Glauber dynamics in the continuum: Generating functionals approach ∗

Glauber dynamics in the continuum: Generating functionals approach ^∗

Maria Jo˜ ao Oliveira

(Universidade Aberta/CMAF)

∗ Joint work with Yu. G. Kondratiev and D. L. Finkelshtein (CAOT, 2012)

Framework

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

The configuration space:

Γ := {γ ⊂ R ^d : |γ ∩ K| < ∞, ∀ compact K ⊂ R ^d } Each γ ∈ Γ is identified with a Radon measure:

Γ ∋ γ 7→ X

x∈γ

δ _x (configuration)

Framework

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

The configuration space:

Γ := {γ ⊂ R ^d : |γ ∩ K| < ∞, ∀ compact K ⊂ R ^d } Each γ ∈ Γ is identified with a Radon measure:

Γ ∋ γ 7→ X

x∈γ

δ _x (configuration) Interpretation:

✔ Mathematical Physics: particles

✔ Biology, Ecology: individuals of a population

✔ Economics: agents

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

At each random moment of time particles randomly may...

✔ ... Appear (or born)

✔ ... Disappear (or die)

✔ ... Move (continuously or hopping)

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

At each random moment of time particles randomly may...

✔ ... Appear (or born)

✔ ... Disappear (or die)

✔ ... Move (continuously or hopping) Glauber Dynamics:

(LF )(γ) := X

x∈γ

F (γ\x)−F (γ)
+z

Z

R ^d

e ^−E(x,γ) F (γ∪x)−F (γ)

dx

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

At each random moment of time particles randomly may...

✔ ... Appear (or born)

✔ ... Disappear (or die)

✔ ... Move (continuously or hopping) Glauber Dynamics:

(LF )(γ) := X

x∈γ

F (γ \x) − F (γ)
+z

Z

R ^d

e ^−E(x,γ) F (γ∪x)−F (γ)

dx

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

At each random moment of time particles randomly may...

✔ ... Appear (or born)

✔ ... Disappear (or die)

✔ ... Move (continuously or hopping) Glauber Dynamics:

(LF )(γ) := X

x∈γ

F (γ\x)−F (γ)
+z

Z

R ^d

e ^−E(x,γ) F (γ ∪x) − F (γ)
dx

where z > 0 (activity parameter), E (x, γ) = X

y∈γ

φ(x − y) ∈ [0, +∞]

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

✔ Kolmogorov equation:

d

dt F _t = LF t

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

✔ Kolmogorov equation:

d

dt F _t = LF t

✔ Fokker-Planck equation:

d

dt µ _t = L ^∗ µ _t

 d dt

Z

Γ

F (γ) dµ t (γ) = Z

Γ

(LF )(γ)dµ t (γ)



Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

... An alternative approach:

Assume that for each t ≥ 0 there is a family k _t ⁽ⁿ⁾ : (R ^d ) ⁿ → R ⁺ ₀ , n ∈ N such that

Z

Γ

X

{x ₁ ,...,x _n }⊂γ

G(x ₁ , . . . , x _n ) dµ _t (γ)

= 1 n!

Z

(R ^d ) ⁿ

G(x ₁ , . . . , x _n )k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n ) dx ₁ . . . dx _n

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

... An alternative approach:

Assume that for each t ≥ 0 there is a family k _t ⁽ⁿ⁾ : (R ^d ) ⁿ → R ⁺ ₀ , n ∈ N such that

Z

Γ

X

{x ₁ ,...,x _n }⊂γ

G(x ₁ , . . . , x _n ) dµ _t (γ)

= 1 n!

Z

(R ^d ) ⁿ

G(x ₁ , . . . , x _n )k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n ) dx ₁ . . . dx _n

(Correlation functions or n-factorial moments of µ t )

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Now take G(x ₁ , . . . , x _n ) = θ(x ₁ ) . . . θ(x _n ), n ∈ N, and sum n:

Z

Γ

X ∞ n=0

X

{x ₁ ,...,x n }⊂γ

G(x ₁ , . . . , x _n )

| {z }

Y

x∈γ

(1 + θ(x))

dµ _t (γ)

=

X ∞ n=0

1 n!

