Lecture 1: Introduction
Naoki KAWASHIMAISSP, U. Tokyo

April 8, 2024

Statistical Mechanics I: Lecture 1

April 8, 2024

1 / 21In this lecture, we see ...
Historically, the statistical mechanics was developed by Boltzmann toexplain macroscopic phenomena from the 1st principle, i.e., Newton’slaw, Schrödinger equation, etc.
However, many cooperative phenomena seem to have good
explanation without referring to the 1st principles.
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 are reflecting the universality of many-bodysystems.
In particular, the universality holds exactly in the critical phenomena.
(universality of critical phenomena)

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Very brief review of conventional statistical mechanics
Equi-probability principle: P (S) is constant independent of S.
Micro-canonical ensemble: P (S|E) = δE,E(S) /W (E)Equilibrium with heat-bath: (A: the system, B: the heat-bath)
PAB ((SA , SB )|E) = δE,EA (SA )+EB (SB ) (WAB (E))−1
X δE,E +E
WB (E − EA (SA ))A
B
PA (SA ) ==WAB (E))
WAB (E)SB

Entropy and temperature:
S(E) = log W (E) (extensive),β ≡ 1/T ≡∂S(intensive)∂E

Canonical ensemble:
PA (SA ) ∝ e−βB EA (SA )Statistical Mechanics I: Lecture 1

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Phenomena described by or related to Ising model
Ferromagnetism ... exchange coupling
Ferroelectrics ... optical phonon
Binary alloys ... change in band structure
Gases ... van der Waals force
Voters’ decision making model ... human psychology
Percolation ... trees catching fire
Potts model ... generalization to higher symmetry

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Ferromagnets
For a ferromagnetic insurator, the magnetic contribution to the totalenergy can be (at least approximately) written as
X XXX
H=−Jαβ Siα Sjβ − D(Siz )2 − H
Sizij α,β=x,y,zi(1)iwhere Siα 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 model
X
X
HI = −J
Si S j − HSi
(2)
i(ij)

represents the ferromagnet at least qualitatively.
Statistical Mechanics I: Lecture 1April 8, 2024

5 / 21Gases — Real gas
Real gas is described by Schrödinger equation,HΨ(x1 , x2 , · · · , xN ) = EΨ(x1 , x2 , · · · , xN ).

(3)

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

X p2ii

2mi+X

V (xi , xj ).

(4)

(ij)

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Gases — Lenard-Jones model
In some circumstances, we can neglect quantum nature of atoms andtreat them as classical particle with no internal degree of freedom(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′
VLJ (x, x ) = 4ϵ−(5)rrwhere r ≡ |x − x′ |.

Statistical Mechanics I: Lecture 1

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7 / 21Gases — Lattice gas
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 = −ϵni nj − µni(6)iji

where 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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Voters’ decision making model
We consider a group of voters behaving in the following way:
1
2

3

4

Everyone is wondering whether he should vote for Biden or Trump.
Everyone is showing his current preference by wearing a blue cap (forBiden) or a red one (for Trump).
At each time t, everyone is looking around himself, and observe thecurrent average preference m(t) (−1 ≤ m ≤ 1, m = 1 for perfectpreference for Biden and m = −1 for Trump).
Observing m(t) influences him in deciding his next preference: the colorof his cap next time is blue with probability (1 + tanh(βm(t)))/2 whereβ is a constant representing vorters’ sensitivity to others’ opinions.

Then, we can show that m(t) obeys the following equation of motion:m(t + 1) = tanh(βm(t))(7)

This is exactly identical to the equation of motion of themagnetization of the Ising model.

