These are the slide that I will be presenting next week in Darmstadt for teh Winter School on Spatial Models in Statistical Mechanics. Basically, gives the main info on how to do my PhD.

1. Large Deviations on the Hydrodynamic Limit of
the TASEP to Burgers’s Equation
H. G. Duhart, P. M¨rters, J. Zimmer
o
University of Bath
24-28 February 2014

2. General overview
The big picture to consider is the following:
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

3. General overview
The big picture to consider is the following:
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

4. The TASEP
The Totally Asymmetric Simple Exclusion Process is one of
the simplest interacting particle systems. It was introduced by
Liggett in 1975.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

5. The TASEP
The Totally Asymmetric Simple Exclusion Process is one of
the simplest interacting particle systems. It was introduced by
Liggett in 1975.
We will consider Ω = {0, 1}N as its state space and its
generator:
Lf (η) = α(1 − η1 ) f (η 1 ) − f (η)
ηk (1 − ηk+1 ) f (η k,k+1 ) − f (η)
+
k∈N
Where η 1 is the same vector η with its first component
changed and η k,k+1 is the same vector η but with its
components k and k + 1 interchanged.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

6. The TASEP
One can think intuitively as a semi-infinite lattice that has a
particle reservoir to its left and produces particles into the
system with rate α ∈ (0, 1) and each of these particles jump
to the site on its right with rate 1 unless the site is occuppied.
There can be at most one particle per site.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

7. The TASEP
One can think intuitively as a semi-infinite lattice that has a
particle reservoir to its left and produces particles into the
system with rate α ∈ (0, 1) and each of these particles jump
to the site on its right with rate 1 unless the site is occuppied.
There can be at most one particle per site.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

8. The TASEP
One can think intuitively as a semi-infinite lattice that has a
particle reservoir to its left and produces particles into the
system with rate α ∈ (0, 1) and each of these particles jump
to 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 ourselves
to only looking at its first N sites, the unique invariant
probability measure is given by the Matrix Product Ansatz.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

9. The MPA
Joining the results of Derrida, et. al. (1993) and Sasamoto
and Williams (2012):
Theorem
Let ξ be the process in equilibrium of the first N sites of a
semi-infinite TASEP with entry rate α ∈ (0, 1). If there exist
matrices D and E and vectors w | and |v such that
DE
α w |E
cD|v
= D +E
= w|
= |v ,
then the distribution of ξ is given by
P[ξ = η] =
w|
N
k=1 ηk D
Horacio G. Duhart
+ (1 − ηk )E |v
w |(D + E )N |v
LDP: TASEP to Burgers’s Equation

10. The MPA
In the previous theorem, c = E[η1 (1 − η2 )] is the expected
current and can be explicitly calculated:

 α(1 − α) if α ≤ 1



2
c=


1
1


if α >
4
2
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

11. The MPA
In the previous theorem, c = E[η1 (1 − η2 )] is the expected
current and can be explicitly calculated:

 α(1 − α) if α ≤ 1



2
c=


1
1


if α >
4
2
It can be said more about this matrices and the vectors, but
this is good enough for the moment.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

12. General overview
The big picture to consider is the following:
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

13. Burgers’s Equation
The Burgers’s equation is the quasilinear partial differential
equation .
PDE
∂
∂u
+
F (u) = 0
∂t
∂x
B. C.
Horacio G. Duhart
u(x, 0) = g (x)
LDP: TASEP to Burgers’s Equation

14. Burgers’s Equation
The Burgers’s equation is the quasilinear partial differential
equation .
PDE
∂
∂u
+
F (u) = 0
∂t
∂x
B. C.
u(x, 0) = g (x)
We’re interested in the function F (u) = (2α − 1)u(1 − u) and
g (x) =
Horacio G. Duhart
1 if x < 0
0 if x < 0
LDP: TASEP to Burgers’s Equation

15. Solution to Burgers’s Equation
This PDE can be solved analitically by the method of
characteristics but there’s a region without solution, a
rarefaction fan.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

16. Solution to Burgers’s Equation
This PDE can be solved analitically by the method of
characteristics but there’s a region without solution, a
rarefaction fan.
We can force a unique solution via the vanishing viscosity
method or requiring an entropy condition. Both solutions
coincide.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

17. Solution to Burgers’s Equation
This PDE can be solved analitically by the method of
characteristics but there’s a region without solution, a
rarefaction fan.
We can force a unique solution via the vanishing viscosity
method or requiring an entropy condition. Both solutions
coincide.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

18. General overview
The big picture to consider is the following:
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

19. Hydrodynamic limit
Consider a TASEP {ξt }t≥0 and define the sequence of
random measures on the real numbers
ρN (t) =
Horacio G. Duhart
1
N
ξk (Nt)δ k
k∈N
N
LDP: TASEP to Burgers’s Equation

20. Hydrodynamic limit
Consider a TASEP {ξt }t≥0 and define the sequence of
random measures on the real numbers
ρN (t) =
1
N
ξk (Nt)δ k
k∈N
N
Let ρ : [0, ∞) → [0, 1] be the unique solution to Burgers’s
Equation.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

21. Hydrodynamic limit
Consider a TASEP {ξt }t≥0 and define the sequence of
random measures on the real numbers
ρN (t) =
1
N
ξk (Nt)δ k
k∈N
N
Let ρ : [0, ∞) → [0, 1] be the unique solution to Burgers’s
Equation.
Theorem (Benassi and Fouque 1987)
P
ρN (t) −→ ρ(x, t)dx
N→∞
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

