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Lecture 3 PHYS 416 Tuesday Aug 31 Fall 2021

A. Follow-up: SZR question, Quantum coins partition functionB. Section 2.1 Random walk examplesC. Problem 2.1 Random walks in grade space

This week’s reading assignment: Sethna Ch 2.1, 2.2

1

Continuing our discussion of Worksheet 1, “Quantum Dice & Coins,” last question….

Let’s label all possible sequences of Heads & Tails for M flips by the index a. Each of these sequences has a certain probability, pa, which also depends on whether the sequence is “legal” for Distinguishable, Boson, or Fermion coins.

Let’s define the Partition Function Z to be “the sum over all states of the probability of finding that state.” (This is not necessarily unity, as we will see.)

Consider for Distinguishable coins:

ZDist = pα =α=1

2M

∑ 12M=

α=1

2M

∑ 12M+12M+ ⋅⋅⋅+

12M= 2M 1

2M⎛

⎝⎜⎜⎜⎞

⎠⎟⎟⎟⎟=1

Here we use that fact that all sequences are legal, there are 2M sequences, and all are equally likely.

2

Now consider Bosons. There are still 2M possible sequences, but only a small subset are allowed (“legal”). Now the partition function is:

ZBosons = pα =α=1

2M

∑ 12M+

12M+ ...+ 1

2M+0+0+ ...0

= M +1( ) 12M=M +1

2M

Here we used the fact that under Boson rules there are only M+1 legal states, even though all possible sequences of coin flips occur. The legal states are all equally likely with probability 1/2M. Comparing the probability under Boson rules versus the probability under Distinguished rules:

PBoson(α)=PDist (α)

Z= PDist (α)

2M

M +1=12M

2M

M +1=

1M +1

3

Let’s consider one more example of a partition function. Assume that each “Heads” is a higher energy state than each “Tails.” For example, consider spins in a magnetic field, which can either point up or down, but one orientation has a higher energy than the other.

The energy of a given sequence, then, depends on how many Heads there are. (We set the energy of Tails to zero, and assume Heads has a positive energy.)

According to the Boltzmann Factor, then, the probability of each sequence a depends on its energy. Also, the probability depends on a temperature T, so the partition function also depends on temperature:

ZBoltzmann = pα =α=1

2M

∑ exp(−εαkBT)

α=1

2M

∑

4

Note that the Boltzmann factor is usually presented as a ratio of probabilities. This the probability of a state with energy e1 compared to the probability of e0 is given as:

P(1)P(0)

= e−(ε1−ε2 )/kBT

You need the partition function, however, to get the absolute probability:

P(1)= e−ε1/kT

Z = e−ε1/kT

e−εα /kTα∑

Note: P(α)= Z Z =1α∑

5

sN = lii=1

N∑

( ) ( )2 221

11 1 12s = = - +

s22 = 2 = 14 −2( )2 + 12 0( )2 + 14 2( )2 ,s32 = 3= 18 −3( )2 + 38 −1( )2 + 38 1( )2 + 18 3( )2

Now lets talk again about flipping coins. We want to analyze the sum of heads and tails as a function of the number of flips.

(Discussion based on my notes moves to the blackboard….)

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7

C(n,r)= n!(n−r)!r!

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Copyright Oxford University Press 2006 v2.0 --

28 Random walks and emergent properties

Fig. 2.2 Random walk: scale in-variance. Random walks form ajagged, fractal pattern which looks thesame when rescaled. Here each suc-ceeding walk is the first quarter ofthe previous walk, magnified by a fac-tor of two; the shortest random walkis of length 31, the longest of length32 000 steps. The left side of Fig. 1.1is the further evolution of this walk to128 000 steps.

Copyright Oxford University Press 2006 v2.0 --

28 Random walks and emergent properties

Fig. 2.2 Random walk: scale in-variance. Random walks form ajagged, fractal pattern which looks thesame when rescaled. Here each suc-ceeding walk is the first quarter ofthe previous walk, magnified by a fac-tor of two; the shortest random walkis of length 31, the longest of length32 000 steps. The left side of Fig. 1.1is the further evolution of this walk to128 000 steps.

