Part 7 Mean Field Theory

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Perimeter Institute Statistical Physics Lecture Notes part 7. Mean Field Theory Version 1,7 9/11/09 LeoKadanof

Part VII: More is the SameIssues

Abstract phases of matter

phase diagram water

basic physics in phase transitions Gibbs definition; Ehrenfest classification

Magnetic Phase Diagrama qualitative change in Behaviorfrom Gibbs to singularities

Mean Field Theory

more is the same for Ising model Mean Field Theory is Only Partially Right

Calculate resultssimplified phase diagram

Graph of Order Parameterbehavior in neighborhood of critical point

go after magnetizationgo after energygo after correlations

critical opalescencebasic equationcorrelation functionsusceptability

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More is the SamePhase Transitions and Mean Field Theories

Leo P. Kadanoffemail:[email protected]

Abstract

This talk summarizes concepts derived from the study of phasetransitions mostly within condensed matter physics. In its

original form, the talk was aimed equally at condensed matterphysicists and philosophers of science. The latter group areparticularly interested in the logical structure of science. Thistalk bears some traces of its history. The key technical ideasgo under the names of ``singularity'', `order parameter'',

``mean field theory'', ``variational method'', and ``correlationlength''. The key ideas here go under the names of mean fieldtheory, phase transitions, universality, variationalmethod, and scaling.

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mailto:[email protected]:[email protected]


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IssuesMatter exists in differentphases, different states of

matter with qualitativelydifferent properties: Thesephases are interesting inmodern physics andprovocative to modern

philosophy. For example, nophase transition can everoccur in a finite system.Thus, in some sense phasetransitions are not products

of the finite world but of thehuman imagination.

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Phase Diagram for Water

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liquid-gas phasetransition line


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Phase Transitions Describe Basic PhysicsIdeas derived from phase transitions and other intellectual products ofcondensed matter physics are crucial for many branches of modernscience and for modern theoretical physics. The basic ideas include:

a. Phase transitions always require an infinite number of degrees offreedom. This infinity may come either from something that happensover an infinite period of time, over an infinite amount of space, or via

the development of some sort of infinite complexity within the system.b. The structure of space and time are important determinants of whatwe see happen. The dimensions of space matter as do whether thesystem in question repeats itself infinitely often. The topology of thesurrounding space and of created structure are quite important.

c. Condensed matter systems provide a good area for study becausethey provide an observable platform for amazing diverse and richphenomena, well beyond the untutored imagination of scientists.

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7/40Perimeter Institute Statistical Physics Lecture Notes part 7. Mean Field Theory Version 1,7 9/11/09 LeoKadanof

Gibbs: A phase transition is a singularity

in thermodynamic behavior.This occurs only in an infinite system

Ehrenfest:

First order= discontinuous jumpin thermodynamic quantities.

Second order has continuousthermodynamic quantities, butinfinity in derivative of

thermodynamic quantities.

.......

P. Ehrenfest 7



Magnetic Phase Diagram

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A Phase Transition is a change from one behavior to another

A first order phase transition involves adiscontinuous jump in some statistical variable.

The discontinuous property is called the orderparameter. Each phase transition has its ownorder parameter. The possible orderparameters range over a tremendous variety ofphysical properties. These properties include

the density of a liquid-gas transition, themagnetization in a ferromagnet, the size of aconnected cluster in a percolation transition,and a condensate wave function in a superfluidor superconductor. A continuous transition

occurs when the discontinuity in the jumpapproaches zero. This section is about thedevelopment of mean field theory as a basis fora partial understanding of phase transitionphenomena.

http://azahar.files.wordpress.com/2008/12/

http://blogs.trb.com/news/local/longisland/politics/blog/2008/04/

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http://blogs.trb.com/news/local/longisland/politics/blog/2008/04/http://blogs.trb.com/news/local/longisland/politics/blog/2008/04/http://blogs.trb.com/news/local/longisland/politics/blog/2008/04/http://blogs.trb.com/news/local/longisland/politics/blog/2008/04/http://azahar.files.wordpress.com/2008/12/snowflake_.jpghttp://azahar.files.wordpress.com/2008/12/snowflake_.jpg



