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Fenmenos crticosExponentes crticosParmetro de ordenTeora de LandauLeyes de escala

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Los fenmenos crticos

Desde 1905, las tcnicas de medidapermitieron detectar saltos abruptos en el

calor especfico de ciertos cuerpos. Por ejemplo, el helio.

Ese mismo comportamiento apareca enotros coeficientes termodinmicos.

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Diversas denominaciones

Las primeras fueron transformaciones, porque se creyeron transicionesde fase.

As, la transformacin lambda se le dio por la forma de la curva.

Despus, Ehrenfest crey que su origen era una discontinuidad en lasegunda derivada del potencial entalpa libre, de ah el nombre detransicin de segundo orden.

La denominacin actual es la de fenmenos crticos.

Este nombre deriva de las similitudes que se han encontrado entre elpunto crtico de los gases y ciertos puntos caractersticos de los lquidosy los slidos, como el paso de helio normal a superfluido y el punto deCurie de los materiales ferromagnticos.

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Universalidad del comportamiento

Los resultados experimentales ha inducido a creer que estecomportamiento es universal.

El fenmeno crtico es una caracterstica general de la naturaleza, que se

refleja con diversos parmetros en distintos cuerpos.

Los sistemas mejor conocidos son los expansivos y los magnticos,aunque hay muchos otros ejemplos.

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Caractersticas comunes

La temperatura crtica, , es aquella en la que se produce el mximodel coeficiente.

Los fenmenos crticos se producen en un intervalo de temperatura

pequeo, T < 5 K. La desviacin relativa de temperatura es:

cT

( )

c

c

T

TTt

=

0, > tTT

c

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Parmetro de orden

Con el fin de describir lo dicho antes, se introduce el parmetro de

orden, , con las siguientes propiedades: Es una caracterstica interna del sistema que no puede imponerse desde

el exterior.

Posee valor en la fase ordenada y se anula en la desordenada.

Debe definirse en cada problema.

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Ejemplo de parmetro de orden

Alrededor del punto crtico de una gas hay dos fases que llegan a confundirse.

El parmetro de orden puede ser la diferencia de densidades de las fases:

vaporlquido =

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Ejemplo de parmetro de orden

Un material ferromagntico pueden tener dos imanaciones remanentes opuestas.

Tras el punto de Curie, se hace paramagntico. El parmetro de orden es:

M=

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Exponentes crticos

Alrededor del punto crtico las propiedades tienden a depender de texponencialmente. Por ejemplo, el parmetro de orden:

( ) ( ) ...}1{ ++= x

tCtB

( )tlm t

=

ln

ln0

donde xtiende a cero cuando tlo hace, y se conoce comoexponente crtico:

Obsrvese que slo existe para t < 0.

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PP

pTGT

TSTc

=

=

2

2

TT

TP

G

VP

V

V

=

=

2

211

TPP PT

G

VT

V

V,

211

=

=

Las discontinuidades en las derivadas primeras de Gimplican divergencias en las derivadas segundas

Divergencias del calor especfico enla transicin:

CVP TTC ,

CT TT

y de la compresibilidad:

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Diagrama P-V Diagrama P-

La forma de la isoterma crtica cerca del punto crtico es:

0)( >

CCC signoPP

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La forma de la curva de coexistencia en el plano-T cerca del punto crtico, para T

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1,3,2/1,0 ====

Para el gas de van der Waals:

TRbvv

aP =

+ )(

2

2~3

1~3

~8~vv

tp

=

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Para un sistema ferromagntico en presencia de campo magntico:

Usando el modelo de Ising:

Diagrama de fase H-TEs una transicin orden-desorden

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En 1937, Lev Landau propuso una teora en la que el potencial

termodinmico del sistema se haca funcin continua del parmetro de

orden.Para un sistema expansivo se cumple:

( ),,TpGG =

0,

=

Tp

G

Teora de Landau

y en el equilibrio:

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Teora de Landau

En el entorno del punto crtico, Landau acept como vlido el desarrollo:

( ) ( ) = GG

( ) ( ) ...,,,432

+++++= CBApTGpTG o

( ) ( ) ...,,, 42 +++= CApTGpTG o

donde los coeficientes , , ... son funciones de la temperatura y de lapresin. Como la entalpa libre es un mnimo debe ser una funcin par de

, es decir:

por lo que:

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Teora de Landau

Aceptando la cuarta potencia como una aproximacin suficiente, en el

equilibrio:

0=

042 3

,

=+=

CAG

pT

cTT represents the statistical average by which one obtains thermodynamic functions.

