Lecture 1: Introduction
Phase transitions, critical phenomena and universality

Naoki KAWASHIMA1
1 ISSP, U. Tokyo

April 8, 2019

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To begin withHistorically, the statistical mechanics was developed by Boltzmann toexplain macroscopic phenomena from the 1st principle, i.e., Newton’slaw (or Schrödinger equation in the later developments).
However, many cooperative phenomena seem to have good
explanation without referring to the 1st principles. In this lecture, wetake a look at a few examples.

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April 8, 20192 / 21[1-1] Various Phenomena described by Ising model
Ferromagnets
Ferroelectrics
Binary alloys
Gas-liquid transition

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Ferromagnets
For a ferromagnetic insurator, the magnetic contribution to the totalenergy can be (at least approximately) written as
X X
XX
H=−Jαβ Sαi Sβj − D(Szi )2 − H
Sziij α,β=x,y ,zi(1)iwhere Sαi is a generator of SU(2) algebra in some irreducible
representation characterized by the magnitude of spin S = 1/2, 1, 3/2, · · · .
The coupling constant Jαβ = Jδαβ for isotoropic coupling. For somemagnets, the anisotropy is easy-axis type and D is positive, in which case,only two states, Siz = ±S, are important. As a result of these, in somecases one may consider the Ising modelXX
HI = −JSi Sj − HSi(2)(ij)

irepresents the ferromagnet at least qualitatively.
Naoki KAWASHIMA (ISSP)

Statistical Machanics IApril 8, 20194 / 21Real gases
Real gas is described by Schrödinger equation,HΨ(x1 , x2 , · · · , xN ) = E Ψ(x1 , x2 , · · · , xN ).

(3)

The Hamiltonian consists of the kinetic energy and the two-body Coulombinteractions among nuclei and electrons.
H≡

X p2
i

i

2mi

Naoki KAWASHIMA (ISSP)+X

V (xi , xj ).

(4)

(ij)

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Lenard-Jone gas
We can neglect quantum nature of atoms and treat them as classicalparticle with no internal degree of freedom, in some circumstances(e.g., gas-liquid transition at room temperature).
In such cases, we consider a classical model, such as Lenard-Jones(LJ) model σ 12  σ 6−(5)VLJ (x, x0 ) = 4rrwhere r ≡ |x − x0 |.

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Lattice gas model
We may simplify the system even further when we focus on thenature of phase transitions.
For example, by discretizing the space and neglecting the long-range
tail of the Lenard-Jones potential, we obtain the lattice gas modelXX
H = −
ni nj − µni(6)ijiwhere ni = 0, 1 represents absense/presense of a particle at the site i.
(One can easily verify that this is mathematically equivalent to theIsing model with a uniform magnetic field.)

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Universality
The phase diagram one obtains from the LJ model agrees with thephase diagram determined by real experiments. The agreement canbe made even quantitatively accurate for noble gases by choosing
right values for  and σ.
This observation shows that the microscopic mechanism and themacroscopic properties are related to each other only through a few
parameters. We may call this the universality of statisticalmechanical phenomena.
Moreover, when we focus on the critical phenomena, one can infer
even the exact values of real systems from a very simplified model.
For example, the value of the critical index β is estimated for thelattice-gas model to be β ≈ 0.3272, and the experimental result canbe fit well by assuming this estimate.
This observation is an example of the universality of criticalphenomena.
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April 8, 20198 / 21[1-2] Percolation
Statistical mechanics applies to phenomena whose microscopic
elements are not really microscopic
Phenomena with completely different microscopic origin can be
described by the same (type of) model

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April 8, 20199 / 21Forest fire and percolation
In a forest fire, a tree catches fire from a burning tree in its
neighborhood. An important question is whether there is a big clusterof trees in which they are close to each other.
Suppose the forest is a square lattice and that a tree is planted withprobability p on each lattice point.
Let us call the two trees are “connected” when they are nearest
neighbors to each other.
How big is the largest cluster of connected trees? (site-percolationproblem)
In the bond-percolation, every lattice point has a tree, but they areconnected only with probability p.
The largest cluster size is an increasing function of p.
The function has a singular point at p = 0.5. Above this point, thelargest cluster is infinity and remains finite below this point.
(percolation transition).
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Generating function of the bond percolation (1)Let us consider the average cluster size defined byDEχ ≡ Vc2 /Vc(7)

where Vc is the volume (the number of lattice points) of theconnected cluster c.
The over-line denotes the average over all connected clusters,XX
Qc /1.
Qc ≡(8)cc

The angular bracket denotes the statistical average,
PW (G )Q(G )hQ(G )i = GP
G W (G )(9)where the summation runs over all possible connection graphs.
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Generating function of the bond percolation (2)The weight W (G ) is expressed formally as
W (G ) = p |G | (1 − p)NB −|G | = (const.) × v |G |(10)

where |G | is the number of the connections in G , NB is the total
number of the nearest neighbor pairs of sites, and v ≡ p/(1 − p).
To obtain compact expression*of the averagecluster size,+P 2 XV1Pc c =χ =Vc2VNc ccXX
1=p |G | (1 − p)NB −|G |Vc2NcG=∂2∂h(1 − p)NB2XG2∂

v |G |X
c=1(1 − p)NB
ΞBPN
∂h2h→0Naoki KAWASHIMA (ISSP)

