Nucleation for the zero range process on a finite set ofsites

Johel Beltran, PUCP - IMCA

Joint work withM. Jara and C. Landim (IMPA)

J. Beltran () Nucleation for the zero range process 1 / 9

A configuration: η ∈ {0, 1, 2, . . . }S .

Given η and x , y ∈ S , the transition η → ηx ,y occurs at rate

g(η(x))r(x , y)

J. Beltran () Nucleation for the zero range process 2 / 9

A configuration: η ∈ {0, 1, 2, . . . }S .

Given η and x , y ∈ S , the transition η → ηx ,y occurs at rate

g(η(x))r(x , y)

J. Beltran () Nucleation for the zero range process 2 / 9

A configuration: η ∈ {0, 1, 2, . . . }S .

Given η and x , y ∈ S , the transition η → ηx ,y occurs at rate

g(η(x))r(x , y)

J. Beltran () Nucleation for the zero range process 2 / 9

A configuration: η ∈ {0, 1, 2, . . . }S .

Given η and x , y ∈ S , the transition η → ηx ,y occurs at rate

g(η(x))r(x , y)

J. Beltran () Nucleation for the zero range process 2 / 9

A configuration: η ∈ {0, 1, 2, . . . }S .

Given η and x , y ∈ S , the transition η → ηx ,y occurs at rate

g(η(x))r(x , y)

J. Beltran () Nucleation for the zero range process 2 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0;

and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

{r(x , y) ≥ 0 : x , y ∈ S} is such that

r is irreducible.

The uniform distribution on S is invariant for r .

g : {0, 1, 2, . . . } → R+ is such that g(0) = 0;

g(n)→ 1;

for some n0 ∈ N, g(n) > 1 ∀n ≥ n0; and

limn→∞ n(g(n)− 1) = b, with b > 1

g(n) ∼ 1 +b

n

J. Beltran () Nucleation for the zero range process 3 / 9

We keep S fixed and N ↑ ∞.

Conditions on g produce a condensation effect!

J. Beltran () Nucleation for the zero range process 4 / 9

We keep S fixed and N ↑ ∞.

Conditions on g produce a condensation effect!

J. Beltran () Nucleation for the zero range process 4 / 9

We keep S fixed and N ↑ ∞.

Conditions on g produce a condensation effect!

J. Beltran () Nucleation for the zero range process 4 / 9

We are interested in {UNt : t ≥ 0},

UNt := ηtN2/N

∈ ES

whereES := {u ∈ RS

+ :∑x∈S

u(x) = 1} .

Theorem[B., Jara, Landim]For each N ≥ 1, let ηN be a configurations of N particles and let v ∈ ES

such that ηN/N → v . Let {UNt : t ≥ 0} start at ηN/N. Then

{UNt : t ≥ 0} (Law)−−−→ {Ut : t ≥ 0}

where {Ut}t≥0 is a Markov process starting at v .

J. Beltran () Nucleation for the zero range process 5 / 9

We are interested in {UNt : t ≥ 0},

UNt := ηtN2/N ∈ ES

whereES := {u ∈ RS

+ :∑x∈S

u(x) = 1} .

Theorem[B., Jara, Landim]For each N ≥ 1, let ηN be a configurations of N particles and let v ∈ ES

such that ηN/N → v . Let {UNt : t ≥ 0} start at ηN/N. Then

{UNt : t ≥ 0} (Law)−−−→ {Ut : t ≥ 0}

where {Ut}t≥0 is a Markov process starting at v .

J. Beltran () Nucleation for the zero range process 5 / 9

We are interested in {UNt : t ≥ 0},

UNt := ηtN2/N ∈ ES

whereES := {u ∈ RS

+ :∑x∈S

u(x) = 1} .

Theorem[B., Jara, Landim]For each N ≥ 1, let ηN be a configurations of N particles and let v ∈ ES

such that ηN/N → v .

Let {UNt : t ≥ 0} start at ηN/N. Then

{UNt : t ≥ 0} (Law)−−−→ {Ut : t ≥ 0}

where {Ut}t≥0 is a Markov process starting at v .

J. Beltran () Nucleation for the zero range process 5 / 9

We are interested in {UNt : t ≥ 0},

UNt := ηtN2/N ∈ ES

whereES := {u ∈ RS

+ :∑x∈S

u(x) = 1} .

Theorem[B., Jara, Landim]For each N ≥ 1, let ηN be a configurations of N particles and let v ∈ ES

such that ηN/N → v . Let {UNt : t ≥ 0} start at ηN/N.

