 + bxy + cy^2</math> 是 <math>x</math> 和 <math>y</math> 二次齐次函数，而对 <math>a, b,</math> <math>c\ </math>则是一次齐次的。
例如，<math>ax^2 + bxy + cy^2</math> 是 <math>x</math> 和 <math>y</math> 二次齐次函数，而对 <math>a, b,</math> <math>c\ </math>则是一次齐次的。
将 <math>\lambda = 1/x</math> 带入({{EquationNote|1}})，则有齐次性的另一种表达式：如果<math>f (x, y, z,
将 <math>\lambda = 1/x</math> 带入({{EquationNote|1}})，则有齐次性的另一种表达式：如果<math>f (x, y, z,
\ldots)</math>满足关系：{{NumBlk|2=<math>f(x, y, z, \ldots) = x^nf(1, y/x,
\ldots)</math>满足关系：{{NumBlk|2=<math>f(x, y, z, \ldots) = x^nf(1, y/x,
则它是 <math>x, y,
则它是 <math>x, y,
z, \ldots</math> 的 <math>n</math> 次齐次函数。
z, \ldots</math> 的 <math>n</math> 次齐次函数。
即等于 <math>x</math> 的 <math>n</math> 次方乘以某个以比值 <math>y/x, z/x, \ldots</math> 为变量的函数 <math>\phi</math>。
即等于 <math>x</math> 的 <math>n</math> 次方乘以某个以比值 <math>y/x, z/x, \ldots</math> 为变量的函数 <math>\phi</math>。
如果<math>f (x, y, z, \ldots)</math> 对 <math>x, y, z, \ldots</math> 是 <math>n</math> 次齐次的，则它满足欧拉定理：{{NumBlk|:|<math>x\frac{\partial f}{\partial x}+y\frac{\partial
如果<math>f (x, y, z, \ldots)</math> 对 <math>x, y, z, \ldots</math> 是 <math>n</math> 次齐次的，则它满足欧拉定理：{{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}}}}
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.</math>|{{EquationRef|3}}}}
在[[热力学|'''热力学 Thermodynamics''']]中，如果一个系统的标度增加 <math>\lambda</math> 倍而其强度量不发生变化，仅是该系统所有化学组分的广度量（如熵 <math>S\ </math>，能量 <math>E\ </math>，体积 <math>V\ </math>和质量 <math>m_1, m_2, \ldots</math> 等）也增加相同倍数。则有广度函数 <math>S(E, V, m_1, m_2, \ldots)</math> 在广义论证中满足齐次关系：{{NumBlk|:|<math>S(\lambda E, \lambda V, \lambda
在[[热力学|'''热力学 Thermodynamics''']]中，如果一个系统的标度增加 <math>\lambda</math> 倍而其强度量不发生变化，仅是该系统所有化学组分的广度量（如熵 <math>S\ </math>，能量 <math>E\ </math>，体积 <math>V\ </math>和质量 <math>m_1, m_2, \ldots</math> 等）也增加相同倍数。则有广度函数 <math>S(E, V, m_1, m_2, \ldots)</math> 在广义论证中满足齐次关系：{{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}}}}
{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).</math>|{{EquationRef|4}}}}
以 <math>T</math>，<math>p</math>，<math>\mu_i</math> 分别表示温度，压力和不同组分 <math>i\ </math>的化学势，根据热力学关系 <math>\partial
以 <math>T</math>，<math>p</math>，<math>\mu_i</math> 分别表示温度，压力和不同组分 <math>i\ </math>的化学势，根据热力学关系 <math>\partial
S/\partial E = 1/T\ </math>， <math>\partial S/\partial V = p/T\ </math>，和 <math>\partial S/\partial m_i = - \mu_i/T\ </math>：再由欧拉定理可得：{{NumBlk|:|<math>\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
S/\partial E = 1/T\ </math>， <math>\partial S/\partial V = p/T\ </math>，和 <math>\partial S/\partial m_i = - \mu_i/T\ </math>：再由欧拉定理可得：{{NumBlk|:|<math>\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
\cdots) =S,</math>|{{EquationRef|5}}}}
\cdots) =S,</math>|{{EquationRef|5}}}}
任何广度函数 <math>X(T, p, m_1, m_2, \ldots)\ </math>（如体积 <math>V\ </math>或者吉布斯自由能<math>E+pV-TS\ </math>）在等温等压状态下，对 <math>m_i</math> 都是一次齐次的，因此：{{NumBlk|:|<math>X = m_1 \frac{\partial X}{\partial
任何广度函数 <math>X(T, p, m_1, m_2, \ldots)\ </math>（如体积 <math>V\ </math>或者吉布斯自由能<math>E+pV-TS\ </math>）在等温等压状态下，对 <math>m_i</math> 都是一次齐次的，因此：{{NumBlk|:|<math>X = m_1 \frac{\partial X}{\partial
