Combinatorics and Statistical Physics:

a story of hopping particles

Lauren K. Williams

Harvard University and MSRI

1

Combinatorics and statistical physics: a story of hopping particles

Program

1. Background on a model from statistical mechanics: the

asymmetric exclusion process (ASEP). Motivation from traffic flow

and biology.

2. Some combinatorial objects coming from geometry:

Γ

-diagrams

and permutation tableaux

3. A surprising relationship between 1 and 2.

4. Applications and connections to other things ...

Lauren K. Williams 2

Combinatorics and statistical physics: a story of hopping particles

Understanding traffic

Consider a one-way road with cars entering at one end and exiting

at the other end. We’d like to understand the behavior of traffic in

this simple situation, addressing questions such as the following:

• Current: how many cars move past one spot in unit time?

• Density: what is the average number of cars present?

• Distribution: what is the probability that we see a particular

configuration of cars?

• Lines: a line of cars can form at either end of the road if cars

enter too fast or exit too slowly.

• Shocks: this is a distinct transition at some point on the road

from low density to high density traffic. This is when one needs to

suddenly slow down because of the buildup of cars ahead.

Lauren K. Williams 3

Combinatorics and statistical physics: a story of hopping particles

Traffic flow

Lauren K. Williams 4

http://ops.fhwa.dot.gov/opssecurity/dev-mx/images/fig09.jpg

Combinatorics and statistical physics: a story of hopping particles

A model for traffic: the asymmetric exclusion process

We’d like a model simple enough to mathematically analyze, yet

rich enough to exhibit various traffic phenomena.

Fix a one-dimensional lattice of n sites, and represent cars by

particles which occupy the sites. Choose a parameter q (0 ≤ q ≤ 1).

• New particles can enter the lattice from the left at rate α, and

particles can exit from the right at rate β.

• The model is asymmetric in the sense that the probability of a

particle jumping left is q times the probability of jumping right.

• Exclusion: at most one particle on each site

We’ll depict particles as • or 1 and empty sites as ◦ or 0.

Lauren K. Williams 5

Combinatorics and statistical physics: a story of hopping particles

The asymmetric exclusion process

• Introduced by biologists (MacDonald, Gibbs, Pipkin) in 1968,

and independently by a mathematician (Spitzer) in 1970. Much

more work on the ASEP by Liggett, Derrida, Spohn, Sasamoto,

Lebowitz, Ferrari, etc. (More than 300 papers on arXiv!)

Let Bn be the set of all 2n words of length n on letters {◦, •}.

The ASEP is the Markov chain on Bn with transition probabilities:

• If X = A•◦B and Y = A◦•B then PX,Y = 1

n+1 and PY,X = q

n+1 .

• If X = ◦B and Y = •B then PX,Y = α

n+1 .

• If X = B• and Y = B◦ then PX,Y = β

n+1 .

• Otherwise PX,Y = 0 for Y #= X and PX,X = 1 − ∑

X !=Y PX,Y .

Lauren K. Williams 6

Combinatorics and statistical physics: a story of hopping particles

ASEP model

The state diagram of the ASEP model for n = 2.

1/3q/3

α/3 β/3

α/3β/3

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Combinatorics and statistical physics: a story of hopping particles

Some features of the ASEP

Hydrodynamics: particle density in ASEP evolves according to

Burger’s equation.

The ASEP exhibits boundary-induced phase transitions. (Here,

q = 0.)

1 2

1 2

1

1

HIGH DENSITY

MAXIMAL CURRENTDENSITY

LOW

Lauren K. Williams 8

Combinatorics and statistical physics: a story of hopping particles

(a) α = 0.2, β = 1 (b) α = 1, β = 0.2 (c) α = β = 1

Lauren K. Williams 9 http://front.math.ucdavis.edu/9910.0270

Combinatorics and statistical physics: a story of hopping particles

ASEP in the context of biology

• Sequence alignment can be mapped onto the asymmetric

exclusion process: Drasdo, Hwa, Lassig, Bundschuh, etc.

• The ASEP models the nuclear pore complex, a multiprotein

machine that manages the transport of material into and out of the

nucleus (through a single-file pore represented by the 1D lattice):

Colvin, etc.