22. General overview
The big picture to consider is the following:
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

23. Large Deviations
Formally,
Definition (Large deviation principle)
Let X be a Polish space. Let {Pn }n∈N be a sequence of probability
of measures on X . We say {Pn }n∈N satisfies a large deviation
principle with rate function I if the following three conditions meet:
i) I is a rate function.
ii)
lim sup
n→∞
1
log Pn [F ] ≤ − inf I (x) ∀F ⊂ X closed
x∈F
n
iii)
lim inf
n→∞
1
log Pn [G ] ≥ − inf I (x)
x∈G
n
Horacio G. Duhart
∀G ⊂ X open
LDP: TASEP to Burgers’s Equation

24. Large Deviations
Simply put, a sequence of random variables {Xn }n∈N satisfies a
LDP with rate function I is
P[Xn ≈ x] ≈ exp{−nI (x)}
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

25. Large Deviations
Simply put, a sequence of random variables {Xn }n∈N satisfies a
LDP with rate function I is
P[Xn ≈ x] ≈ exp{−nI (x)}
That is, the rate function may be interpreted as the exponential
rate at which the probability of the random variables being a
specific value converges to zero.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

26. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

27. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

28. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Derrida, Evans, Hakim, and Pasquier (1993) find the invariant
measure ofthe ASEP via the MPA.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

29. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Derrida, Evans, Hakim, and Pasquier (1993) find the invariant
measure ofthe ASEP via the MPA.
Jensen (2000) porposed a rate function satisfying the upper
bound condition of a LDP for a TASEP on a periodic lattice.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

30. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Derrida, Evans, Hakim, and Pasquier (1993) find the invariant
measure ofthe ASEP via the MPA.
Jensen (2000) porposed a rate function satisfying the upper
bound condition of a LDP for a TASEP on a periodic lattice.
Enaud and Derrida (2003) proved a LDP for the
hydrodynamic limit of the WASEP.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

31. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Derrida, Evans, Hakim, and Pasquier (1993) find the invariant
measure ofthe ASEP via the MPA.
Jensen (2000) porposed a rate function satisfying the upper
bound condition of a LDP for a TASEP on a periodic lattice.
Enaud and Derrida (2003) proved a LDP for the
hydrodynamic limit of the WASEP.
Bodineau and Derrida (2008) conjectured on how to find the
rate function for the TASEP as a limiting case of the WASEP.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

32. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Derrida, Evans, Hakim, and Pasquier (1993) find the invariant
measure ofthe ASEP via the MPA.
Jensen (2000) porposed a rate function satisfying the upper
bound condition of a LDP for a TASEP on a periodic lattice.
Enaud and Derrida (2003) proved a LDP for the
hydrodynamic limit of the WASEP.
Bodineau and Derrida (2008) conjectured on how to find the
rate function for the TASEP as a limiting case of the WASEP.
Sasamoto and Williams (2012) approximate the semi-infinite
ASEP by an “unphysical” finite ASEP.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

33. Previous results
Benassi and Fouque (1987) proved that the sequence of
empirical measures of the ASEP converges to Burgers’s
equation in its hydrodynamical limit.
G¨rtner (1987) proved the corresponding limit for the WASEP.
a
Derrida, Evans, Hakim, and Pasquier (1993) find the invariant
measure ofthe ASEP via the MPA.
Jensen (2000) porposed a rate function satisfying the upper
bound condition of a LDP for a TASEP on a periodic lattice.
Enaud and Derrida (2003) proved a LDP for the
hydrodynamic limit of the WASEP.
Bodineau and Derrida (2008) conjectured on how to find the
rate function for the TASEP as a limiting case of the WASEP.
Sasamoto and Williams (2012) approximate the semi-infinite
ASEP by an “unphysical” finite ASEP.
Angeletti, Touchette, Bertin, and Abry (2014) show examples
on how to find LDP for systems admitting a MPA.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

34. What’s next?
Using the techniques from the previous two papers we believe
we can find a LDP in the boundary of a semi-infinite TASEP.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

35. What’s next?
Using the techniques from the previous two papers we believe
we can find a LDP in the boundary of a semi-infinite TASEP.
1
In fact we believe that the case α ≤ is “easy”. But we still
2
need to write down the details.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

36. What’s next?
Using the techniques from the previous two papers we believe
we can find a LDP in the boundary of a semi-infinite TASEP.
1
In fact we believe that the case α ≤ is “easy”. But we still
2
need to write down the details.
1
Progress in the case α > is being done and we expect
2
results soon.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

37. What’s next?
Using the techniques from the previous two papers we believe
we can find a LDP in the boundary of a semi-infinite TASEP.
1
In fact we believe that the case α ≤ is “easy”. But we still
2
need to write down the details.
1
Progress in the case α > is being done and we expect
2
results soon.
Finally, we should be able to join the boundary to the bulk
with Jensen’s result, or at least get a rate function satisfying
the upper bound for the semi-infinite TASEP.
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

38. What’s next?
Using the techniques from the previous two papers we believe
we can find a LDP in the boundary of a semi-infinite TASEP.
1
In fact we believe that the case α ≤ is “easy”. But we still
2
need to write down the details.
1
Progress in the case α > is being done and we expect
2
results soon.
Finally, we should be able to join the boundary to the bulk
with Jensen’s result, or at least get a rate function satisfying
the upper bound for the semi-infinite TASEP.
But we’re not there yet...
Horacio G. Duhart
LDP: TASEP to Burgers’s Equation

39. Spatial Models in Statistical Mechanics
Winter School, 24 - 28 February 2014, TU Darmstadt