m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots ,</math>|{{EquationRef|6}}}}
m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots ,</math>|{{EquationRef|6}}}}
是重要的一类关系。
是重要的一类关系。
引言交代了经典热力学语境下的标度关系。在其后发展出的统计力学中，“标度律”一词指代热力学函数和相关函数在临界点附近的齐次形式，以及这些函数中指数之间的关系。
引言交代了经典热力学语境下的标度关系。在其后发展出的统计力学中，“标度律”一词指代热力学函数和相关函数在临界点附近的齐次形式，以及这些函数中指数之间的关系。
当 <math>t = T/T_c-1\ </math>时，铁磁物质临近居里点（临界点），磁场强度 <math>H\</math>，磁化强度 <math>M\</math>以及 <math>t = T/T_c-1\ </math>满足{{NumBlk|:|<math>H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid
当 <math>t = T/T_c-1\ </math>时，铁磁物质临近居里点（临界点），磁场强度 <math>H\</math>，磁化强度 <math>M\</math>以及 <math>t = T/T_c-1\ </math>满足{{NumBlk|:|<math>H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid
^{1/\beta})</math>|{{EquationRef|7}}}}
^{1/\beta})</math>|{{EquationRef|7}}}}
其中 <math>j(x)</math> 是“标度”函数，<math>\beta</math> 和<math>\delta</math> 是临界点指数。因此由({{EquationNote|2}})和({{EquationNote|7}})，当铁磁物质趋近于临界点时<math>(H\rightarrow 0</math> 且 <math>t\rightarrow 0)\ </math>，<math>\mid H\mid</math> 是 <math> t </math> 和 <math>\mid M\mid
其中 <math>j(x)</math> 是“标度”函数，<math>\beta</math> 和<math>\delta</math> 是临界点指数。因此由({{EquationNote|2}})和({{EquationNote|7}})，当铁磁物质趋近于临界点时<math>(H\rightarrow 0</math> 且 <math>t\rightarrow 0)\ </math>，<math>\mid H\mid</math> 是 <math> t </math> 和 <math>\mid M\mid
<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]]
当<math>\mid H\mid = 0+</math> 且 <math>t<0\ </math>，<math>M</math> 是自发磁化率，由({{EquationNote|7}})可得<math>\mid M\mid = (-\frac{t}{b})^\beta\ </math>，其中 <math>\beta</math> 为临界指数。在临界等温线<math>(t=0)\ </math>，当 <math>M\rightarrow 0</math> 时，我们有<math>H \sim cM\mid
当<math>\mid H\mid = 0+</math> 且 <math>t<0\ </math>，<math>M</math> 是自发磁化率，由({{EquationNote|7}})可得<math>\mid M\mid = (-\frac{t}{b})^\beta\ </math>，其中 <math>\beta</math> 为临界指数。在临界等温线<math>(t=0)\ </math>，当 <math>M\rightarrow 0</math> 时，我们有<math>H \sim cM\mid
方程({{EquationNote|7}})和({{EquationNote|8}})都是标度律的范例，({{EquationNote|7}})是齐次性的表述，({{EquationNote|8}})作为指数关系式则是这种齐次性的结果。
方程({{EquationNote|7}})和({{EquationNote|8}})都是标度律的范例，({{EquationNote|7}})是齐次性的表述，({{EquationNote|8}})作为指数关系式则是这种齐次性的结果。
由于<math>M = -(\partial
由于<math>M = -(\partial
F/\partial H)_T\ </math>，等温条件下自由能 <math>F</math> 可以通过积分由({{EquationNote|7}})式得出，且相应的热容 <math>C_H = -(\partial ^2
F/\partial H)_T\ </math>，等温条件下自由能 <math>F</math> 可以通过积分由({{EquationNote|7}})式得出，且相应的热容 <math>C_H = -(\partial ^2
F/\partial T^2)_H\ </math>。由({{EquationNote|7}})式可知，在<math>H=0</math> 时 <math>C_H</math> 在临界点处依 <math>\mid t\mid ^{-\alpha}</math> 比例发散（其中 <math>t\rightarrow 0-</math> 和 <math>t\rightarrow
F/\partial T^2)_H\ </math>。由({{EquationNote|7}})式可知，在<math>H=0</math> 时 <math>C_H</math> 在临界点处依 <math>\mid t\mid ^{-\alpha}</math> 比例发散（其中 <math>t\rightarrow 0-</math> 和 <math>t\rightarrow
0+ </math> 各有不同的系数），临界点指数 <math>\alpha</math> 与<math>\beta</math> 和 <math>\gamma</math> 满足以下标度律：{{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 discontinuity is still present.
