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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis      0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> (    6h  `}  LUSQdkYkHrh7h_    0,  `  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    0t ^ `  X*  0\H ^   Z*  0$H ^ ` H Z*H  0޽h ? 3380___PPT10.p6-O  ؞!jg 0  (  r  S Hpp`  H   0 H@ 0 +  {% Youjin Deng 09.12.5&(2&H  0޽h ? 33___PPT10i.q@+D='  = @B +> 0 p>(  P  C (A.Diagram   0  Y)Let start with the phase diagram of water**  <$ P5& QPhenomena observed near the critical point are referred to as Critical phenomena.RRH  0޽h ? 3380___PPT10.`sСj[  0  (  r  S po `  `  c $A2 ?? `  c $A3 ??`  `  c $A5 ??P@F `  c $A4 ?? `  c $A6 ??   `   c $A8 ?? `  `   c $A7  ?? `p  H  0޽h ? 3380___PPT10.as SE{  0 P"(  x  c $Ip0     0 NA '? ?0 ' `  c $A: ??0 `  c $A  ??0 |   H  0޽h ? 33___PPT10i.qR+D='  = @B +k  0 z,(  ,r , S U`   ` , c $A  ??@   H , 0޽h ? 33___PPT10i.q0,+D='  = @B +   0 zr (  r  S P  X  0A?P H  0޽h ? 3380___PPT10.(sP 0 p\(  \R \ s *A@cH \ 0޽h ? 33___PPT10i.Sr1S+D='  = @B +I 0 `X`(  `X ` 0A? X ` 0A? Q`H ` 0޽h ? 33___PPT10i.Vr0p+D='  = @B + 0 P(  X  0A?/ H  0޽h ? 33___PPT10i. s%+D='  = @B + 0 .&p(  X  0A?` &  <@Q pReduced temperature vs. reduced density in the gas-liquid coexistence region, for eight different substances. qqpH  0޽h ? 33___PPT10i. s +D='  = @B +  0 (  r  S `P  H  0޽h ? 33___PPT10i. s|+D='  = @B +  0 0Q(  0r 0 S D`P#   ` 0 c $A .??`G . 0 <| +  GFourier transformation:` 0 c $A /??  / 0 < 6  @And the relation` 0 c $A 0??P @ 0H 0 0޽h ? 33___PPT10i.qց+D='  = @B +k  0 z4(  4r 4 S 4[P S   ` 4 c $A ??+T H 4 0޽h ? 33___PPT10i.q`+D='  = @B + 0 <(  <x < c $pp`    < 0@ 0 +  {% Youjin Deng 09.12.5&(2&H < 0޽h ? 33___PPT10i.q@+D='  = @B +  0 D(  Dr D S l&0   H D 0޽h ? 33___PPT10i.q@3]+D='  = @B +"  0 91 @(  @r @ S @   ` @ c $A ;??P3 ; @ </H@nF LFor each site i, using the mean field approximation, the Hamiltonian becomesMM` @ c $A <??p  < @ <aH   FThe partition sum:` @ c $A =??`  =  @ < @  F Note that  `  @ c $A >??   >  @ <H  @  ;And let`  @ c $A ??? @>  ?H @ 0޽h ? 33___PPT10i.q\+D='  = @B + 0 H(  H` H c $A @??P 0  @H H 0޽h ? 33___PPT10i.qРu+D='  = @B +O  0 f^ L(  Lr L S  `S   D L C Aa$H L 0޽h ? 33___PPT10i.qD+D='  = @B +U  0 ld0P(  Px P c $`S   D P C Ab@0H P 0޽h ? 33___PPT10i.qD+D='  = @B +U  0 ld@T(  Tx T c $( `S   D T C Ac@PTH T 0޽h ? 33___PPT10i.qD+D='  = @B + 0 d(  d d <" B P Behavior close to critical point!!` d c $A _??0@ _ d <PbPx  Q!Magnetization in zero field (h=0)""` d c $A b??P6 bH d 0޽h ? 33___PPT10i.Yr +D='  = @B + 0 l(  l l <8 LWhen T=Tc and h finite` l c $A c??P0 c l <! x LWhen KKc and h finite` p c $A e??` eH p 0޽h ? 33___PPT10i.^r[+D='  = @B + 0 t2(  t t 00 P  2Energy density e and C for h=0` t c $A f??  fH t 0޽h ? 33___PPT10i.arPf!+D='  = @B + 0 @|(  |] | 05z  Note that, Mean field theory does not in general give a correct description of critical behavior, since that nonuniform spin configurations have been excluded, thus, the effect of fluctuations has been ignored. Although, unsatisfactory in many aspects, the mean field theory captures a few important features of critical phenomena, it gives us a qualitative understanding of the major mechanisms behind the critical phenomena.m 5H | 0޽h ? 