Talk Nice

Phase transitions on the scaling limits of the

symmetric slowed exclusion

Patr´ıcia Gon¸calves PUC - Rio de Janeiro

University of Nice 17 October 2013

joint work with

♣ Tertuliano Franco (Federal University of Bahia),

Outline

1. Slowed exclusion process: generator and invariant states.

2. Scaling limits for the density of particles: Hydrodynamics and Fluctuations.

3. Central limit theorem for the current and tagged particle.

4. Occupation time.

Slowed exclusion process

• ηt is a Markov process with space stateΩ = {0, 1}Tn.

• for x ∈ Tn,η(x) = 1 if the site is occupied, otherwise η(x) = 0.

• The generator is given on f : Ω → R by: Lnf(η) =



x∈Tn

an_x,x+1f(ηx,x+1) − f (η),

where ηx,x+1 is the configuration obtained fromη by exchanging the occupation variables η(x) and η(x + 1).

• Let νρbe the Bernoulli product measures of constant parameter

ρ ∈ [0, 1]: the occupation variables (η(x))x are independent and

Slowed exclusion process

Above for β ∈ [0, ∞] and α > 0, we consider

an_x,x+1 =



αn−β_{, if x = −1 ,}

1, otherwise . Then:

Hydrodynamic Limit

For each configuration η we denote by πn(η; du) the empirical measure: πn(η; du) = 1 n  x∈Tn η(x)δx/n(du)

and π_tn(η, du) := πn(η_tn2, du).

Assume a L.L.N. for {π₀n}_n∈N to ρ0(u)du under the initial

distribution of the system, then for any t > 0 the L.L.N. holds for

{πn

t}n∈N to ρ(t, u)du under the corresponding distribution of the system at time t, where ρ(t, u) evolves according to the

Hydrodynamic Limit: initial state

Definition

Fixρ0: T → [0, 1]. A sequence of probability measures {μn}n∈N on Ω is said to be associated to a profile ρ0 : T → [0, 1] if, for every

δ > 0 and every continuous function H : T → R, it holds that

lim n→∞μn  η : 1 n  x∈Tn H(x_n) η(x) −  TH(u) ρ0(u)du  > δ = 0 .

Hydrodynamic Limit: Law of Large Numbers

Theorem

Fix β ∈ [0, ∞] and ρ0 : T → [0, 1], and let {μn}n∈N be associated

to ρ0. Then, for any t ∈ [0, T ], for every δ > 0 and every

continuous function H : T → R: lim n→∞Pμn  η.: 1_n  x∈Tn Hx_nη_tn2(x)−  TH(u)ρ(t, u)du  > δ = 0 , where:

for β ∈ [0, 1), ρ(t, ·) is the unique weak solution of (1); for β = 1, ρ(t, ·) is the unique weak solution of (2); for β ∈ (1, ∞], ρ(t, ·) is the unique weak solution of (3).

Hydrodynamic Limit: Law of Large Numbers

Theorem

Fix β ∈ [0, ∞] and ρ0 : T → [0, 1], and let {μn}n∈N be associated

to ρ0. Then, for any t ∈ [0, T ], for every δ > 0 and every

continuous function H : T → R: lim n→∞Pμn  η.: 1_n  x∈Tn Hx_nη_tn2(x)−  TH(u)ρ(t, u)du  > δ = 0 , where:

for β ∈ [0, 1), ρ(t, ·) is the unique weak solution of (1);

for β = 1, ρ(t, ·) is the unique weak solution of (2); for β ∈ (1, ∞], ρ(t, ·) is the unique weak solution of (3).

Hydrodynamic Limit: Law of Large Numbers

Theorem

Fix β ∈ [0, ∞] and ρ0 : T → [0, 1], and let {μn}n∈N be associated

to ρ0. Then, for any t ∈ [0, T ], for every δ > 0 and every

continuous function H : T → R: lim n→∞Pμn  η.: 1_n  x∈Tn Hx_nη_tn2(x)−  TH(u)ρ(t, u)du  > δ = 0 , where:

for β ∈ [0, 1), ρ(t, ·) is the unique weak solution of (1); for β = 1, ρ(t, ·) is the unique weak solution of (2);

Hydrodynamic Limit: Law of Large Numbers

Theorem

Fix β ∈ [0, ∞] and ρ0 : T → [0, 1], and let {μn}n∈N be associated

to ρ0. Then, for any t ∈ [0, T ], for every δ > 0 and every

continuous function H : T → R: lim n→∞Pμn  η.: 1_n  x∈Tn Hx_nη_tn2(x)−  TH(u)ρ(t, u)du  > δ = 0 , where:

for β ∈ [0, 1), ρ(t, ·) is the unique weak solution of (1); for β = 1, ρ(t, ·) is the unique weak solution of (2); for β ∈ (1, ∞], ρ(t, ·) is the unique weak solution of (3).