0+ </math> 各有不同的系数），临界点指数 <math>\alpha</math> 与<math>\beta</math> 和 <math>\gamma</math> 满足以下标度律[9]：{{NumBlk|:|<math>\alpha +2\beta +\gamma=2. </math>|{{EquationRef|9}}}}
当 <math>2\beta+\gamma=2</math>，则有 <math>\alpha =0</math>，这通常意味着对数发散而不是幂律发散，并且在 <math>t=0+</math> 和 <math>t=0-</math> 之间存在叠加有限不连续。[4]在二维伊辛模型中，仅有对数关系而这种不连续是不存在的；而在平均场近似中情形相反。
=3</math> and (as noted above) <math>\alpha =0\ .</math>
({{EquationNote|8}})和({{EquationNote|9}})分别来自里斯和斯考特的贡献。它们大概是历史上最早版本的临界指数关系。在此之后，Domb和Sykes以及Fisher注意到指数 <math>\gamma</math> 实际上比平均场值<math>\gamma =1</math> 大。而在更早之前，Guggenheim的对应状态分析就清楚地表明 <math>\beta</math>值更靠近1/3而非平均场值的1/2。之后在 <math>\gamma
=== 临界指数 Critical exponents ===
({{EquationNote|8}})和({{EquationNote|9}})分别来自Rice [10]和Scott [11]的贡献。它们大概是历史上最早版本的临界指数关系。在此之后，Domb和Sykes[12]以及Fisher[13]注意到指数 <math>\gamma</math> 实际上比平均场值<math>\gamma =1</math> 大。而在更早之前，Guggenheim的对应状态分析就清楚地表明 <math>\beta</math>值更靠近1/3而非平均场值的1/2。之后在 <math>\gamma
=1</math> 和 <math>\beta \simeq 1/3\ </math>的假设下，里斯由({{EquationNote|8}})式总结出 <math>\delta = 1+1/\beta
=1</math> 和 <math>\beta \simeq 1/3\ </math>的假设下，里斯由({{EquationNote|8}})式总结出 <math>\delta = 1+1/\beta
\simeq 4</math>（如今已知正确值接近5）。同时斯考特由({{EquationNote|9}})式得出 <math>\alpha =1-2\beta \simeq 1/3</math>（正确值接近1/10）。另外平均场值 <math>\delta
\simeq 4</math>（如今已知正确值接近5）。同时斯考特由({{EquationNote|9}})式得出 <math>\alpha =1-2\beta \simeq 1/3</math>（正确值接近1/10）。另外平均场值 <math>\delta
=3</math>，<math>\alpha =0\ </math>。
=3</math>，<math>\alpha =0\ </math>。
:<math>\label{eq:10}
:<math>\label{eq:10}
铁磁体和流体中的长程空间相关函数在临界点附近也表现出齐次性。简单起见，考虑磁场强度 <math>H=0</math> 且温度接近临界点的情况，关联函数 <math>h(r,t)</math> 作为空间分离 <math>r</math>（假设很大）的函数，如下所示：{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi). </math>|{{EquationRef|10}}}}
铁磁体和流体中的长程空间相关函数在临界点附近也表现出齐次性。简单起见，考虑磁场强度 <math>H=0</math> 且温度接近临界点的情况，关联函数 <math>h(r,t)</math> 作为空间分离 <math>r</math>（假设很大）的函数，如下所示：{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi). </math>|{{EquationRef|10}}}}
其中 <math>d</math> 是空间维度，<math>\eta</math> 是另一临界点指数，<math>\xi</math> 是关联长度（相关关系的指数衰减长度），当趋近于临界点时，其发散过程满足：{{NumBlk|:|<math>\xi\sim \mid t\mid ^{-\nu}</math>|{{EquationRef|11}}}}
其中 <math>d</math> 是空间维度，<math>\eta</math> 是另一临界点指数，<math>\xi</math> 是关联长度（相关关系的指数衰减长度），当趋近于临界点时，其发散过程满足：{{NumBlk|:|<math>\xi\sim \mid t\mid ^{-\nu}</math>|{{EquationRef|11}}}}
其中 <math>\nu</math> 是另外的临界指数。因此 <math>h(r,t)</math>（<math>H=0</math>）是 <math>r</math> 和 <math>\mid t\mid