33___PPT10i.rp+D='  = @B + 0 h(  hx h c $=pp`    h 0 9@ 0 +  {% Youjin Deng 09.12.5&(2&H h 0޽h ? 33___PPT10i.q@+D='  = @B +  0 (  r  S J#   r  S XK@Pc   `  c $A ?? p y  H  0޽h ? 33___PPT10i."s'+D='  = @B +$  0 $(  r  S R    r  S S `  H  0޽h ? 3380___PPT10.#sਢ 0 (  X  0A?0qH  0޽h ? 3380___PPT10.$sP  0 (  r  S @h    r  S i  `  c $A ?? @N  H  0޽h ? 3380___PPT10.%sP  0  (  r  S t    r  S u `  `  c $A" ??@  `  c $A! ??E `  c $A$ ??  p `  c $A ??p@  `   c $A# ??@  `   c $A& ?? p  H  0޽h ? 33___PPT10i.(sPJ+D='  = @B +)  0 @8  (  x  c $|    x  c $T `  `   c $A' ??`@ `   c $A( ?? ` `   c $A% ??`  `   c $A) ??   H  0޽h ? 33___PPT10i.(sPJ+D='  = @B +  0 0 8(  x  c $H   H x  c $H H `  c $A+ ??00  `   c $A, ??( `   c $A- ??0  `   c $A* ??   `   c $A/ ??   H  0޽h ? 33___PPT10i.(sPJ+D='  = @B +  0 P (  x  c $    x  c $،  `   c $A0 ??`  H  0޽h ? 33___PPT10i.(sPJ+D='  = @B +  0 `(  x  c $    X  0A1?@@   0蜍  The first is the Josephson law, and the third is the Fisher law. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   F 0 pF(    0  Y)Let start with the phase diagram of water**  < P5& QPhenomena observed near the critical point are referred to as Critical phenomena.RRX  0A9?`nF H  0޽h ? 3380___PPT10.`sСj[rS%U]11 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   [ 0 [(  X  0A.?   <i7"  W!The phase diagram for ferromagnet H  0޽h ? 3380___PPT10.s0jrE^'wED^1>G(  : <'Equation Equation.DSMT40*MathType 6.0 Equation  Equation Equation.DSMT40*MathType 6.0 Equation.Equation Equation.DSMT40*MathType 6.0 Equation/Equation Equation.DSMT40*MathType 6.0 Equation0Equation Equation.DSMT40*MathType 6.0 EquationEquation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation<Equation Equation.DSMT40*MathType 6.0 Equation=Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation\Equation Equation.DSMT40*MathType 6.0 Equation ]Equation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@``   2%    !$%')*+,./0 23 5 679 : 2$xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+2$6% ;~`,2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!2$͓\8WYC 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: And susceptibility ,h is the external field$ 0py 2Critical exponent (In the language of Ising model)33 & = Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-%&456 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   w 0 w(  X  0A.?   <i ) s!The phase diagram for ferromagnet"" H  0޽h ? 3380___PPT10.s0jr^'\E^1jG(  : >'KvWlX{Z;lx\5b ?8Q9~x8*x&h<ü*|LL3}e5"JqY?SȎ9;|,NLyfA.`0QEAR2^mծ]77unCRLRy$UyU[|IdjŻD#HHAd~8(,e'P_O$|Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation ]Equation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@`` $  2%    !$%')*+,./0 23 5 679 :$ 2$xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+2$6% ;~`,2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!