• The ASEP models the process of translation in protein synthesis:

Macdonald, Gibbs, Pipkin, etc.

Lauren K. Williams 10

Combinatorics and statistical physics: a story of hopping particles

The ASEP as a model for protein synthesis

In the translation step, ribo-

somes “read” the codons of mes-

senger RNA (mRNA) as the ri-

bosomes move along an mRNA

chain. Three steps: initiation,

where ribosomes attach them-

selves one at a time at the “start”

end of the mRNA; elongation,

where ribosomes move down the

chain in a series of steps; termi-

nation, where they detach them-

selves at the “stop” codon.

Lauren K. Williams 11

http://www.accessexcellence.org/RC/VL/GG/images/protein_synthesis.gif

Combinatorics and statistical physics: a story of hopping particles

Stationary Distribution of the ASEP model

The ASEP has a unique stationary distribution – that is, it has a

unique left eigenvector of the transition matrix associated with

eigenvalue 1. This is called the steady state.

1/3q/3

α/3 β/3

α/3β/3

(Solve for prob.’s, say when α = β = 1.)

Lauren K. Williams 12

Combinatorics and statistical physics: a story of hopping particles

Some combinatorics:

Γ

-diagrams and permutation tableaux

Definition: A

Γ

-diagram is a partition λ = (λ1, . . . , λn) (where

λi ≥ 0) together with a filling with 0’s and 1’s such that:

• There is no 0 which has a 1 above it in the same column and a 1

to its left in the same row.

0 0 1 1 1 1 1 0 0 1 0 1 1 0 0

Lauren K. Williams 13

Combinatorics and statistical physics: a story of hopping particles

Γ

-diagrams and permutation tableaux

0 0 1 1 1 1 1 0 0 1 0 1 1 0 0

•

Γ

-diagrams were introduced by Postnikov and shown to

correspond to cells in a cell decomposition of the totally

nonnegative part of the Grassmannian Gr+kn (subset of the real

Grassmannian with all Plucker coordinates positive).

• Gr+kn is really coming from representation theory. More generally,

Lusztig has introduced the totally non-negative part of any real

flag variety – deep connections to Lusztig’s canonical basis, and to

the cluster algebras of Fomin and Zelevinsky.

• Surprisingly, this combinatorics is also related to the asymmetric

exclusion process.

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Combinatorics and statistical physics: a story of hopping particles

Permutation tableaux

Definition: We say that a

Γ

-diagram is a permutation tableau if:

• Each column of the rectangle contains at least one 1.

There is a nice bijection from perm-tableaux to permutations which

carries statistics on tableaux to statistics on permutations

(Steingrimsson-W.).

0 0 1 1 1 1 1 0 0 1 0 1 1 0 0

. . . . .

. . .

Lauren K. Williams 15

Combinatorics and statistical physics: a story of hopping particles

Enumeration: q-Eulerian numbers

Theorem(W.) There is an explicit rank-generating function for cells

in Gr+kn. As a consequence, one gets an enumeration formula for

permutation tableaux:

The weight wt(T ) of a permutation tableau T is the number of 1’s

minus the number of columns.

Let Êk,n(q) = ∑

T q wt(T ), summing over all perm-tableaux T with

k rows and n − k columns.

Theorem(W.): Let [i] := 1 + q + q2 + · · · + qi−1.

Êk,n(q) = q k−k2

k−1 ∑

i=0

(−1)i[k − i]q nqki−k

((

n

i

)

qk−i +

(

n

i − 1

))

.

Additionally, Êk,n(q) specializes at q = −1, 0, 1 to binomial

numbers, Narayana numbers, and Eulerian numbers.

Lauren K. Williams 16

Combinatorics and statistical physics: a story of hopping particles

Corteel’s result

Theorem (Corteel): Let α = β = 1. In the steady state, the

probability that the ASEP model with n sites is in a configuration

with precisely k particles is:

Êk+1,n+1(q)

Zn

Here, Zn is the partition function for the model – the sum of the

probabilities of all possible states. So Zn = ∑n

k=0 Êk+1,n+1(q).

Question: Corteel’s result doesn’t say anything about the location

of the particles. How can we