Large Deviations on the Hydrodynamic Limit ofthe TASEP to Burgers’s Equation

H. G. Duhart, P. Morters, J. Zimmer

University of Bath

24-28 February 2014

General overview

The big picture to consider is the following:

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

General overview

The big picture to consider is the following:

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The TASEP

The Totally Asymmetric Simple Exclusion Process is one ofthe simplest interacting particle systems. It was introduced byLiggett in 1975.

We will consider Ω = 0, 1N as its state space and itsgenerator:

Lf (η) = α(1− η1)(f (η1)− f (η)

)+∑k∈N

ηk(1− ηk+1)(

f (ηk,k+1)− f (η))

Where η1 is the same vector η with its first componentchanged and ηk,k+1 is the same vector η but with itscomponents k and k + 1 interchanged.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The TASEP

The Totally Asymmetric Simple Exclusion Process is one ofthe simplest interacting particle systems. It was introduced byLiggett in 1975.

We will consider Ω = 0, 1N as its state space and itsgenerator:

Lf (η) = α(1− η1)(f (η1)− f (η)

)+∑k∈N

ηk(1− ηk+1)(

f (ηk,k+1)− f (η))

Where η1 is the same vector η with its first componentchanged and ηk,k+1 is the same vector η but with itscomponents k and k + 1 interchanged.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The TASEP

One can think intuitively as a semi-infinite lattice that has aparticle reservoir to its left and produces particles into thesystem with rate α ∈ (0, 1) and each of these particles jumpto the site on its right with rate 1 unless the site is occuppied.There can be at most one particle per site.

It is not an ergodic process. However if we restrict ourselvesto only looking at its first N sites, the unique invariantprobability measure is given by the Matrix Product Ansatz.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The TASEP

One can think intuitively as a semi-infinite lattice that has aparticle reservoir to its left and produces particles into thesystem with rate α ∈ (0, 1) and each of these particles jumpto the site on its right with rate 1 unless the site is occuppied.There can be at most one particle per site.

It is not an ergodic process. However if we restrict ourselvesto only looking at its first N sites, the unique invariantprobability measure is given by the Matrix Product Ansatz.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The TASEP

One can think intuitively as a semi-infinite lattice that has aparticle reservoir to its left and produces particles into thesystem with rate α ∈ (0, 1) and each of these particles jumpto the site on its right with rate 1 unless the site is occuppied.There can be at most one particle per site.

It is not an ergodic process. However if we restrict ourselvesto only looking at its first N sites, the unique invariantprobability measure is given by the Matrix Product Ansatz.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The MPA

Joining the results of Derrida, et. al. (1993) and Sasamotoand Williams (2012):

Theorem

Let ξ be the process in equilibrium of the first N sites of asemi-infinite TASEP with entry rate α ∈ (0, 1). If there existmatrices D and E and vectors 〈w | and |v〉 such that

DE = D + Eα〈w |E = 〈w |cD|v〉 = |v〉,

then the distribution of ξ is given by

P[ξ = η] =〈w |

∏Nk=1 ηkD + (1− ηk)E |v〉〈w |(D + E )N |v〉

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The MPA

In the previous theorem, c = E[η1(1− η2)] is the expectedcurrent and can be explicitly calculated:

c =

α(1− α) if α ≤ 1

2

1

4if α >

1

2

It can be said more about this matrices and the vectors, butthis is good enough for the moment.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

The MPA

In the previous theorem, c = E[η1(1− η2)] is the expectedcurrent and can be explicitly calculated:

c =

α(1− α) if α ≤ 1

2

1

4if α >

1

2

It can be said more about this matrices and the vectors, butthis is good enough for the moment.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

General overview

The big picture to consider is the following:

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Burgers’s Equation

The Burgers’s equation is the quasilinear partial differentialequation .

PDE∂u

∂t+

∂

∂xF (u) = 0

B. C. u(x , 0) = g(x)

We’re interested in the function F (u) = (2α− 1)u(1− u) and

g(x) =

1 if x < 00 if x < 0

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Burgers’s Equation

The Burgers’s equation is the quasilinear partial differentialequation .

PDE∂u

∂t+

∂

∂xF (u) = 0

B. C. u(x , 0) = g(x)

We’re interested in the function F (u) = (2α− 1)u(1− u) and

g(x) =

1 if x < 00 if x < 0

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Solution to Burgers’s Equation

This PDE can be solved analitically by the method ofcharacteristics but there’s a region without solution, ararefaction fan.

We can force a unique solution via the vanishing viscositymethod or requiring an entropy condition. Both solutionscoincide.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Solution to Burgers’s Equation

This PDE can be solved analitically by the method ofcharacteristics but there’s a region without solution, ararefaction fan.We can force a unique solution via the vanishing viscositymethod or requiring an entropy condition. Both solutionscoincide.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Solution to Burgers’s Equation

This PDE can be solved analitically by the method ofcharacteristics but there’s a region without solution, ararefaction fan.We can force a unique solution via the vanishing viscositymethod or requiring an entropy condition. Both solutionscoincide.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

General overview

The big picture to consider is the following:

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Hydrodynamic limit

Consider a TASEP ξtt≥0 and define the sequence ofrandom measures on the real numbers

ρN(t) =1

N

∑k∈N

ξk(Nt)δ kN

Let ρ : [0,∞)→ [0, 1] be the unique solution to Burgers’sEquation.