$2$S\0G9QCz"* 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: And susceptibility ,h is the external field$ 0py 2Critical exponent (In the language of Ising model)33 & = Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-%&456 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   {  0 P"(  x  c $'p0     0 NA '? ?0 ' `  c $A; >??`W >`  c $A  ??0 |   H  0޽h ? 33___PPT10i.qR+D=' = @B +xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. NLe/ҧ#htGxƎgt2Fkl.y*=Pzf*"&4v驘1I?`;p(8x(8v~ g.ǚ"X g.sDZ<8>8_]!pgFX7C ] oQl ;/Y,ԃ{g'oc`[0џ@2r.a$IJS_LMfV=m֝Sϫt/(\jh=p$zz,Z;2$@wۙʌ5OM'Equation Equation.DSMT40*MathType 6.0 Equation  Equation Equation.DSMT40*MathType 6.0 Equation.Equation Equation.DSMT40*MathType 6.0 Equation/Equation Equation.DSMT40*MathType 6.0 Equation0Equation Equation.DSMT40*MathType 6.0 EquationEquation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation<Equation Equation.DSMT40*MathType 6.0 Equation=Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@`` $  2%    !$%')*+,./0 23 5 679 :$ $2$2dx͸3< ?d2$Iힿ&Xk2$E(^=s 2${wV2 zQ" 2$lGWO=KdR$(exE}kяI32$g 6Y9Of9|2$d:TλFM2$O) 00pJ902$q1F3sG:y22$R߄E!a 2$]fƦ֠-C 2$P`EB%sX%'J78W2$pƙ.R$>xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+2$6% ;~`,2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!$2$S\0G9QCz"* 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %c,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: $n 0py 2Critical exponent (In the language of Ising model)33 & = Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-%&456 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis     0 *"P(  x  c $'0     0 NA '? ?0 ' `  c $A; >?? >H  0޽h ? 33___PPT10i.qR+D=' = @B +xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. NLe/ҧ#htGxƎgt2Fkl.y*=Pzf*"&4v驘1I?`;p(8x(8v~ g.ǚ"X g.sDZ<8>8_]!pgFX7C ] oQl ;/Y,ԃ{g'oc`[0џ@2r.a$IJS_LMfV=m֝Sϫt/(\jh=p$zz,Z;2$@wۙʌ5OM'Equation Equation.DSMT40*MathType 6.0 Equation  Equation Equation.DSMT40*MathType 6.0 Equationw޳3sD D@W l$ D/4xPˉ D@ ." 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!b$Uvc!"` q^ 2$S\0G9QCz"* 2$ (]B:K[8cz b$"4sA)b6} 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: $n 0py`(a) 2Critical exponent (In the language of Ising model)33 & = Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-%&456 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis    0 (  N  C &A:Ising1`P  <P  LFor magnetic susceptibility `  c $A< ??? t ?H  0޽h ? 3380___PPT10.s@~^ 0 ^(    <;P fFor magnetization m L  C $A= Ising] H  0޽h ? 3380___PPT10.sP xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. NLe/ҧ#htGxƎgt2Fkl.y*=Pzf*"&4v驘1I?`;p(8x(8v~ g.ǚ"X g.sDZ<8>8_]!pgFX7C ] oQl ;/Y,ԃ{g'oc`[0џ@2r.a$IJS_LMfV=m֝Sϫt/(\jh=p$zz,Z;2$@wۙʌ5OMӚVs 6)İCg~Y7#q,HyP oFjq7EXpxUDǜHf+ ;}ͮ@<8+sܞ7'JlbV8^#i4We"B<23rVZ$a$۱ZݛL׍؋d'@|s, uT}`1#6BբUߝ9T_tz^C$wgF3V(C2'd%l 3<0 "l9jiHnK8 uZSg}~,W01τksyc>Oղ~oGoEBBov:ItXǁrPNf+mXC/\+/?*mbrYE%ϿQDF!ù x휬Uvt_wgW|ьBnƓ/uU_ ;~]ˑҰDh Ds"Ad| /(&Cm!%~(5D9(ƥE` "f '\R}LD3wOd[T \]q+@4"{yG9=v8:6sy-leWB~@'߃+V҃ `3ī 􅶩w{ѫߌ2+2";f@秿|>.p0|_qD|T'ڔQ F^*s(ʑo?