Copyright Oxford University Press 2006 v2.0 --

28 Random walks and emergent properties

Fig. 2.2 Random walk: scale in-variance. Random walks form ajagged, fractal pattern which looks thesame when rescaled. Here each suc-ceeding walk is the first quarter ofthe previous walk, magnified by a fac-tor of two; the shortest random walkis of length 31, the longest of length32 000 steps. The left side of Fig. 1.1is the further evolution of this walk to128 000 steps.

Random walk scale invariance…..

9

Copyright Oxford University Press 2006 v2.0 --

28 Random walks and emergent properties

Fig. 2.2 Random walk: scale in-variance. Random walks form ajagged, fractal pattern which looks thesame when rescaled. Here each suc-ceeding walk is the first quarter ofthe previous walk, magnified by a fac-tor of two; the shortest random walkis of length 31, the longest of length32 000 steps. The left side of Fig. 1.1is the further evolution of this walk to128 000 steps.

Copyright Oxford University Press 2006 v2.0 --

28 Random walks and emergent properties

Fig. 2.2 Random walk: scale in-variance. Random walks form ajagged, fractal pattern which looks thesame when rescaled. Here each suc-ceeding walk is the first quarter ofthe previous walk, magnified by a fac-tor of two; the shortest random walkis of length 31, the longest of length32 000 steps. The left side of Fig. 1.1is the further evolution of this walk to128 000 steps.

Copyright Oxford University Press 2006 v2.0 --

28 Random walks and emergent properties

Fig. 2.2 Random walk: scale in-variance. Random walks form ajagged, fractal pattern which looks thesame when rescaled. Here each suc-ceeding walk is the first quarter ofthe previous walk, magnified by a fac-tor of two; the shortest random walkis of length 31, the longest of length32 000 steps. The left side of Fig. 1.1is the further evolution of this walk to128 000 steps.

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Copyright Oxford University Press 2006 v2.0 --

2 What is statistical mechanics?

Fig. 1.1 Random walks. The mo-tion of molecules in a gas, and bacteriain a liquid, and photons in the Sun, aredescribed by random walks. Describingthe specific trajectory of any given ran-dom walk (left) is not feasible. Describ-ing the statistical properties of a largenumber of random walks is straightfor-ward (right, showing endpoints of manywalks starting at the origin). The deepprinciple underlying statistical mechan-ics is that it is often easier to under-stand the behavior of these ensemblesof systems.

matter physics. There we use it to explain metals, insulators, lasers,stellar collapse, and the microwave background radiation patterns fromthe early Universe.Monte Carlo methods allow the computer to find ensemble averages

in systems far too complicated to allow analytical evaluation. Thesetools, invented and sharpened in statistical mechanics, are used every-where in science and technology—from simulating the innards of particleaccelerators, to studies of traffic flow, to designing computer circuits. InChapter 8, we introduce Monte Carlo methods, the Ising model, and themathematics of Markov chains.Phases. Statistical mechanics explains the existence and properties of

phases. The three common phases of matter (solids, liquids, and gases)have multiplied into hundreds: from superfluids and liquid crystals, tovacuum states of the Universe just after the Big Bang, to the pinned andsliding ‘phases’ of earthquake faults. We explain the deep connectionbetween phases and perturbation theory in Section 8.3. In Chapter 9we introduce the order parameter field, which describes the properties,excitations, and topological defects that emerge in a given phase.Fluctuations and correlations. Statistical mechanics not only de-

scribes the average behavior of an ensemble of systems, it describesthe entire distribution of behaviors. We describe how systems fluctuateand evolve in space and time using correlation functions in Chapter 10.There we also derive powerful and subtle relations between correlations,response, and dissipation in equilibrium systems.Abrupt Phase Transitions. Beautiful spatial patterns arise in sta-

tistical mechanics at the transitions between phases. Most such tran-sitions are abrupt; ice is crystalline and solid until (at the edge of theice cube) it becomes unambiguously liquid. We study the nucleation

11

Copyright Oxford University Press 2006 v2.0 --

2.2 The diffusion equation 29

using renormalization-groupmethods, originally developed to study con-tinuous phase transitions.