Statistical Machanics I

e −hVch→0April 8, 201912 / 21

Generating function of the bond percolation (3)The generating function of bond-percolation
XXΞBP ≡v |G |e −hVc .
G(11)c∂ΞBP (h)|h→0 = −N(1 − p)−NB ,Using ∂hχ=−Naoki KAWASHIMA (ISSP)∂2ΞBP∂h2



h→0∂
ΞBP
∂hStatistical Machanics I
(12)
h→0

April 8, 201913 / 21

Relation among percolation, Ising and Potts models
We have seen a few examples in which the statistical mechanics isapplied beyond the tight connection to the microscopic mechanisms.
In the first set of examples, various phenomena was described by theIsing model whereas in the latter the percolation model was essential.
Now, it may be good to know that these apparently unrelated models
can be also related to each other at least in a mathematical level.

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Potts model
We first generalize the Ising model to the Potts model. The extension
is made by replacing binary variables in the Ising model by q-valuedones.
XX
Hq (S) ≡ −J
δSi ,Sj − H
δSi ,1(ij)iwhereS ≡ {Si },and Si = 1, 2, · · · , q

It is easy to verify that the q = 2 Potts model is identical to the Isingmodel after trivial redefinitions of J and H.

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Fortuin-Kasteleyn formula (1)By defining K ≡ βJ, h ≡ βH, the partition function isX
X Y KδYZq ≡e −βHq =e Si ,Sje hδSi ,1SS

(13)

i(ij)

By introducing a one-bit auxiliary variable gij = 0, 1 for every pair ofnearest-neighbor sites:XKδ
e Si ,Sj = 1 + (e K − 1)δSi ,Sj ≡v (gij )δ(gij |Si , Sj )(14)
gij =0,1

wherev (0) = 1,and v (1) = e K − 1.

(15)

δ(gij |Si , Sj ) ≡ δgij ,0 + δgij ,1 δSi ,Sj

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Fortuin-Kasteleyn formula (2)With N1 (S) being the number of sites where Si = 1,XYX
Zq =v (gij )δ(gij |Si , Sj )e hN1 (S)S(17)(ij) gij

By using a simplifying notation
YY
V (G ) ≡v (gij ) and ∆(G |S) ≡δ(gij |Si , Sj )
(ij)

(18)

(ij)

we obtainXX
Zq =V (G )∆(G |S)e hN1 (S)=SGX

V (G )

GNaoki KAWASHIMA (ISSP)

X(19)∆(G |S)e hN1 (S)

(20)

S

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Fortuin-Kasteleyn formula (3)G is the set of local graph variables, i.e., G ≡ {gij }.
∆(G |S) is a binary valued function that represents “matching” of Gand S, i.e., if any two variables in S are the same whenever they areconnected in G , ∆(G |S) = 1, otherwise ∆(G |S) = 0.
For each connected cluster in G , let one of local variables Si (i ∈ c)represent the cluster degree of freedom and use the symbol Sc forsuch a representative,
XX PY∆(G |S)e hN1 (S) =e h c Vc δSc ,1 =(e hVc +(q −1)). (21){Sc }ScThus, we have arrived at the Fortuin-Kasteleyn formula of thepartition functin of the Potts model,
XY(22)
Zq =v |G | (e hVc + (q − 1)).
G

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Fortuin-Kasteleyn formula (4)When H = 0,
X
Zq =v |G | q Nc (G ) .

(23)

G

The generating function of the bond-percolation can be derived fromEq.(22) in the limit  ≡ q − 1 → 0:XYZq =v |G | (e hVc + )cG!
Y≈ XGc= X|G | hN

GNaoki KAWASHIMA (ISSP)v |G |v

ee hVcXe −hVccXe−hVc= e hN ΞBP .

cStatistical Machanics I

April 8, 201919 / 21[1-3] Summary
Many cooperative macroscopic phenomena do have good explanation
without referring to details of the microscopic mechanisms.
The essential macroscopic properties can be understood by models interms of intermediate-scale degrees of freedom.
Often the same model can describe the essence of multiple
phenomena with completely different microscopic origins.
These observations may be phrased as the universality of statisticalmechanical phenomena.
In particular, the universality holds exactly in the critical phenomena.
(universality of critical phenomena)

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April 8, 201920 / 21

Homework
Following the same type of argument leading to the Fortuin-Kasteleyn
formula, show that the susceptibilityχ ≡ β hM 2 i − hMi2(24)whereM≡XSi(25)iof the Ising model at H = 0 is proportional to the average size of theconnected clusters.
Submit your report at the beginning of the next lecture.

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April 8, 201921 / 21