其中 <math>\nu</math> 是另外的临界指数。因此 <math>h(r,t)</math>（<math>H=0</math>）是 <math>r</math> 和 <math>\mid t\mid
^{-\nu}</math> 的<math>-(d-2+\eta)\ </math>次齐次方程。标度函数 <math>G(x)</math> 具有以下性质(在常数比例因子范围内)：{{NumBlk|:|<math>G(x) \sim \left\{
^{-\nu}</math> 的<math>-(d-2+\eta)\ </math>次齐次方程。标度函数 <math>G(x)</math> 具有以下性质(在常数比例因子范围内)：{{NumBlk|:|<math>G(x) \sim \left\{
\infty \\ 1, & x\rightarrow 0 . \end{array} \right. </math>|{{EquationRef|12}}}}
\infty \\ 1, & x\rightarrow 0 . \end{array} \right. </math>|{{EquationRef|12}}}}
因此，在任何靠近临界点的恒温热力学状态下，当 <math>r\rightarrow \infty</math> 时，<math>h</math> 随<math>r</math> 的增加依 <math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ </math> 成比例衰减（参见'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论 Ornstein-Zernike theory]'''）。如果相反，在固定的大 <math>r\ </math> 条件下，迫近临界点（<math>\xi \rightarrow \infty</math>），会有 <math>h(r)</math> 作为逆幂 <math>r^{-(d-2+\eta)}\ </math> 随 <math>r</math> 衰减，这也修正了在此极限条件下奥恩斯泰因-泽尔尼克理论中出现的 <math>r^{-(d-2)}</math>。标度律({{EquationNote|1=10}})及标度函数 <math>G(x)</math> 内插于这些极限之间。
因此，在任何靠近临界点的恒温热力学状态下，当 <math>r\rightarrow \infty</math> 时，<math>h</math> 随<math>r</math> 的增加依 <math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ </math> 成比例衰减（参见'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论 Ornstein-Zernike theory]'''）。如果相反，在固定的大 <math>r\ </math> 条件下，迫近临界点（<math>\xi \rightarrow \infty</math>），会有 <math>h(r)</math> 作为逆幂 <math>r^{-(d-2+\eta)}\ </math> 随 <math>r</math> 衰减，这也修正了在此极限条件下奥恩斯泰因-泽尔尼克理论中出现的 <math>r^{-(d-2)}</math>。标度律({{EquationNote|1=10}})及标度函数 <math>G(x)</math> 内插于这些极限之间。
在流体研究中，由数密度 <math>\rho</math> 和 等温压缩率 <math>\chi</math>，我们可以得到一个'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论]'''的精确表达式：{{NumBlk|:|<math>\rho kT
在流体研究中，由数密度 <math>\rho</math> 和 等温压缩率 <math>\chi</math>，我们可以得到一个'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论]'''的精确表达式：{{NumBlk|:|<math>\rho kT
\chi =1+\rho \int h(r) \rm{d}\tau</math>|{{EquationRef|13}}}}
\chi =1+\rho \int h(r) \rm{d}\tau</math>|{{EquationRef|13}}}}
其中 <math>k</math> 是'''[[玻尔兹曼常数]]'''，<math>\rm{d} \tau</math> 是体积元，积分区域是整个空间。对铁磁体也有相同的关系成立，包含磁化率 <math>\chi</math>，<math>\rho</math> 与临界密度 <math>\rho_c</math> 的差值，以及磁化强度 <math>M\ </math>。在临界点处，<math>\chi</math> 无穷大，且对应积分式也发散，因为衰减长度 <math>\xi</math> 也是无穷大的。而密度 <math>\rho</math>为有限正常数 <math>\rho_c</math>，<math>T</math> 为 <math>T_c\ </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 ({{EquationNote|1=9}}),