1985 1990 1995 2000 2005Year

1

1.5

2

S&P

500

inde

x / a

vg. r

etur

n S & PRandom

Fig. 2.3 S&P 500, normalized.Standard and Poor’s 500 stock indexdaily closing price since its inception,corrected for inflation, divided by theaverage 6.4% return over this time pe-riod. Stock prices are often mod-eled as a biased random walk (Exer-cise 2.11). Notice that the fluctua-tions (risk) in individual stock priceswill typically be much higher. By av-eraging over 500 stocks, the randomfluctuations in this index are reduced,while the average return remains thesame; see [111, 112]. For comparison, aone-dimensional multiplicative randomwalk is also shown.

Polymers. Finally, random walks arise as the shapes for polymers.Polymers are long molecules (like DNA, RNA, proteins, and many plas-tics) made up of many small units (called monomers) attached to oneanother in a long chain. Temperature can introduce fluctuations in theangle between two adjacent monomers; if these fluctuations dominateover the energy,8 the polymer shape can form a random walk. Here each

8Polymers do not always form randomwalks. Polymeric plastics at low tem-perature can form crystals; functionalproteins and RNA often pack tightlyinto well-defined shapes. Molten plas-tics and denatured proteins, though,do form self-avoiding random walks.Double-stranded DNA is rather stiff;the step size for the random walk ofDNA in solution is many nucleic acidslong (Exercise 2.10).

step does not increase the time, but rather the number of the monomeralong the chain.The random walks formed by isolated polymers are not the same as

those in our first two examples; they are in a different universality class.This is because the polymer cannot intersect itself; a walk that wouldcause two monomers to overlap is not allowed. Polymers undergo self-avoiding random walks. In two and three dimensions, it turns out thatthe effects of these self-intersections is not a small, microscopic detail,but changes the properties of the random walk in an essential way.9 One

9 Self-avoidance is said to be a rel-evant perturbation that changes theuniversality class. In (unphysical)spatial dimensions higher than four,self-avoidance is irrelevant; hypotheti-cal hyper-polymers in five dimensionswould look like regular random walkson long length scales.

can show that these forbidden intersections would often arise on far-separated regions of the polymer, and that they change the dependenceof squared radius 〈s2N 〉 on the number of segments N (Exercise 2.10).In particular, the power law

√〈s2N 〉 ∼ Nν changes from the ordinary

random walk value ν = 1/2 to a higher value (ν = 3/4 in two dimensionsand ν ≈ 0.588 in three dimensions [116]). Power laws are central to thestudy of scale-invariant systems; ν is our first example of a universalcritical exponent (Chapter 12).

2.2 The diffusion equation

In the continuum limit of long length and time scales, simple behav-ior emerges from the ensemble of irregular, jagged random walks; theirevolution is described by the diffusion equation:10

10In the remainder of this chapter wespecialize for simplicity to one dimen-sion. We also change variables from thesum s to position x.

∂ρ

∂t= D∇2ρ = D

∂2ρ

∂x2. (2.7)

The diffusion equation can describe the evolving density ρ(x, t) of a localcloud of perfume as the molecules random walk through collisions withthe air molecules. Alternatively, it can describe the probability density ofan individual particle as it random walks through space; if the particlesare non-interacting, the probability distribution of one particle describesthe density of all particles.In this section, we derive the diffusion equation by taking a continuum

limit of the ensemble of random walks. Consider a general, uncorrelatedrandom walk where at each time step∆t the particle’s position x changesby a step $:

x(t+∆t) = x(t) + $(t). (2.8)

Let the probability distribution for each step be χ($).11 We will assume

11 In our two examples the distribu-tion χ(") is discrete; we can write it us-ing the Dirac δ-function. (note 3 onp. 7). For coin flips, χ(") = 1/2δ(" +1)+1/2δ("−1); for the drunkard, χ(!) =δ(|!|−L)/(2πL), evenly spaced aroundthe circle.that χ has mean zero and standard deviation a, so the first few mo-

ments12 of χ are 12The nth moment of a function ρ(x)is defined to be 〈xn〉 =

∫xnρ(x) dx.

Copyright Oxford University Press 2006 v2.0 --

2.2 The diffusion equation 29

using renormalization-groupmethods, originally developed to study con-tinuous phase transitions.