Theorem (Benassi and Fouque 1987)

ρN(t)P−→

N→∞ρ(x , t)dx

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Hydrodynamic limit

Consider a TASEP ξtt≥0 and define the sequence ofrandom measures on the real numbers

ρN(t) =1

N

∑k∈N

ξk(Nt)δ kN

Let ρ : [0,∞)→ [0, 1] be the unique solution to Burgers’sEquation.

Theorem (Benassi and Fouque 1987)

ρN(t)P−→

N→∞ρ(x , t)dx

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Hydrodynamic limit

Consider a TASEP ξtt≥0 and define the sequence ofrandom measures on the real numbers

ρN(t) =1

N

∑k∈N

ξk(Nt)δ kN

Let ρ : [0,∞)→ [0, 1] be the unique solution to Burgers’sEquation.

Theorem (Benassi and Fouque 1987)

ρN(t)P−→

N→∞ρ(x , t)dx

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

General overview

The big picture to consider is the following:

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Large Deviations

Formally,

Definition (Large deviation principle)

Let X be a Polish space. Let Pnn∈N be a sequence of probabilityof measures on X . We say Pnn∈N satisfies a large deviationprinciple with rate function I if the following three conditions meet:

i) I is a rate function.

ii)

lim supn→∞

1

nlog Pn[F ] ≤ − inf

x∈FI (x) ∀F ⊂ X closed

iii)

lim infn→∞

1

nlog Pn[G ] ≥ − inf

x∈GI (x) ∀G ⊂ X open

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Large Deviations

Simply put, a sequence of random variables Xnn∈N satisfies aLDP with rate function I is

P[Xn ≈ x ] ≈ exp−nI (x)

That is, the rate function may be interpreted as the exponentialrate at which the probability of the random variables being aspecific value converges to zero.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Large Deviations

Simply put, a sequence of random variables Xnn∈N satisfies aLDP with rate function I is

P[Xn ≈ x ] ≈ exp−nI (x)

That is, the rate function may be interpreted as the exponentialrate at which the probability of the random variables being aspecific value converges to zero.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.

Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.

Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.

Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.

Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.

Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.

Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.

Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Previous results

Benassi and Fouque (1987) proved that the sequence ofempirical measures of the ASEP converges to Burgers’sequation in its hydrodynamical limit.Gartner (1987) proved the corresponding limit for the WASEP.Derrida, Evans, Hakim, and Pasquier (1993) find the invariantmeasure ofthe ASEP via the MPA.Jensen (2000) porposed a rate function satisfying the upperbound condition of a LDP for a TASEP on a periodic lattice.Enaud and Derrida (2003) proved a LDP for thehydrodynamic limit of the WASEP.Bodineau and Derrida (2008) conjectured on how to find therate function for the TASEP as a limiting case of the WASEP.Sasamoto and Williams (2012) approximate the semi-infiniteASEP by an “unphysical” finite ASEP.Angeletti, Touchette, Bertin, and Abry (2014) show exampleson how to find LDP for systems admitting a MPA.

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

What’s next?

Using the techniques from the previous two papers we believewe can find a LDP in the boundary of a semi-infinite TASEP.

In fact we believe that the case α ≤ 1

2is “easy”. But we still

need to write down the details.

Progress in the case α >1

2is being done and we expect

results soon.

Finally, we should be able to join the boundary to the bulkwith Jensen’s result, or at least get a rate function satisfyingthe upper bound for the semi-infinite TASEP.

But we’re not there yet...

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

What’s next?

Using the techniques from the previous two papers we believewe can find a LDP in the boundary of a semi-infinite TASEP.

In fact we believe that the case α ≤ 1

2is “easy”. But we still

need to write down the details.

Progress in the case α >1

2is being done and we expect

results soon.

Finally, we should be able to join the boundary to the bulkwith Jensen’s result, or at least get a rate function satisfyingthe upper bound for the semi-infinite TASEP.

But we’re not there yet...

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

What’s next?

Using the techniques from the previous two papers we believewe can find a LDP in the boundary of a semi-infinite TASEP.

In fact we believe that the case α ≤ 1

2is “easy”. But we still

need to write down the details.

Progress in the case α >1

2is being done and we expect

results soon.

Finally, we should be able to join the boundary to the bulkwith Jensen’s result, or at least get a rate function satisfyingthe upper bound for the semi-infinite TASEP.

But we’re not there yet...

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

What’s next?

Using the techniques from the previous two papers we believewe can find a LDP in the boundary of a semi-infinite TASEP.

In fact we believe that the case α ≤ 1

2is “easy”. But we still

need to write down the details.

Progress in the case α >1

2is being done and we expect

results soon.

Finally, we should be able to join the boundary to the bulkwith Jensen’s result, or at least get a rate function satisfyingthe upper bound for the semi-infinite TASEP.

But we’re not there yet...

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

What’s next?

Using the techniques from the previous two papers we believewe can find a LDP in the boundary of a semi-infinite TASEP.

In fact we believe that the case α ≤ 1

2is “easy”. But we still

need to write down the details.

Progress in the case α >1

2is being done and we expect

results soon.

Finally, we should be able to join the boundary to the bulkwith Jensen’s result, or at least get a rate function satisfyingthe upper bound for the semi-infinite TASEP.

But we’re not there yet...

Horacio G. Duhart LDP: TASEP to Burgers’s Equation

Spatial Models in Statistical Mechanics

Winter School, 24 - 28 February 2014, TU Darmstadt