其中 <math>k</math> 是'''[[玻尔兹曼常数]]'''，<math>\rm{d} \tau</math> 是体积元，积分区域是整个空间。对铁磁体也有相同的关系成立，包含磁化率 <math>\chi</math>，<math>\rho</math> 与临界密度 <math>\rho_c</math> 的差值，以及磁化强度 <math>M\ </math>。在临界点处，<math>\chi</math> 无穷大，且对应积分式也发散，因为衰减长度 <math>\xi</math> 也是无穷大的。而密度 <math>\rho</math>为有限正常数 <math>\rho_c</math>，<math>T</math> 为 <math>T_c\ </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 ({{EquationNote|1=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>
液-气平衡时的表面张力 <math>\sigma</math>，或共存的、相反磁化畴之间的界面单位面积上的类似过剩自由能，在临界点(居里点)与 <math>(-t)^\mu</math>（<math>\mu</math>对应此处临界点指数）成比例消失。界面区域的厚度与关联长度 <math>\xi</math> 的数量级相当，因此 <math>\sigma/\xi</math> 是与界面区域相关的单位体积自由能。在它的大小和它的奇异临界点行为中，每单位体积的自由能和在体相中是一样的，从体相中，依据关于温度的两个微分可以得出热容。因此，<math>\sigma/\xi</math> 依 <math>(-t)^{2-\alpha}\ </math> 成比例消失；再联系({{EquationNote|1=9}})式可以得到另一个标度关系：{{NumBlk|:|<math>\mu + \nu = 2-\alpha= \gamma +2\beta,</math>|{{EquationRef|15}}}}
液-气平衡时的表面张力 <math>\sigma</math>，或共存的、相反磁化畴之间的界面单位面积上的类似过剩自由能，在临界点(居里点)与 <math>(-t)^\mu</math>（<math>\mu</math>对应此处临界点指数）成比例消失。界面区域的厚度与关联长度 <math>\xi</math> 的数量级相当，因此 <math>\sigma/\xi</math> 是与界面区域相关的单位体积自由能。在它的大小和它的奇异临界点行为中，每单位体积的自由能和在体相中是一样的，从体相中，依据关于温度的两个微分可以得出热容。因此，<math>\sigma/\xi</math> 依 <math>(-t)^{2-\alpha}\ </math> 成比例消失；再联系({{EquationNote|1=9}})式可以得到另一个标度关系[15]：{{NumBlk|:|<math>\mu + \nu = 2-\alpha= \gamma +2\beta,</math>|{{EquationRef|15}}}}
其他标度关系 [16,17].
=== 临界点指数与空间维度 ===
临界点指数取决于维数 <math>d\ </math>。人们发现，将 <math>d</math> 视为具有任意大小的连续变量可以解释说明这一观点。在一类被称为超标度的临界点指数关系中，可以清楚地看到 <math>d</math>。关联长度 <math>\xi</math> 为密度或磁化波动的相干长度。决定其大小的是体积 <math>\xi ^d</math> 中与自发波动有关的过剩自由能，且一定是 <math>kT\ </math> 阶的，在临界点处具有有限值 <math>kT_c</math> 。但在这样的微元体中，典型的波动只会产生共轭相。则自由能 <math>kT</math> 为创建区域 <math>\xi^{d-1}\ </math>的界面 <math>\sigma \xi^{d-1}\ </math>的自由能。因此，当接近临界点时，<math>\sigma \xi^{d-1}</math> 具有 <math>kT_c\ </math> 阶的有限极限。再由指数 <math>\mu</math> 和 <math>\nu\ </math>的定义可得超标度关系：{{NumBlk|:|<math>\mu = (d-1)\nu,</math>|{{EquationRef|16}}}}
临界点指数取决于维数 <math>d\ </math>。人们发现，将 <math>d</math> 视为具有任意大小的连续变量可以解释说明这一观点。在一类被称为超标度的临界点指数关系中，可以清楚地看到 <math>d</math>。关联长度 <math>\xi</math> 为密度或磁化波动的相干长度。决定其大小的是体积 <math>\xi ^d</math> 中与自发波动有关的过剩自由能，且一定是 <math>kT\ </math> 阶的，在临界点处具有有限值 <math>kT_c</math> 。但在这样的微元体中，典型的波动只会产生共轭相。则自由能 <math>kT</math> 为创建区域 <math>\xi^{d-1}\ </math>的界面 <math>\sigma \xi^{d-1}\ </math>的自由能。因此，当接近临界点时，<math>\sigma \xi^{d-1}</math> 具有 <math>kT_c\ </math> 阶的有限极限。再由指数 <math>\mu</math> 和 <math>\nu\ </math>的定义可得超标度关系：{{NumBlk|:|<math>\mu = (d-1)\nu,</math>|{{EquationRef|16}}}}