J. Willard Gibbs:

gets to: phase transitions are a property of infinite systemsproof: .... consider Ising model for example

problem

defined by

H/(kT) = K

nn

rs + h

r

r

free energydefined by

F/(kT) = ln

{r=1}

expH{r}/(kT)

H is a smooth function of K and h.Since a finite sum of exponentials of smooth functionsis a positive smooth function, it follows thatthe free energy is smooth too.

theory starts from: phase transitions are singularities in free energy

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Ising model, spin, simplifiedatomone spin in a magnetic fieldstatistical average:

Mean Field Theory: more is the same

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spin in a magnetic field, dimension d

focus on one spin, at r: that spin feels h and Ks nn to rs

= 1

< >= tanh(h)

H= B= kTh

one spin

many spins

H/ kT= K r

nn

s+ h rr


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more is the same

statistical average: = tanh h

one spin

many spins

focus on one spin

statistical average:

Heff

/ (kT) = r[h

r+K ]

heff

= [h+ Kz< >]

< >= tanh(heff

)z=number of nn

or, if there is space variation, heff= hr+Ks nn to r

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Mean Field Theory is Only Partially Right

As we shall discuss in detail, mean field theory gives an interesting

and instructive theory of phase transitions, but one which is onlypartially right. Near the critical point, for lower dimensionalsystems, including three dimensions, fluctuations dominate thesystem behavior and mean field theory gives the wrong answer,badly wrong. Very near first order phase transitions, fluctuations

also count, but in a less obvious manner.However in high dimensions, usually above four, mean field theorygives a good picture of phase transitions. It also has features whichpoint the way toward the right theory.

It is also simple to use

Mean field theory says that spin moves in the average field produced byall other spins. But actual value is often larger in magnitude than mean

value and fluctuates in sign. Net result is error, with unknown sign. Thesame ideas can be applied to lots of problems. (In particle physics meanfield theory often goes with the words one loop approximation.)

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Calculate results of mean field theory

Numerical method: I calculate at high temperatures, smallK, by using Newtons method starting from =h. I thenincrease K step by step at fixed h, and find in each step byusing the last step as the starting point for Newtons method.

I then calculate -F using the fact that h(-F) = and theknown value of the free energy at high temperatures.

One of the useful results that emerges from this numericalcalculation is the value of the coupling Kwhich produces the firstsplitting in the two h=0 magnetization curves. This bifurcation

occurs at Kz=1. Consequently we identify this value of K as thecritical one. Since K is a physical coupling strength divided bytemperature, we can write Kx in terms of the temperature andthe critical temperature as

Kz=Tc/T.

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simplified phase diagram for ferromagnet

The basic variables defining the state of the system

are the magnetic field and the temperature. Wedescribe what is happening by looking at themagnetization. The magnitude of the magnetizationmeasures the extent to which the spins in thesystem are lined up with each other. Its signdescribes the direction of the alignment. If thetemperature is sufficiently low, the system has anon-zero magnetization even at zero magneticfield. At these lower temperatures, the zero-fieldmagnetization has two possible values, for the twopossible directions in which the spins may alignthemselves.

The heavy line is the locus of points at which this spontaneous magnetization is non-zero. As one crosses this line, there is a discontinuous jump in the magnetization,which maintains its magnitude but reverses its direction. This jump is a first orderphase transition. Typically, this jump decreases in size as the temperature gets higheruntil, at some critical point, the jump goes continuously to zero. This point is thenthe position of a continuous phase transition.

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order parameter in meanfield transition

First order phase transition critical point

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

H=0

H=0.2H=0.5

H=0-

H=-0.2

H=-0.5magnetiza

tion

Tc/T

In legend H should be h

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Behavior in neighborhood of critical Point

The mean field equation is = tanh heff with heff= hr+Ks nn to r.