Heat equation with periodic boundary conditions:



∂tρ(t, u) = Δρ(t, u) , t ≥ 0, u ∈ T ,

ρ(0, u) = ρ0(u), u∈ T . (1)

Definition

ρ : [0, T ] × T → [0, 1] is a weak solution if for any t ∈ [0, T ] and

any H ∈ C1,2([0, T ] × T), 

Tρ(t, u)H(t, u) − ρ(0, u)H(0, u)du

−  _t 0  Tρ(s, u)  ∂sH(s, u) + ΔH(s, u) du ds = 0 .

Heat equation with a type of Robin’s boundary conditions:

ρ : [0, T ] × T → [0, 1] is a weak solution if ρ ∈ L2_{(0, T ; H}1_{) and} for all t ∈ [0, T ] and for all H ∈ C1,2_{([0, T ] × [0, 1]),}



Tρ(t, u)H(t, u) − ρ(0, u)H(0, u)du −  _t 0  Tρ(s, u)∂sH(s, u)duds − t 0  Tρ(s, u)ΔH(s, u)duds −  _t 0(ρs(0)∂u H_s(0)−ρ_s(1)∂_uH_s(1))ds +  _t 0 α(ρs(0) − ρs(1))(Hs(0) − Hs(1))ds = 0.

Heat equation with Neumann’s boundary conditions:

ρ : [0, T ] × T → [0, 1] is a weak solution if ρ ∈ L2_{(0, T ; H}1_{) and} for all t ∈ [0, T ] and for all H ∈ C1,2([0, T ] × [0, 1]),



Tρ(t, u)H(t, u) − ρ(0, u)H(0, u)du

−  _t 0  Tρ(s, u)  ∂sH(s, u) + ΔH(s, u) du ds −  _t 0(ρs(0)∂u H_s(0) − ρ_s(1)∂_uH_s(1)) ds = 0 .

Equilibrium fluctuations

Take the process with state space {0, 1}Z and fixρ ∈ [0, 1]. For

t ∈ [0, T ], define the density fluctuation field on functions f ∈ S_β(R) as Yn t(f ) := √1_n  x∈Z f _x n {η_tn2(x) − ρ}. Theorem

The sequence of processes {Y_tn}_n∈N converges in distribution, as n goes to ∞, with respect to the Skorohod topology of

D([0, T ], S

β(R)) to a gaussian process Yt in C([0, T ], Sβ(R)),

which is the stationary solution of the Ornstein-Uhlenbeck equation

dY_t = Δ_βY_tdt +



2χ(ρ)∇_βdB_t, where B_t is aS_β(R)-valued Brownian motion.

Equilibrium fluctuations

This means that the trajectories of Y_t are in C([0, T ], S_β(R)), Y0 is a white noise of variance χ(ρ), namely if for any f ∈ S_β(R), the real-valued random variable Y0(f ) has a normal distribution of mean zero and variance χ(ρ) f 2.

Definition

For β ∈ [0, ∞], we define the operators Δ_β and∇_β onS_β(R) by

∇βH(u) =  H(1)(u), if u = 0 , H(1)(0+), if u = 0 , and ΔβH(u) =  H(2)(u), if u = 0 , H(2)(0+), if u = 0 ,

Equilibrium fluctuations

Definition

Let S(R\{0}) be the space of functions H ∈ C∞(R\{0}), that are continuous from the right at x = 0, with

H k, := supx∈R\{0}|(1 + |x|) H(k)(x)| < ∞ , for all integers

k,  ≥ 0, and H(k)(0−) = H(k)(0+), for all k integer, k ≥ 1. Let Sβ(R) be the subset of S(R\{0}) composed of functions H

satisfying

For β ∈ [0, 1), H(0−) = H(0+) .