再由({{EquationNote|1=15}})有[16]：{{NumBlk|:|<math>d\nu = 2-\alpha = \gamma+2\beta,</math>|{{EquationRef|17}}}}
再由({{EquationNote|1=15}})有：{{NumBlk|:|<math>d\nu = 2-\alpha = \gamma+2\beta,</math>|{{EquationRef|17}}}}
结合({{EquationNote|1=8}})和({{EquationNote|1=14}})得到[18]：{{NumBlk|:|<math>2-\eta = \frac{\delta -1}{\delta +1} d. </math>|{{EquationRef|18}}}}
结合({{EquationNote|1=8}})和({{EquationNote|1=14}})得到：{{NumBlk|:|<math>2-\eta = \frac{\delta -1}{\delta +1} d. </math>|{{EquationRef|18}}}}
标度律({{EquationNote|1=8}})，({{EquationNote|1=9}})，({{EquationNote|1=14}})和({{EquationNote|1=15}})没有明显的和空间维数相联系，而({{EquationNote|1=16}})-({{EquationNote|1=18}})则是依赖于 <math>d</math> 的指数关系式，且仅对 <math>d<4\ </math>成立。对于 <math>d=4</math>，热力学函数中依据平均场理论给出的以相关函数参数为指数的项是主导项。而本身在 <math>d<4</math> 时，包含依赖于 <math>d</math> 的指数的主导项，虽然依然存在，但是已经被取代而变成次要项。
标度律({{EquationNote|1=8}})，({{EquationNote|1=9}})，({{EquationNote|1=14}})和({{EquationNote|1=15}})没有明显的和空间维数相联系，而({{EquationNote|1=16}})-({{EquationNote|1=18}})则是依赖于 <math>d</math> 的指数关系式，且仅对 <math>d<4\ </math>成立。对于 <math>d=4</math>，热力学函数中依据平均场理论给出的以相关函数参数为指数的项是主导项。而本身在 <math>d<4</math> 时，包含依赖于 <math>d</math> 的指数的主导项，虽然依然存在，但是已经被取代而变成次要项。
在理想玻色气体的解析溶解模型中，可以清楚地看到临界点性质从 <math>d<4</math> 到 <math>d=4</math> 再到 <math>d>4</math> 的变化过程。在 <math>d \le 2\ </math> 的情形下，不存在相变或者临界点。当 <math>d>2</math> 时，对于所有 <math>\rho \Lambda ^d
在理想玻色气体的解析溶解模型中，可以清楚地看到临界点性质从 <math>d<4</math> 到 <math>d=4</math> 再到 <math>d>4</math> 的变化过程。在 <math>d \le 2\ </math> 的情形下，不存在相变或者临界点。当 <math>d>2</math> 时，对于所有 <math>\rho \Lambda ^d
\ge \zeta (d/2)\ </math>，化学势 <math>\mu</math>（此处不要与表面张力指数 <math>\mu</math> 混淆）都会变为零。其中 <math>\rho</math> 是密度，<math>\Lambda</math> 是热德布罗意波长，即 <math>h/\sqrt {2\pi mkT}</math>（其中 <math>h</math> 是普朗克常数，<math>m</math> 是原子质量），<math>\zeta (s)</math> 是黎曼 <math>\zeta</math> 函数。当由下 <math>\rho \Lambda^d \rightarrow
\ge \zeta (d/2)\ </math>，化学势 <math>\mu</math>（此处不要与表面张力指数 <math>\mu</math> 混淆）都会变为零。其中 <math>\rho</math> 是密度，<math>\Lambda</math> 是热德布罗意波长，即 <math>h/\sqrt {2\pi mkT}</math>（其中 <math>h</math> 是普朗克常数，<math>m</math> 是原子质量），<math>\zeta (s)</math> 是黎曼 <math>\zeta</math> 函数。当由下 <math>\rho \Lambda^d \rightarrow
\\ \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}}}}
当 <math>2<d<4</math> 时，平均场指数 <math>-\mu</math> 依然存在，但是主导指数则是<math>(-\mu)^{d/2-1}\ </math>；当 <math>d>4</math> 时，奇异指数 <math>(-\mu)^{d/2-1}</math> 依然存在，但是主导指数为 <math>-\mu\ </math>。