We assume that we are near the critical point both the field and themagnetization are small. Then we can expand the equation for in apower series in heff = tanh heffheff-heff3/3We also assume slow variation in space and that the temperature is close tocriticality. To lowest order heffKz . The lowest order result

is used in the cubic term and the rest is evaluated exactly as=hr+ [1-t]+Ks nn to r[- ] -3/3Here, t is the temperature deviation from criticality t=1-Tc/T. Finally replace

the remaining K by its critical value, 1/z and expand to second order in thedifference s-r to obtain

0=hr -t + a22 /z -3/3This is the near-critical mean field theory in the Weiss model of aferromagnet. It is the continuum version. On the lattice

a22 ------> s nn to r[- ]17


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We assume neighboring spins are uncorrelated

-/(kT) = Kz + other effects

nearTc 2

= 0 above Tc 2

=-tz belowHence specific heat, d /dT has a jump at Tc. It looks like

Conclusions from Weiss Mean Field TheoryFor example, go after when h=0, no

variation in space

H/ kT= K r

nn

s+ h rr

TTc

dE/dT

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Critical Opalescence and Fluctuations

In the early 20th century experimenters observed what is called

critical opalescence. As the critical point is approached, a fluidwhich is otherwise clear and transparent is seen to becomemilky and reflects light. This phenomenon was explained bySmoluchowski (1908) and Einstein (1910)

Einstein particularly showed that the scattering came from

density fluctuations, and noted that these fluctuations diverged atthe critical point. Soon thereafter, in 1916 and 1914, Ornsteinand Zernike put together a theory which show that thefluctuations came from large regions of correlated fluctuationsand derived a qualitatively correct mean field theory treatment

of the phenomenon.This development would prove to be particularly importantbecause spatial and correlational structure would turn out to bethe key to understanding phase transitions.

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Weiss Mean Field Theory for CorrelationsFor example, go after , when h=0, no variation in space, T>Tc

0=hr-t + s nn to r[- ]/z -3/3neglect cubic term, differentiate equation with respect to hu

0=(d hr /d hu )-t (d/d hu ) + s nn to r[d/d hu- d/d hu]/z

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d/d hu is correlation function g(r-u)= ,which then obeys

0=r,u-tg(r-u) + s nn to r[g(s-u)-g(r-u)]/z

define G(q)=exp[-iq.(r-u)] g(r-u) so thatr

0=1 +[-t + 2i=1,2...d (cos qi-1)/z ]G(q)


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Fourier transform of correlation function obeys

0=1 +{-t +2 i=1,2...d [cos(aqi)-1]/z }G(q)

We can now solve for G(q) = 1/{t+2i=1,2...d [1 -cos (aqi)]/z}and invert the Fourier transform to get

g(r-s)=dq exp[iq.(r-s)]/(2)d 1/{t+2i=1,2...d [1 -cos (aqi)]/z}

Here the integrals over each component of q extend from -/ato /a. To get a result for large spatial separation expand tosecond order in q and extend the integrations to cover all q

g(r-s)=dq {exp[iq.(r-s)]/(2)d } 1/[t+a2q2/z]

One can perform the integral to find that

g(r-s)~ exp[-|r-s|/]/ |r-s|d-2

when the exponent is of order one, and d >2. The correlation

length,, is given by =a/(zt)1/2

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The result is then of the form

g(r-s)~ exp[-|r-s|/]/ |r-s|d-2

The correlation length,, is given by =a/(zt)1/2

Correlation function looks like

This result is important. It says that the correlations in spaceextend over longer distances as one gets closer to criticality.

So the fact that the phase transition will only occur in an infinitesystem partially shows up in mean field theory. The correlationextending to infinity is a crucial fact in phase transitions. However,mean field theory allows a phase transition in a periodic but finite

system.Also, mean field theory gets an answer which is somewhat incorrectin the neighborhood of the critical point. Near the critical point

~a/(t) but =1 in d=2 and 0.64 in three dimensions.

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Conclusions from Weiss Mean Field Theory:The magnetic susceptibility

The quantity / h, calculated at fixed T, is called the magneticsusceptibility. It is given by the Fourier transform of the spincorrelation function as / h=G(0). Thus. we know its value inmean field theory, which is

/ h ~ 1/t.

So the infinite correlation length has as its direct consequence

the infinity in this important thermodynamic derivative.