For β = 1, H(1)(0+) = H(1)(0−) = α(H(0+) − H(0−)) . For β ∈ (1, +∞], H(1)(0+) = H(1)(0−) = 0 .

Equilibrium fluctuations

Definition

Let S(R\{0}) be the space of functions H ∈ C∞(R\{0}), that are continuous from the right at x = 0, with

H k, := supx∈R\{0}|(1 + |x|) H(k)(x)| < ∞ , for all integers

k,  ≥ 0, and H(k)(0−) = H(k)(0+), for all k integer, k ≥ 1. Let Sβ(R) be the subset of S(R\{0}) composed of functions H

satisfying

For β ∈ [0, 1), H(0−) = H(0+) .

For β = 1, H(1)(0+) = H(1)(0−) = α(H(0+) − H(0−)) . For β ∈ (1, +∞], H(1)(0+) = H(1)(0−) = 0 .

Equilibrium fluctuations

Definition

Let S(R\{0}) be the space of functions H ∈ C∞(R\{0}), that are continuous from the right at x = 0, with

H k, := supx∈R\{0}|(1 + |x|) H(k)(x)| < ∞ , for all integers

k,  ≥ 0, and H(k)(0−) = H(k)(0+), for all k integer, k ≥ 1. Let Sβ(R) be the subset of S(R\{0}) composed of functions H

satisfying

For β ∈ [0, 1), H(0−) = H(0+) .

For β = 1, H(1)(0+) = H(1)(0−) = α(H(0+) − H(0−)) . For β ∈ (1, +∞], H(1)(0+) = H(1)(0−) = 0 .

Equilibrium fluctuations

Definition

Let S(R\{0}) be the space of functions H ∈ C∞(R\{0}), that are continuous from the right at x = 0, with

H k, := supx∈R\{0}|(1 + |x|) H(k)(x)| < ∞ , for all integers

k,  ≥ 0, and H(k)(0−) = H(k)(0+), for all k integer, k ≥ 1. Let Sβ(R) be the subset of S(R\{0}) composed of functions H

satisfying

For β ∈ [0, 1), H(0−) = H(0+) .

For β = 1, H(1)(0+) = H(1)(0−) = α(H(0+) − H(0−)) . For β ∈ (1, +∞], H(1)(0+) = H(1)(0−) = 0 .

The current

For u ∈ R define the current through the bond {un − 1, un}, as J_un(t), that is J_un(t) counts the total number of jumps from the site un − 1 to the site un minus the total number of jumps from the site un to the site un − 1 in the time interval [0, tn2_].

+

-The current

For every t ≥ 0 and every u ∈ R, Ju_√n(t)

n −−−→_n→∞ Ju(t) in the sense of

finite-dimensional distributions, where J_u(t) is a gaussian process with mean zero and variance given by

• for β ∈ [0, 1), Eνρ[(Ju(t))2] = 2χ(ρ)  t π; • for β = 1, Eνρ[(Ju(t))2] = 2χ(ρ)  t π +Φ2t(2u+4αt) e 4αu+4α2 t_−Φ2 t(2u) 2α ; • for β ∈ (1, +∞], Eνρ[(Ju(t))2] = 2χ(ρ)  t π  1− e−u2/t  + 2u Φ2t(2u) , where Φ2t(x) :=  _+∞ x e−u2/4t √ 4πt du.

A tagged particle

Start {η_t : t ≥ 0} from ν_ρ conditioned to have a particle at the site

un. If Xn

u(t) is the position at the time tn2 of that tagged

particle, then {X_un(t) ≥ k} =J_un(t) ≥ un+k−1_ x=un ηtn2(x) .

A tagged particle

Theorem

For allβ ∈ [0, ∞], every u ∈ R and t ≥ 0 X_√un(t)

n −−−−→_n→+∞ Xu(t) in

the sense of finite-dimensional distributions, where X_u(t) = J_u(t)/ρ in law. In particular, the variance of X_u(t) is given by

• for β ∈ [0, 1), Eνρ[(Xu(t))2] = 2 χ(ρ) ρ2  t π; • for β = 1, Eνρ[(Xu(t))2] = 2 χ(ρ) ρ2  t π + Φ2t(2u+4αt) e 4αu+4α2 t 2α ; • for β ∈ (1, +∞], Eνρ[(Xu(t))2] = 2 χ(ρ) ρ2  t π  1− e−u2/t  + 2u Φ2t(2u) .