当 <math>2<d<4</math> 时，平均场指数 <math>-\mu</math> 依然存在，但是主导指数则是<math>(-\mu)^{d/2-1}\ </math>；当 <math>d>4</math> 时，奇异指数 <math>(-\mu)^{d/2-1}</math> 依然存在，但是主导指数为 <math>-\mu\ </math>。
这一行为也反映在'''重整化群理论 renormalization-group theory'''中[19-21]。最简单的情形是，重整化群流中有两个相互竞争的不动点，一点与依赖 <math>d</math> 的指数相关，同时满足与 <math>d</math> 无关的标度关系和超标度关系，另一点则与平均场理论的 <math>d</math> 无关指数相关[21]。前者决定了当 <math>d<4\ </math> 时的主导临界点行为。<math>d=4</math> 时，这两个不动点重合，指数现在是平均场理论的指数，但在平均场幂律中增加了对数因子。对于 <math>d>4</math>，两固定点再次分开，此时主导临界点行为源自平均场理论的指数。综上所述，两固定点产生的影响覆盖所有 <math>d\ </math>的取值范围，但是随着 <math>d\ </math>取值的变化，主导临界点行为会在二者之间切换。
===齐次性的成因与块自旋===
来自于重整化群理论的卡丹诺夫块自旋（图2）[5]为({{EquationNote|1=7}})和({{EquationNote|1=10}})中的齐次性以及由它们推导出的指数关系提供了物理解释[19,20]。
在格子自旋模型（[[伊辛模型 Ising Model|'''伊辛模型''']]）中，假设有许多自旋块，每一个的线性尺寸为 <math>L\ </math>，因此包含 <math>L^d</math>，而 <math>L</math> 远小于发散关联长度 <math>\xi</math>（图2）。
在格子自旋模型（[[伊辛模型 Ising Model|'''伊辛模型''']]）中，假设有许多自旋块，每一个的线性尺寸为 <math>L\ </math>，因此包含 <math>L^d</math>，而 <math>L</math> 远小于发散关联长度 <math>\xi</math>（图2）。
[[Image:scaling_laws_widom_nocaption_Fig2.png|thumb|300px|right|Block spins|链接=Special:FilePath/Scaling_laws_widom_nocaption_Fig2.png]]
[[Image:scaling_laws_widom_nocaption_Fig2.png|thumb|300px|right|Block spins]]
每一个自旋块通过与相邻块的共同边界相互作用。在对应的重标度模型中可以将它们视作单独的自旋。每个块的大小都是有限的，因此其内部的自旋只对系统的自由能提供解析项。自由能密度（单位自旋自由能）中包含临界点奇点及其指数的部分源自于自旋块间的相互作用。设自由能密度为<math>f(t,H)\ </math>，它是温度（由 <math>t=T/T_c-1</math>）和磁场强度 <math>H\ </math>的函数。在重标度后的图像中，相关长度与原始图像中相同，但以格子间距的数量来度量，前者比后者小 <math>L\ </math>倍。因此，重标度模型实际上比原始模型离临界点更远。当逼近临界点时，<math>H</math> 和 <math>t</math>趋近于0，重标度模型中的有效 <math>H</math> 和<math>t</math> 为 <math>L^xH</math> 和 <math>L^yt</math>，其中 <math>x</math> 和 <math>y\ ,</math> 是正指数。从原始模型的角度来看，每个块的自旋对自由能奇异部分的贡献是 <math>L^df(t,H)\ </math>，而对重标度模型来说，则是 <math>f(L^yt, L^xH)\ </math>。因此有：{{NumBlk|1=:|2=<math>f(L^yt,
每一个自旋块通过与相邻块的共同边界相互作用。在对应的重标度模型中可以将它们视作单独的自旋。每个块的大小都是有限的，因此其内部的自旋只对系统的自由能提供解析项。自由能密度（单位自旋自由能）中包含临界点奇点及其指数的部分源自于自旋块间的相互作用。设自由能密度为<math>f(t,H)\ </math>，它是温度（由 <math>t=T/T_c-1</math>）和磁场强度 <math>H\ </math>的函数。在重标度后的图像中，相关长度与原始图像中相同，但以格子间距的数量来度量，前者比后者小 <math>L\ </math>倍。因此，重标度模型实际上比原始模型离临界点更远。当逼近临界点时，<math>H</math> 和 <math>t</math>趋近于0，重标度模型中的有效 <math>H</math> 和<math>t</math> 为 <math>L^xH</math> 和 <math>L^yt</math>，其中 <math>x</math> 和 <math>y\ ,</math> 是正指数。从原始模型的角度来看，每个块的自旋对自由能奇异部分的贡献是 <math>L^df(t,H)\ </math>，而对重标度模型来说，则是 <math>f(L^yt, L^xH)\ </math>。因此有：{{NumBlk|1=:|2=<math>f(L^yt,