Statistical Mechanics I: Lecture 1

April 8, 2024

9 / 21Universality
The magnet, the gas, and the voters may behave according to thesame model.
This observation shows that completely different microscopic
mechanisms may lead to an identical statistical mechanical model,
and the microscopic mechanism influences the macroscopic properties
only through a few parameters. This is a manifestation of theuniversality, one of the major subject of the present course.
Moreover, when we focus on the critical phenomena, one can infereven 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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Percolation
Statistical mechanics applies to phenomena whose microscopic
elements are not really microscopic
Phenomena with completely different microscopic origin can bedescribed by the same (type of) model

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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-percolation
problem)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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Average cluster size in percolation
Let us consider the average cluster size defined by*+P 2 XV1Vc2 ,Vc ≡ Pc c =Nc Vcc(8)where Vc is the volume (the number of lattice points) of theconnected cluster c in G.
The angular bracket denotes the statistical average,
PW (G)Q(G)⟨Q(G)⟩ = GP
G W (G)(9)where the summation runs over all possible graphs.

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Generating function of bond percolation
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 average cluster size,*+XX
1 X |G|
1Vc2 =p (1 − p)NB −|G|Vc2
Vc =NNcc
G==XX∂2NB|G|
(1−p)
v
e−hVc
2
∂hc
G2
∂h→01
ΞBP
(1 − p)NBN∂h2h→0Statistical Mechanics I: Lecture 1April 8, 2024

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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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q-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) ≡ −JδSi ,Sj − H
δSi ,1(ij)iwhereS ≡ {Si },andSi = 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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Introducing the bond variables G in Potts model
By defining K ≡ βJ, h ≡ βH, the partition function isX
X Y Kδ
Y−1−βHq
Si ,SjZq ≡e=eeh(δSi ,1 −q )S(11)iS (ij)

By introducing a one-bit auxiliary variable gij = 0, 1 for every pair ofnearest-neighbor sites:XKδ
e Si ,Sj = 1 + (eK − 1)δSi ,Sj ≡v(gij )δ(gij |Si , Sj )(12)
gij =0,1

wherev(0) = 1,andv(1) = eK − 1.

(13)

δ(gij |Si , Sj ) ≡ δgij ,0 + δgij ,1 δSi ,Sj(14)

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Partition function as summation w.r.t. S and G
With N1 (S) being the number of sites where Si = 1,PXYX−1
Zq =v(gij )δ(gij |Si , Sj )eh i (δσi ,1 −q )(15)

S (ij) gij

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

(16)

(ij)

we obtain
Zq =XXS=X
GV (G)∆(G|S)ehPi (δσi ,1 −q−1 )(17)

GV (G)

X∆(G|S)eh

Pi (δσi ,1 −q−1 )

(18)

S

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Tracing out spin degree of freedom S
G is the set of local graph variables, i.e., G ≡ {gij }.
∆(G|S) = 0, 1 represents “mismatching” or “matching” between Gand S, respectively.
For each cluster in G, let Sc be one of local variables Si (i ∈ c),PX
X P−1h i (δσi ,1 −q −1 )∆(G|S)e=eh c Vc (δSc ,1 −q )S{Sc }= e−hN qY−1(ehVc + (q − 1))(19)

cThus, we have arrived at the Fortuin-Kasteleyn formula of thepartition functin of the Potts model,
XY−1(20)
Zq = e−hN qv |G| (ehVc + (q − 1)).
cG

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Fortuin-Kasteleyn formula reveals Ising/percolation relation
The generating function of the bond-percolation can be derived fromEq.(20) in the limit ϵ ≡ q − 1 → 0:X
Y−hN q −1|G|
Z1+ϵ = ev(ehVc + ϵ)cG!
≈ Z1 + ϵe−hN q= Z1 + ϵX

X−1v |G|v |G|

YehVc

Gc

X

e−hVc = ϵ ΞBP .

X

e−hVc

ccG

Zq − Z1.
q→1+0 q − 1⇒ ΞBP = limq-Potts model and percolation are related.

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Exercise 1.1: Following the same type of argument leading to theFortuin-Kasteleyn formula, show for the Ising model at H = 0 that thesusceptibilityXχ ≡ β ⟨M 2 ⟩ − ⟨M ⟩2(M ≡Si ),iis proportional to the average size of the connected clusters.

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