1985 1990 1995 2000 2005Year

1

1.5

2

S&P

500

inde

x / a

vg. r

etur

n S & PRandom

Fig. 2.3 S&P 500, normalized.Standard and Poor’s 500 stock indexdaily closing price since its inception,corrected for inflation, divided by theaverage 6.4% return over this time pe-riod. Stock prices are often mod-eled as a biased random walk (Exer-cise 2.11). Notice that the fluctua-tions (risk) in individual stock priceswill typically be much higher. By av-eraging over 500 stocks, the randomfluctuations in this index are reduced,while the average return remains thesame; see [111, 112]. For comparison, aone-dimensional multiplicative randomwalk is also shown.

Polymers. Finally, random walks arise as the shapes for polymers.Polymers are long molecules (like DNA, RNA, proteins, and many plas-tics) made up of many small units (called monomers) attached to oneanother in a long chain. Temperature can introduce fluctuations in theangle between two adjacent monomers; if these fluctuations dominateover the energy,8 the polymer shape can form a random walk. Here each

8Polymers do not always form randomwalks. Polymeric plastics at low tem-perature can form crystals; functionalproteins and RNA often pack tightlyinto well-defined shapes. Molten plas-tics and denatured proteins, though,do form self-avoiding random walks.Double-stranded DNA is rather stiff;the step size for the random walk ofDNA in solution is many nucleic acidslong (Exercise 2.10).

step does not increase the time, but rather the number of the monomeralong the chain.The random walks formed by isolated polymers are not the same as

those in our first two examples; they are in a different universality class.This is because the polymer cannot intersect itself; a walk that wouldcause two monomers to overlap is not allowed. Polymers undergo self-avoiding random walks. In two and three dimensions, it turns out thatthe effects of these self-intersections is not a small, microscopic detail,but changes the properties of the random walk in an essential way.9 One

9 Self-avoidance is said to be a rel-evant perturbation that changes theuniversality class. In (unphysical)spatial dimensions higher than four,self-avoidance is irrelevant; hypotheti-cal hyper-polymers in five dimensionswould look like regular random walkson long length scales.

can show that these forbidden intersections would often arise on far-separated regions of the polymer, and that they change the dependenceof squared radius 〈s2N 〉 on the number of segments N (Exercise 2.10).In particular, the power law

√〈s2N 〉 ∼ Nν changes from the ordinary

random walk value ν = 1/2 to a higher value (ν = 3/4 in two dimensionsand ν ≈ 0.588 in three dimensions [116]). Power laws are central to thestudy of scale-invariant systems; ν is our first example of a universalcritical exponent (Chapter 12).

2.2 The diffusion equation

In the continuum limit of long length and time scales, simple behav-ior emerges from the ensemble of irregular, jagged random walks; theirevolution is described by the diffusion equation:10

10In the remainder of this chapter wespecialize for simplicity to one dimen-sion. We also change variables from thesum s to position x.

∂ρ

∂t= D∇2ρ = D

∂2ρ

∂x2. (2.7)

The diffusion equation can describe the evolving density ρ(x, t) of a localcloud of perfume as the molecules random walk through collisions withthe air molecules. Alternatively, it can describe the probability density ofan individual particle as it random walks through space; if the particlesare non-interacting, the probability distribution of one particle describesthe density of all particles.In this section, we derive the diffusion equation by taking a continuum

limit of the ensemble of random walks. Consider a general, uncorrelatedrandom walk where at each time step∆t the particle’s position x changesby a step $:

x(t+∆t) = x(t) + $(t). (2.8)

Let the probability distribution for each step be χ($).11 We will assume

11 In our two examples the distribu-tion χ(") is discrete; we can write it us-ing the Dirac δ-function. (note 3 onp. 7). For coin flips, χ(") = 1/2δ(" +1)+1/2δ("−1); for the drunkard, χ(!) =δ(|!|−L)/(2πL), evenly spaced aroundthe circle.that χ has mean zero and standard deviation a, so the first few mo-

ments12 of χ are 12The nth moment of a function ρ(x)is defined to be 〈xn〉 =

∫xnρ(x) dx.

12