$n}l rٓli2 ba6$A(TJSY-\ӻ_3 Kӹ,7: C?Aif{L|[RG%lcg51ij1,:a4DXj-hEkdKVUD,hMQbQ1 0s({D(&09|'2YAHι9!l}WݮdiҝS8lS3AgTi6qpc¶ : `!72dx͸3<  SMT40*MathType 6.0 Equation.Equation Equation.DSMT40*MathType 6.0 Equation/Equation Equation.DSMT40*MathType 6.0 Equation0Equation Equation.DSMT40*MathType 6.0 EquationEquation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation<Equation Equation.DSMT40*MathType 6.0 Equation=Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation?_Equation Equation.DSMT40*MathType 6.0 Equation@bEquation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@``   4'    !$%')*+,./0 23 5 679 :<> b$l?L+ Nf 2$2dx͸3< ?d2$Iힿ&Xk2$E(^=s 2${wV2 zQ" 2$lGWO=KdR$(exE}kяI32$g 6Y9Of9|2$d:TλFM2$O) 00pJ902$q1F3sG:y22$R߄E!a 2$]fƦ֠-C 2$P`EB%sX%'J78W2$pƙ.R$>xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+2$6% ;~`,2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!b$Uvc!"` q^ 2$S\0G9QCz"* 2$ (]B:K[8cz b$"4sA)b6} $2$b6~C`Ï@O 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: $n 0py`(a) 2Critical exponent (In the language of Ising model)33 & = Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-%&456 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis    0  (    <; P@ BFor magnetization L  C $A= Ising] `  c $A? @??  @H  0޽h ? 3380___PPT10.sP xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. NLe/ҧ#htGxƎgt2Fkl.y*=Pzf*"&4v驘1I?`;p(8x(8v~ g.ǚ"X g.sDZ<8>8_]!pgFX7C ] oQl ;/Y,ԃ{g'oc`[0џ@2r.a$IJS_LMfV=m֝Sϫt/(\jh=p$zz,Z;2$@wۙʌ5OMӚVs 6)İCg~Y7#q,HyP oFjq7EXpxUDǜHf+ ;}ͮ@<8+sܞ7'JlbV8^#i4We"B<23rVZ$a$۱ZݛL׍؋d'@|s, uT}`1#6BբUߝ9T_tz^C$wgF3V(C2'd%l 3<0 "l9jiHnK8 uZSg}~,W01τksyc>Oղ~oGoEBBov:ItXǁrPNf+mXC/\+/?*mbrYE%ϿQDF!ù x휬Uvt_wgW|ьBnƓ/uU_ ;~]ˑҰDh Ds"Ad| /(&Cm!%~(5D9(ƥE` "f '\R}LD3wOd[T \]q+@4"{yG9=v8:6sy-leWB~@'߃+V҃ 87\udgSn޸ ?)YMEYǓp|D&>a`ra>N9F'uc)=V3Y&itpm:tX:漞Cow݉|'9#wwnu 3Hժ;ɭX??Lph(d/q3+H maہsY@R`mcGj|~/?/emxon-dDb1 l^}*w/a| ޕqS7w^.m>rHW~)Lt'l@Xp |]7pAA5>ͺ? g(8X5,N 4 GE} cw]@pIkVN8qX3':h~a#!A<"F/WLy.^Ƣ/i%xfBl+- 6,^ 0s];ZR&fmV{{,{ Np扵퇘}LV>VW-٢bDx _ Ј\cD&!xɜ|'|^+ƘO&Cw7iyߺܸ},Wlmbi\Nގm59,Tkp Q628DpTpfNt4r$Z\"_a x|)~ZEb1H(   @'Equation Equation.DSMT40*MathType 6.0 Equation  Equation Equation.DSMT40*MathType 6.0 Equation.Equation Equation.DSMT40*MathType 6.0 Equation/Equation Equation.DSMT40*MathType 6.0 Equation0Equation Equation.DSMT40*MathType 6.0 EquationEquation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation<Equation Equation.DSMT40*MathType 6.0 Equation=Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation?_Equation Equation.DSMT40*MathType 6.0 Equation@bEquation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@``   4'    !$%')*+,./0 23 5 679 :<> b$l?L+ Nf 2$2dx͸3< ?d2$Iힿ&Xk2$E(^=s 2${wV2 zQ" 2$lGWO=KdR$(exE}kяI32$g 6Y9Of9|2$d:TλFM2$O) 00pJ902$q1F3sG:y22$R߄E!a 2$]fƦ֠-C 2$P`EB%sX%'J78W2$pƙ.R$>xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+2$6% ;~`,2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!b$Uvc!"` q^ 2$S\0G9QCz"* 2$ (]B:K[8cz b$"4sA)b6} $2$b6~C`Ï@O 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4ddd [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation?_Equation Equation.DSMT40*MathType 6.0 Equationd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: $n 0py`(a) 2Critical exponent (In the language of Ising model)33 & = Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-%&456 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. NLe/ҧ#htGxƎgt2Fkl.y*=Pzf*"&4v驘1I?`;p(8x(8v~ g.ǚ"X g.sDZ<8>8_]!pgFX7C ] oQl ;/Y,ԃ{g'oc`[0џ@2r.a$IJS_LMfV=m֝Sϫt/(\jh=p$zz,Z;2$@wۙʌ5OMӚVs 6)İCg~Y7#q,HyP oFjq7EXpxUDǜHf+ ;}ͮ@<8+sܞ7'JlbV8^#i4We"B<23rVZ$a$۱ZݛL׍؋d'@|s, uT}`1#6BբUߝ9T_tz^C$wgF3V(C2'd%l 3<0 "l9jiHnK8 uZSg}~,W01τksyc>Oղ~oGoEBBov:ItXǁrPNf+mXC/\+/?*mbrYE%ϿQDF!ù x휬Uvt_wgW|ьBnƓ/uU_ ;~]ˑҰDh Ds"Ad| /(&Cm!%~(5D9(ƥE` "f '\R}LD3wOd[T \]q+@4"{yG9=v8:6sy-leWB~@'߃+V҃ 87\udgSn޸ ?)YMEYǓp|D&>a`ra>N9F'uc)=V3Y&itpm:tX:漞Cow݉|'9#wwnu 3Hժ;ɭX??Lph(d/q3+H maہsY@R`mcGj|~/?/emxon-dDb1 l^}*w/a| ޕqS7w^.m>rHW~)Lt'l@Xp |]7pAA5>ͺ? g(8X5,N 4 GE} cw]@pIkVN8qX3':h~a#!A<"F/WLy.^Ƣ/i%xfBl+- 6,^ 0s];ZR&fmV{{,{ Np扵퇘}LV>VW-٢bDx _ Ј\cD&!xɜ|'|^+ƘO&Cw7iyߺܸ},Wlmbi\Nގm59,Tkp Q628DpTpfNt4r \k_Xbq b1H(   @'Equation Equation.DSMT40*MathType 6.0 Equation  Equation Equation.DSMT40*MathType 6.0 Equation.Equation Equation.DSMT40*MathType 6.0 Equation/Equation Equation.DSMT40*MathType 6.0 Equation0Equation Equation.DSMT40*MathType 6.0 EquationEquation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation<Equation Equation.DSMT40*MathType 6.0 Equation=Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation?_Equation Equation.DSMT40*MathType 6.0 Equation@bEquation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@``   7'    !$)*+,.0 23 5 679 :<>@AB b$l?L+ Nf 2$2dx͸3< ?d2$Iힿ&Xk2$E(^=s 2${wV2 zQ" 2$lGWO=KdR$(exE}kяI32$g 6Y9Of9|2$d:TλFM2$O) 00pJ902$q1F3sG:y22$R߄E!a 2$]fƦ֠-C 2$P`EB%sX%'J78W2$pƙ.R$>xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+2$6% ;~`,2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!b$Uvc!"` q^ 2$S\0G9QCz"* 2$ (]B:K[8cz b$"4sA)b6} $2$b6~C`Ï@O 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: $n 0py`(a) 2Critical exponent (In the language of Ising model)33 & %&6 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( 8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   c* Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-d+e,  0  (  r  S ! `  `  c $A2 ?? `  c $A3 ??`P `  `  c $A5 ??P@F `  c $A4 ?? `  c $A6 ??   `   c $A8 ?? `  `   c $A7  ?? `p  H  0޽h ? 33___PPT10i.as SE+D=' = @B +  0 x(  x  c $4P  X  0A?P H  0޽h ? 3380___PPT10.(sP 0 (  X  0A?/ H  0޽h ? 33___PPT10i. s%+D=' = @B + 0 .& (  X  0A?` &  <tG p! pReduced temperature vs. reduced density in the gas-liquid coexistence region, for eight different substances. qqpH  0޽h ? 33___PPT10i. s +D=' = @B +xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. NLe/ҧ#htGxƎgt2Fkl.y*=Pzf*"&4v驘1I?`;p(8x(8v~ g.ǚ"X g.sDZ<8>8_]!pgFX7C ] oQl ;/Y,ԃ{g'oc`[0џ@2r.a$IJS_LMfV=m֝Sϫt/(\jh=p$zz,Z;2$@wۙʌ5OMӚVs 6)İCg~Y7#q,HyP oFjq7EXpxUDǜHf+ ;}ͮ@<8+sܞ7'JlbV8^#i4We"B<23rVZ$a$۱ZݛL׍؋d'@|s, uT}`1#6BբUߝ9T_tz^C$wgF3V(C2'd%l 3<0 "l9jiHnK8 uZSg}~,W01τksyc>Oղ~oGoEBBov:ItXǁrPNf+mXC/\+/?*mbrYE%ϿQDF!ù x휬Uvt_wgW|ьBnƓ/uU_ ;~]ˑҰDh Ds"Ad| /(&Cm!%~(5D9(ƥE` "f '\R}LD3wOd[T \]q+@4"{yG9=v8:6sy-leWB~@'߃+V҃ 87\udgSn޸ ?)YMEYǓp|D&>a`ra>N9F'uc)=V3Y&itpm:tX:漞Cow݉|'9#wwnu 3Hժ;ɭX??Lph(d/q3+H maہsY@R`mcGj|~/?