Occupation time

Definition

The occupation time of the origin is defined as Γn(t) := 1

n3/2  _tn2

Occupation time

Theorem

As n goes to infinity, the sequence of processes

{Γn(t) : t ∈ [0, T ]}n∈N converges in distribution with respect to

the uniform of C([0, T ], R) to a mean-zero gaussian process {Γ(t) : t ∈ [0, T ]}, with variance given by:

• for β ∈ [0, 1), EΓ(t)2 = 4 3χ(ρ)√_πt3/2. • for β = 1, EΓ(t)2= 4 3 χ(ρ)_√ π t3/2+ 2_√χ(ρ) 4πt _t 0 _s 0 Fα(s − r)drds where F_α(t) = 1 2t  _+∞ 0 z e−z2/4t−2αz dz • for β ∈ (1, +∞), EΓ(t)2= 8 3χ(ρ)√_π t3/2.

Hydrodynamic Equations

Theorem

For α > 0, let ρα: [0, T ] × [0, 1] → [0, 1] be the unique weak solution of the heat equation with Robin’s boundary conditions:

⎧ ⎪ ⎨ ⎪ ⎩ ∂tρα(t, u) = Δρα(t, u) , ∂uρα(t, 0) = ∂uρα(t, 1) = α(ρα(t, 0) − ρα(t, 1)) , ρα_{(0, u) = ρ0(u).}

Then, lim_α→0ρα = ρ0 _{and lim}

α→∞ρα = ρ∞, in

L2([0, T ] × [0, 1]), where ρ0 is the unique weak solution of the heat equation with Neumann’s boundary conditions and ρ∞ is the unique weak solution of the heat equation with periodic boundary conditions.

References

Franco, T.; Gonc¸alves, P.; Neumann, A. Hydrodynamical

behavior of symmetric exclusion with slow bonds, Annales de

l’Institut Henri Poincar´e: Probability and Statistics, 49, no. 2, 402–427 (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase Transition

of a Heat Equation with Robin’s Boundary Conditions and Exclusion Process, to appear in Transactions of the American

Mathematical Society (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase transition

in equilibrium fluctuations of symmetric slowed exclusion,

Stochastic Processes and their Applications, 123, no. 12, 4156–4185, (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Occupation times

References

Franco, T.; Gonc¸alves, P.; Neumann, A. Hydrodynamical

behavior of symmetric exclusion with slow bonds, Annales de

l’Institut Henri Poincar´e: Probability and Statistics, 49, no. 2, 402–427 (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase Transition

of a Heat Equation with Robin’s Boundary Conditions and Exclusion Process, to appear in Transactions of the American

Mathematical Society (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase transition

in equilibrium fluctuations of symmetric slowed exclusion,

Stochastic Processes and their Applications, 123, no. 12, 4156–4185, (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Occupation times

References

Franco, T.; Gonc¸alves, P.; Neumann, A. Hydrodynamical

behavior of symmetric exclusion with slow bonds, Annales de

l’Institut Henri Poincar´e: Probability and Statistics, 49, no. 2, 402–427 (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase Transition

of a Heat Equation with Robin’s Boundary Conditions and Exclusion Process, to appear in Transactions of the American

Mathematical Society (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase transition

in equilibrium fluctuations of symmetric slowed exclusion,

Stochastic Processes and their Applications, 123, no. 12, 4156–4185, (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Occupation times

References

Franco, T.; Gonc¸alves, P.; Neumann, A. Hydrodynamical

behavior of symmetric exclusion with slow bonds, Annales de

l’Institut Henri Poincar´e: Probability and Statistics, 49, no. 2, 402–427 (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase Transition

of a Heat Equation with Robin’s Boundary Conditions and Exclusion Process, to appear in Transactions of the American

Mathematical Society (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Phase transition

in equilibrium fluctuations of symmetric slowed exclusion,

Stochastic Processes and their Applications, 123, no. 12, 4156–4185, (2013).

Franco, T., Gonc¸alves, P. and Neumann, A.: Occupation times