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Mean Field Theory of FluidsI did not describe the development of mean field theory inanything like historical order. Van der Waals derived the firstmean field theory in the 1870s. He was interested inunderstanding the data of Andrews (1869) which showed a phasetransition line separating a liquid and a vapor phase that thenends in a critical point in which the two become identically thesame. Pierre Curie noticed that the behavior of a ferromagnethas a close analogy to that of a fluid in 1895. Later in 1907,Pierre Weiss worked this remark into a theory like the onementioned here. Fluids no doubt came first because they weremore familiar and because experiments existed.

T. Andrews, `On the continuity of the gaseous and liquid states of matter,'' Phil. Trans. Roy. Soc.

159, 575-590(1869). Reprinted in: T. Andrews,,The Scientific Papers, Macmillan, London (1889).

J. D. van der Waals, thesis Leiden, 1873.

P. Curie, Ann. Chem. Phys. 5, 289 (1895). P. Weiss, J. Phys. 6, 661 (1907).

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van der Waals derivation

The starting point was the perfect gas lawP = N T perfect gas law,

which of course shows no phase transition. Van der Waals thenintroduced two corrections to describe what he had inferred aboutfluids and the atoms or molecules which formed them. First, he argued

that the molecules could not approach each other too closely becauseof an inferred short-ranged repulsive interaction among the molecules.He probably based his understanding of this repulsion upon the fact thatit is very hard to compress liquids like water. This repulsive effect shouldreduce the volume available to the molecules by an amount proportional

to the number of molecules in the system. Thus, in the perfect gas lawshould be replaced by the available or effective volume,-Nb, where bwould be the excluded volume around each molecule.

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van der waals second effect

The second effect is more subtle. The pressure, P is a force per unit areaproduced by the molecules hitting the walls of the container. However, van

der Waals inferred that there was an attractive interaction pulling eachmolecule towards its neighbors. This attraction is the fundamental reasonwhy a drop of liquid can hold together and form an almost spherical shape.Again we see that familiar facts can be translated into theoreticalunderstanding. As the molecules move toward the walls they are pulledback by the molecules they have left behind them, and their velocity is

reduced. Because of this reduced velocity, their impacts impart lessmomentum to the walls The equation of state contains the pressure asmeasured at the wall, P. This pressure is the one produced by molecularmotion inside the liquid, NT/(-Nb), minus the correction term comingfrom the interaction between the molecules near the walls. Thatcorrection term is proportional to the density of molecules squared. Insymbols the correction is a(N/)2 where a is proportional to the strengthof the interaction between molecules. Van der Waals' correctedexpression for the pressure is thus

P= NkT/(-Nb)-a(N/)2

Here, a and b are parameters that are different for different fluids.27


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Many Different Phase Transitions:

liquid -gas

paramagnetic to ferromagnetic

(un)mixing of solids and liquids

superconducting, .....

van der Waals:(1873) Different simpleliquids-gas transitions have very similar

thermodynamic properties. Derives

mean field theory of liquid.

Curie-Weiss(1907) mean field theoryof magnets.

But, each different phase transition

calls for its own theory.

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Aft d W l


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After van derWaals

Many Different Mean field theoriesThe theory of phase transitions involving the unmixing of fluids was developedby van der Waals himself, while such unmixing in solids was described byW. L.

Bragg and E. J. Williams. Literally dozens of such theories were defined,culminating in the theory of superconductivity ofBardeen, Cooper andSchrieffer. These theories are all different in that they have different physicalquantities playing the roles we have given to the magnetic field, or T-Tc, orserving as the order parameters. The order parameter is the quantity that canundergo a discontinuous jump in the first order transition, and describes the

symmetry group of the physical situation. The magnetization is the orderparameter of the ferromagnet, the density is the order parameter in the liquid-gas transition. Much effort and ingenuity has gone into the discovery anddescription of the order parameter in other phase transitions. In the anti-ferromagnetic transition the order parameter is a magnetization that points inopposite directions upon alternating lattice sites. In ferroelectrics, it is the

electric field within the material. The superfluid and superconducting transitionhave as their order parameter the quantum wave function for a macroscopicallyoccupied state. Liquid crystals have order parameters reflecting possibledifferent kinds of orientation of the molecules within a liquid. The descriptionof these different manifestations of the phase transition concept reflect morethan a century of work in condensed matter physics, physical chemistry, etc.