Then

{UNt : t ≥ 0} (Law)−−−→ {Ut : t ≥ 0}

where {Ut}t≥0 is a Markov process starting at v .

J. Beltran () Nucleation for the zero range process 5 / 9

We are interested in {UNt : t ≥ 0},

UNt := ηtN2/N ∈ ES

whereES := {u ∈ RS

+ :∑x∈S

u(x) = 1} .

Theorem[B., Jara, Landim]For each N ≥ 1, let ηN be a configurations of N particles and let v ∈ ES

such that ηN/N → v . Let {UNt : t ≥ 0} start at ηN/N. Then

{UNt : t ≥ 0} (Law)−−−→ {Ut : t ≥ 0}

where {Ut}t≥0 is a Markov process starting at v .

J. Beltran () Nucleation for the zero range process 5 / 9

Let {ex : x ∈ S} be the canonical basis for RS .

Suppose v ∈ E and v(x) > 0 if and only if x ∈ A.Then for 0 ≤ t ≤ τ , (Ut(x))x∈A follows the generator

LAφ(u) =∑x ,y∈A

{b

cA(r , x , y)

u(y)∂ex + aA(r , x , y)∂ex∂ey

}

where cA(r , x , x) < 0 and cA(r , x , y) > 0 for x 6= y .

J. Beltran () Nucleation for the zero range process 6 / 9

Let {ex : x ∈ S} be the canonical basis for RS .Suppose v ∈ E and v(x) > 0 if and only if x ∈ A.

Then for 0 ≤ t ≤ τ , (Ut(x))x∈A follows the generator

LAφ(u) =∑x ,y∈A

{b

cA(r , x , y)

u(y)∂ex + aA(r , x , y)∂ex∂ey

}

where cA(r , x , x) < 0 and cA(r , x , y) > 0 for x 6= y .

J. Beltran () Nucleation for the zero range process 6 / 9

Let {ex : x ∈ S} be the canonical basis for RS .Suppose v ∈ E and v(x) > 0 if and only if x ∈ A.Then for 0 ≤ t ≤ τ , (Ut(x))x∈A follows the generator

LAφ(u) =∑x ,y∈A

{b

cA(r , x , y)

u(y)∂ex + aA(r , x , y)∂ex∂ey

}

where cA(r , x , x) < 0 and cA(r , x , y) > 0 for x 6= y .

J. Beltran () Nucleation for the zero range process 6 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N

→ v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1

and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0

.

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

Fix some v ∈ {ex : x ∈ S}. Then

UNt := ηtN2/N → v , ∀t ≥ 0 .

DefineXNt := ηtNb+1/N , t ≥ 0

and, for each ε > 0 define ∆(ε) := {u ∈ ES : ‖u − ex‖ > ε}.

Theorem[B., Landim]

If ‖ηN/N − v‖ ≤ `N/N for 1� `N � Nb+1

b(κ−1)+1 and the uniformdistribution is reversible for r , then we have

EηN[∫ T

0 1∆(`N/N)(XNt ) dt

]→ 0, for every T > 0 .

The trace of {XNt : t ≥ 0} on ES \∆(`N/N) converges to a Markov

process {Xt : t ≥ 0} on {ex : x ∈ S}.

J. Beltran () Nucleation for the zero range process 7 / 9

References

Tunneling and metastability of continuous time Markov chains; J. B.,C. Landim. Journal of Stat. Phys. Vol 140, No 6, pp 1065 – 1114,(2010).

Metastability of reversible condensed zero range processes on a finiteset; J. B., C. Landim. Probability Theory and Related Fields. Vol152, No 3-4, pp. 781 – 807, (2012).

Metastability for a non-reversible dynamics: the evolution of thecondensate in totally asymmetric zero range processes; C. Landim

J. Beltran () Nucleation for the zero range process 8 / 9

References

Tunneling and metastability of continuous time Markov chains; J. B.,C. Landim. Journal of Stat. Phys. Vol 140, No 6, pp 1065 – 1114,(2010).

Metastability of reversible condensed zero range processes on a finiteset; J. B., C. Landim. Probability Theory and Related Fields. Vol152, No 3-4, pp. 781 – 807, (2012).

Metastability for a non-reversible dynamics: the evolution of thecondensate in totally asymmetric zero range processes; C. Landim

J. Beltran () Nucleation for the zero range process 8 / 9

Thank you

Johel Beltranjohel.beltran@pucp.edu.pe

J. Beltran () Nucleation for the zero range process 9 / 9