A useful quantity that can now be defined is the correlation function

It measures how the value of the order parameter at one point is correlated to its value at

some other point. If decreases very fast with distance, then far away points are relativelyuncorrelated and the system is dominated by its microscopic structure and short-ranged

forces. On the other hand, a slow decrease of would imply that faraway points have a largedegree of correlation or influence on each other. The system thus becomes organised at a

macroscopic level with the possibility of new structure beyond the obvious one dictated by theshort-ranged microscopic forces. As we shall see below, this possibility does actually occur.

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Usually, near the critical point, the correlation function can be written in the form:

Where is the correlation length.

The correlation length is a measure of the range over which fluctuations in one region of

space are correlated with (influence) those in another region. Two points which areseparated by a distance larger than the correlation length will each have fluctuationswhich are relatively independent, that is, uncorrelated. Experimentally, the correlationlength is found to diverge at the critical point. Thus near the critical point, the correlationlength may be written as

The divergence of the correlation length at the critical point means that very far pointsbecome correlated. In other words, the long-wavelength fluctuations dominate. Thus thesystem near a second-order phase transition loses memory of its microscopic structureand begins to display new long-range macroscopic correlations. Exactly at the critical

point, the correlation function therefore displays a power lawbehaviour

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Here drepresents the effective space dimensionality of the system.

The quantities and are examples of what is known as critical exponents.Experiments, supported by renormalization group theory, have shown that systemsundergoing second-order phase transitions can be grouped into universality classes. Withineach universality class, very different systems with widely different critical temperatures, haveapproximately the same critical exponents. The reason for this is precisely the loss ofmemory mentioned above, so that systems with different microscopic structures can give riseto the same long-range behaviour

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Percolation

Percolation is a geometric analog of thermal phase transitions that is interesting on its own.

Consider a square lattice, large enough (ideally infinite) so that we may ignore boundary effectsin our discussion. Let each site of the lattice be empty or occupied with a probablity P: Theoccupation of the sites is decided by a random process, so the sites are independent of each

other. Now for P=0, all sites are empty while for P=1 all sites are occupied. Define a clusterasa group of nearest neighbour sites that are occupied. As Pincreases from 0, a critical point

P=Pcis reached when a large cluster is formed stretching from one edge of the lattice to theopposite edge. The value Pcis called the percolation threshold and at this point there is asignificant change in the properties of the lattice. For example, if sites represent pores in arock, and being occupied means the pores are open, then at the percolation threshold watercan seep through from one end of the rock to the other. There are other physical problems thatcan be studied with a percolation model, such as forest fires, or conductivity of a randomnetwork. Since percolation is a random process each simulation on a lattice for fixed Pwill giverise to different clusters of varied sizes and one must discuss statistical properties of relevant

quantities (such as cluster size) obtained after an averaging. It is found that near thepercolation threshold the physically interesting quantities diverge and show power-law

behaviour similar to that near the crtical point of a second-order phase transition, with P playing

the role analogous to temperature. Therefore for the percolation problem one can again definecritical exponents and show their universality (that is, independence from underlying latticetype). At the critical point, the structure of the clusters becomes fractal, that is, there areclusters of all scales and the self-similarity dimension is fractal. This is perhaps not verysurprising since at the critical point the properties of the system become scale-invariant andobey power-laws.

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R f

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References

Statistical Mechanics, by Kerson Huang.

The Feynman Lectures on Physics, by Feynman, Leighton and Sands.

For self-study, more notes on thermodynamics and statistical mechanics can be found at http://www.srikant.org/core/phy11sep.html

Some simulations of Brownian motion are at

http://www.phy.ntnu.edu.tw/ hwang/gas2D/gas2D.html and

http://www.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html

Atoms are real. See for example

http://www.fourmilab.ch/autofile/www/section.html

Boltzmann at

http://www-groups.dcs.st-andrews.ac.uk/ history/Mathematicians/Boltzmann.html

Robert Brown at http://www.sciences.demon.co.uk/wbbrowna.htm

An article on nanotechnology with useful references may be found at

http://www.scholars.nus.edu.sg/natureslaw/physics/nano.html

Water at http://www.scholars.nus.edu.sg/natureslaw/physics/water.html

A website explaining "why things dont go wrong more often" is at

http://www.secondlaw.com/default.htm

A web demonstration of critical opalescence is at

http://www.physicsofmatter.com/NotTheBook/CriticalOpal/OpalFrame.html

Wilson at http://www.physics.ohio-state.edu/ kgw/kgw.html

A Ising model simulation package is at at

http://bartok.ucsc.edu/peter/java/ising/keep/ising.html .

A small online Ising model simulation is at

http://www.phy.syr.edu/courses/ijmp_c/Ising.html

Ising at http://www.bradley.edu/las/phy/ising.html

An online percolation simulation package is at

http://kzoo.edu/ jant/p705/Percolate.html

Percolation Theory, by D. Stauffer and A. Aharony.