C o m b i n at o r i c s an d S t at i s t i c al P h y s i c...

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  • 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

    Lauren K. Williams 7

  • 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.

    Lauren K. Williams 14

  • 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