/emxon-dDb1 l^}*w/a| ޕqS7w^.m>rHW~)Lt'l@Xp |]7pAA5>ͺ? g(8X5,N 4 GE} cw]@pIkVN8qX3':h~a#!A<"F/WLy.^Ƣ/i%xfBl+- 6,^ 0s];ZR&fmV{{,{ Np扵퇘}LV>VW-٢bDx _ Ј\cD&!xɜ|'|^+ƘO&Cw7iyߺܸ},Wlmbi\Nގm59,Tkp Q628DpTpfNt4r4 TT\`_eb@-jY[]& ne1H(   @'Equation Equation.DSMT40*MathType 6.0 Equation  Equation Equation.DSMT40*MathType 6.0 Equation.Equation Equation.DSMT40*MathType 6.0 Equation/Equation Equation.DSMT40*MathType 6.0 Equation0Equation Equation.DSMT40*MathType 6.0 EquationEquation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation<Equation Equation.DSMT40*MathType 6.0 Equation=Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation?Equation Equation.DSMT40*MathType 6.0 Equation@ Equation Equation.DSMT40*MathType 6.0 Equation_)Equation Equation.DSMT40*MathType 6.0 Equationb*Equation Equation.DSMT40*MathType 6.0 Equationc+Equation Equation.DSMT40*MathType 6.0 Equationd-Equation Equation.DSMT40*MathType 6.0 Equatione.Equation Equation.DSMT40*MathType 6.0 Equationf0Equation Equation.DSMT40*MathType 6.0 Equation7Equation Equation.DSMT40*MathType 6.0 Equation;Equation Equation.DSMT40*MathType 6.0 Equation>Equation Equation.DSMT40*MathType 6.0 Equation@Equation Equation.DSMT40*MathType 6.0 EquationAEquation Equation.DSMT40*MathType 6.0 EquationBEquation Equation.DSMT40*MathType 6.0 EquationCEquation Equation.DSMT40*MathType 6.0 EquationDEquation Equation.DSMT40*MathType 6.0 EquationEEquation Equation.DSMT40*MathType 6.0 EquationGEquation Equation.DSMT40*MathType 6.0 EquationHEquation Equation.DSMT40*MathType 6.0 EquationIEquation Equation.DSMT40*MathType 6.0 EquationJEquation Equation.DSMT40*MathType 6.0 EquationKEquation Equation.DSMT40*MathType 6.0 EquationLEquation Equation.DSMT40*MathType 6.0 EquationNEquation Equation.DSMT40*MathType 6.0 EquationOEquation Equation.DSMT40*MathType 6.0 EquationQEquation Equation.DSMT40*MathType 6.0 EquationUEquation Equation.DSMT40*MathType 6.0 EquationVEquation Equation.DSMT40*MathType 6.0 EquationWEquation Equation.DSMT40*MathType 6.0 EquationXEquation Equation.DSMT40*MathType 6.0 EquationYEquation Equation.DSMT40*MathType 6.0 EquationZEquation Equation.DSMT40*MathType 6.0 Equation [Equation Equation.DSMT40*MathType 6.0 Equation>\Equation Equation.DSMT40*MathType 6.0 Equation?_Equation Equation.DSMT40*MathType 6.0 Equation@bEquation Equation.DSMT40*MathType 6.0 Equation`/ 0DArialЃ 0Ѓ0hhu 0D[SOalЃ 0Ѓ0hhu 0@ .  @n?" dd@  @@``   8&    !$)*+,.0 23 5 679 :<>@C b$l?L+ Nf 2$2dx͸3< ?d2$Iힿ&Xk2$E(^=s 2${wV2 zQ" 2$lGWO=KdR$(exE}kяI32$g 6Y9Of9|2$d:TλFM2$O) 00pJ902$q1F3sG:y22$R߄E!a 2$]fƦ֠-C 2$P`EB%sX%'J78W2$pƙ.R$>xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+$2$ :]f2"iU'2$¹87x82$ge,eJ'e2$f o-{ 2$xN嵗btS U /2$*A_8,'PK'2$OI[}c5i9 R2$TR{ߦ)R>2$L: c%&92Z2$51JTֹ̫9J2$/yOPo>Z82$&S,itht2$ɕ="oI%I,2$g7LBc2|u2$<-"Ňq xVE2$K r; &}r`2$_mVH HG2$<{`.+ Z\2$6p~v=4go~d( 2$@h޻SԪ˙4>2$1o̘׶>*2$j~O&wJ6lh2$Mu7ȪrVr 2$-/a } y)2$d'R_6Ȧ. 2$&Wk§1? 2!72$Do&q-~~t2$_6n*ʌR<;2$ġ!jJPkF2$cKq4sĸX\^@!b$Uvc!"` q^ 2$S\0G9QCz"* 2$ (]B:K[8cz b$"4sA)b6} $2$b6~C`Ï@O 0AA@wj[ ʚ;&4r8ʚ;g4LdLd 02ppp@  <4dddd4x 0h<4!d!d4x 0h80___PPT10 ?  %,Chap.3 Introduction to Critical Phenomena--( S%T& Let denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ^' For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: $n 0py`(a) 2Critical exponent (In the language of Ising model)33 & %&6 {The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( f+8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   c* Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?- 0 0(  X  0A?    <G 5  This table summarizes the experimental values of the critical exponents as well as the results from some theoretical models. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( f+8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   c* Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|Correlation function$  Then we have   (Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( f+8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   c* Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|(Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( f+8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization groupZ:<Scaling hypothesis The scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ? Dimensional Analysis  One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.F!Dimensional Analysis  }Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatM"Dimensional Analysis  FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations (]4GP#Dimensional Analysis  We getR$Dimensional Analysis   c* Only two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. *?-xYoE1y5i(k$8T8 H81Q%n HH/)HP.@%詇FNp!H@-OM|S4 iL aH!N\)K{]s m!W4{ןm 56 L@"N$^9c0L p C Wz+=©r䓆N]BI8z83|5 D F7s\HGuƇ|_wwGwq22 :܅U9R>|Gv='<BAf> ڰcً^X?TikIr tOԎ.C];s=7/-4wǦ_Ν۽]vxr&6HT$(pC*"fp0d\t #bOŏo0uPDb|y. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance,  ?=|(Chap.4 Mean-field theory of Ising model))(   _The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ``/Recall the Hamiltonian of the Ising model to be00   ! Graphical solution when K<1 " Graphical solution when K>1 # Graphical solution when K=Kc=1 (,/12' Chap.5 Universality and Scaling!!( f+8Universality class >As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as?9Explanation of universality  In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. 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" # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z | } ~   {  \      !"#$%&'()*+,-./0[23456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKMe lPQRSTUVWXYZ[\]^_`abcmqstuvwxyz{|}~ !"#$%&'()*+-OLOing õƬ 28Universality classExplanation of universality õƬ 31Scaling hypothesis  õʾĸģǶ OLE  õƬ D _C **yanmin#4<Version<Version_yanminyanmin2${wV2 zQ" 2$lGWO=KdR$(exE}kяI32$g 6Y9Of9|2$d:TλFM2$O) 00pJ902$q1F3sG:y22$R߄E!a  2$]fƦ֠-C  2$P`EB%sX%'J78W2$pƙ.R$>xT¾('fE|R$ʢdur6}iOM v2$)/'/#Bm1W2$K \{jL>2$t1O/,T2$u%](Et'2$y4 饞7*2$Ncsy)f~2$@GH+UPSn^2$(L_I`Rנz+9M2$Gg[Fش+ $2$ :]f2"iU'2$¹87x82$ge,eJ'e2$u5vQRelation between physical observables and correlation function%&6 {The significance of the critical exponents lie in their universality. 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