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Perimeter Institute Statistical Physics Lecture Notes part 7. Mean Field Theory Version 1,7 9/11/09 LeoKadanof 30

Order Parameter, generalized

Landau(~1937)suggested that

phase transitions weremanifestations of a broken

symmetry, and used the order

parameter to measure the extent of

breaking of the symmetry.

in ferromagnet, parameter =

magnetization

in fluid, parameter = density

L.D. Landau


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Generalized Mean Field Schemes IMany different mean-field schemes developed:. Each onehas an order parameter, an average of a microscopic

quantity. Landau generalized this by assuming anexpansion of the free energy in an order parameter,

expansion assumes a small orderparameter (works near critical point) and

small fluctuations (works far away?!)

h is magnetic field

t is proportional to (T-Tc)minimize F in M: result General Solution M(h, (T-Tc))singularity as t,h both go through zero!singularity as h goes through zerofor T< Tc

F= dr[a-hM+tM2+cM4+(M)2

note: nocubic term.This freeenergy

applies to

symmetry ofIsing model

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Why minimize?

Thermodynamics says that the free energy is extremized by the

variation in any macroscopic parameter, e.g. any extensivevariable. This is part of a general idea that the free energy is aprobability, which arises from how the partition function is used.In statistical mechanics only the most probable things happen.This applies to all macroscopic phenomena, Now M(r) is not

quite macroscopic, but Landaus idea was that at the long wave-length part of it it was macroscopic enough so that one couldneglect its fluctuations. This point of view has turned out to bewildly successful.

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Vary M(r) to vary F

F= dr[a-hM+tM2+cM4+(M)2

F= dr M(r) [-h+2tM(r)+4cM(r)3- 2 M(r)]

integrate by parts in gradient term

to minimize F, coefficient ofM(r) must vanish

0= h+2tM(r)+4cM(r)3-2M(r)

Hence we have an equation for M! That is a generalequation for the order parameter in mean field theorywith the Z2 symmetry, i.e. symmetry under sign changeof M.

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For example, go after magnetization whenh=0, no variation in space

t is proportional to (T-Tc)

if t is positive there is but one real solution M=0.

if t is negative we have a possible solution M=0, and also thesolutions

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0=- h+2tM(r)+4cM(r)3-2M(r)0=2tM+4cM3

M= t

2c

One should choose the solution which actually minimizes the free energy. Itturns out that this solution is the one which has the same sign as h.Therefore as h passes through zero for T less than Tc there is a jump in themagnetization proportional to the square root of t.


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Generalized Mean Field Schemes IICrucial part of the solution: for T< Tc , jump in orderparameter goes as with =1/2

This square root (=1/2) appears

to be a Universal result.Mean Field Theory predicts allnear-critical behavior

order parameter in meanfield transition

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

H=0

H=0.2

H=0.5

H=0-

H=-0.2

H=-0.5magnetization

Tc/Torder parameter, jump,and free energy were

crucial concepts

M~ (TcT)


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order parameter and free energy were crucialconcepts

free energy could be expressed in terms on any descriptors ofsystems behavior. It is a minimized by the correct value of anyone of them, We have thus come loose from the particularthermodynamic variables handed to us by our forefathers,

order parameter could be anything which might jump in thetransition.

other variables could be anything at all.

In the meantime Schwinger was working on electromagneticfields for World War II radar. He use variational methods andeffective fields (lumped variables) to build electromagneticcircuits.

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A ?


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A worry?

Mean field theory

givesand = 1/2This power is,however, wrong.

Experiments arecloser to

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M~ (TcT)

M (Tc T)1/3 in 3-D

order parameter: density versusTemperature in liquid gas phasetransition. AfterE. A. GuggenheimJ.Chem. Phys. 13 253 (1945)

1880-1960: No oneworries much aboutdiscrepancies


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vapor liquid


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