L^xH) \equiv L^df(t,H);</math>|3={{EquationRef|20}}}}
L^xH) \equiv L^df(t,H);</math>|3={{EquationRef|20}}}}
由({{EquationNote|1=1}})可得，<math>f(t,H)</math> 是 <math>t</math> 和 <math>H^{y/x}</math> 的 <math>d/y\ </math>次齐次函数。
由({{EquationNote|1=1}})可得，<math>f(t,H)</math> 是 <math>t</math> 和 <math>H^{y/x}</math> 的 <math>d/y\ </math>次齐次函数。
因此，由({{EquationNote|1=2}})得
因此，由({{EquationNote|1=2}})得
2\beta + \gamma =2</math>。
2\beta + \gamma =2</math>。
在重标度模型中，<math>t</math> 变为<math>L^yt\ </math>，<math>r</math> 则为 <math>r/L\ </math>。对于关联函数 <math>h(r,t)\ </math>标度律({{EquationNote|1=10}})，也存在某一指数 <math>p\ </math>使 <math>L^p</math> 成为联系原始模型和重标度模型的因子；所以有：{{NumBlk|1=:|2=<math>h(r,t) \equiv
在重标度模型中，<math>t</math> 变为<math>L^yt\ </math>，<math>r</math> 则为 <math>r/L\ </math>。对于关联函数 <math>h(r,t)\ </math>标度律({{EquationNote|1=10}})，也存在某一指数 <math>p\ </math>使 <math>L^p</math> 成为联系原始模型和重标度模型的因子；所以有：{{NumBlk|1=:|2=<math>h(r,t) \equiv
L^{p}h(r/L,L^yt);</math>|3={{EquationRef|21}}}}
L^{p}h(r/L,L^yt);</math>|3={{EquationRef|21}}}}
即 <math>h(r,t)</math> 是 <math>r</math>和 <math>t^{-1/y}\ </math> 的 <math>p</math> 次齐次函数。再由齐次性表达式({{EquationNote|1=2}})有：{{NumBlk|1=:|2=<math>h(r,t)h(r,t)\equiv r^p G(r/t^{-1/y})</math>|3={{EquationRef|22}}}}
即 <math>h(r,t)</math> 是 <math>r</math>和 <math>t^{-1/y}\ </math> 的 <math>p</math> 次齐次函数。再由齐次性表达式({{EquationNote|1=2}})有：{{NumBlk|1=:|2=<math>h(r,t)h(r,t)\equiv r^p G(r/t^{-1/y})</math>|3={{EquationRef|22}}}}
其中 <math>G\ </math>是标度函数。与({{EquationNote|1=10}})对比，由临界点处的关联长度服从({{EquationNote|1=11}})，我们可得<math>p=-(d-2+\eta)</math> 以及 <math>1/y=\nu\ </math>。由此齐次性表达式 <math>h(r,t)\ </math>得出的标度律<math>(2-\eta)\nu=\gamma\ </math>依然成立，且再由<math>1/y=\nu</math> 和 <math>d/y=2-\alpha</math>，得到超标度律({{EquationNote|1=17}}) — <math>d\nu=2-\alpha\ </math>。
其中 <math>G\ </math>是标度函数。与({{EquationNote|1=10}})对比，由临界点处的关联长度服从({{EquationNote|1=11}})，我们可得<math>p=-(d-2+\eta)</math> 以及 <math>1/y=\nu\ </math>。由此齐次性表达式 <math>h(r,t)\ </math>得出的标度律<math>(2-\eta)\nu=\gamma\ </math>依然成立，且再由<math>1/y=\nu</math> 和 <math>d/y=2-\alpha</math>，得到超标度律({{EquationNote|1=17}}) — <math>d\nu=2-\alpha\ </math>。
因此，块自旋图产生了热力学函数和相关函数的临界点标度关系，以及标度指数之间的 <math>d</math> 无关和 <math>d</math> 依赖关系。重正化群理论证实了块自旋图的本质。
因此，块自旋图产生了热力学函数和相关函数的临界点标度关系，以及标度指数之间的 <math>d</math> 无关和 <math>d</math> 依赖关系。重正化群理论证实了块自旋图的本质[19,20]。
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== References 参考文献 ==
== References 参考文献 ==