Z

(R ^d ) ⁿ

k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n )θ(x ₁ ) . . . θ(x _n ) dx ₁ . . . dx _n

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Now take G(x ₁ , . . . , x _n ) = θ(x ₁ ) . . . θ(x _n ), n ∈ N, and sum n:

Z

Γ

X ∞ n=0

X

{x ₁ ,...,x n }⊂γ

G(x ₁ , . . . , x _n )

| {z }

Y

x∈γ

(1 + θ(x))

dµ _t (γ)

=

X ∞ n=0

1 n!

Z

(R ^d ) ⁿ

k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n )θ(x ₁ ) . . . θ(x _n ) dx ₁ . . . dx _n Bogoliubov Generating Functional (corresponding to µ) :

B _µ (θ) :=

Z

Γ

Y

x∈γ

(1 + θ(x))dµ(γ)

Analyticity

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

✔ Assume that B is an entire functional on L ¹ (σ) (σ = dx)

B (θ ₀ + θ) = X ∞ n=0

1

n! d ⁿ B(θ ₀ ; θ)

d ⁿ B (θ ₀ ; θ ₁ , . . . , θ n ) := ∂ ⁿ

∂z ₁ ...∂z n

B θ ₀ + X n i =1

z _i θ _i

! 

1≤i≤n

Analyticity

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

✔ Assume that B is an entire functional on L ¹ (σ) (σ = dx)

B (θ ₀ + θ) = X ∞ n=0

1

n ! d ⁿ B(θ ₀ ; θ)

d ⁿ B (θ 0 ; θ 1 , . . . , θ _n ) := ∂ ⁿ

∂z ₁ ...∂z _n B θ ₀ + X n i=1

z _i θ _i

! 

1≤i≤n

= Z

(R

)

δ ⁿ B (θ 0 ) δθ ₀ (x ₁ )...δθ ₀ (x n )

Y n i=1

θ _i (x i )dσ ^⊗n (x ₁ , .., x _n )

(The n-th variational derivative of B at the point θ ₀ )

δ ⁿ B (θ ₀ ) δθ ₀ (x ₁ )...δθ ₀ (x n )

L

((R

)

,σ

)

≤ n!  e r

 n

sup

kθ

k

≤r

|B(θ ₀ + θ ^′ )|

Analyticity

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

B _µ (θ) = Z

Γ

Y

x∈γ

(1 + θ(x)) dµ(γ)

Assume that B _µ is entire on L ¹ (σ). Then, k _µ exists and for θ ₀ = 0,

d ⁿ B _µ (θ ₀ ; θ ₁ , . . . , θ _n ) := ∂ ⁿ

∂z ₁ ...∂z n

B _µ θ ₀ + X n i=1

z _i θ _i

! 

1≤i≤n

= Z

(R

)

δ ⁿ B µ (θ ₀ ) δθ ₀ (x ₁ )...δθ ₀ (x n )

| {z }

k

(x

,...,x

)

Y n i=1

θ _i (x i )dσ ^⊗n (x ₁ , .., x n )

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Z

Γ

X

{x ₁ ,...,x n }⊂γ

G(x ₁ , . . . , x _n ) dµ _t (γ)

= 1 n!

Z

(R ^d ) ⁿ

G(x ₁ , . . . , x _n )k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n ) dx ₁ . . . dx _n

(Correlation functions or n-factorial moments of µ _t )

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Summing over n:

Z

Γ

X ∞ n =0

X

{x ₁ ,...,x n }⊂γ

G(x ₁ , . . . , x _n )

| {z }

:=(KG)(γ)

dµ _t (γ)

=

X ∞ n=0

1 n!

Z

(R ^d ) ⁿ

G(x ₁ , . . . , x _n )k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n ) dx ₁ . . . dx _n

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Summing over n:

Z

Γ

X ∞ n =0

X

{x ₁ ,...,x n }⊂γ

G(x ₁ , . . . , x _n )

| {z }

:=(KG)(γ)

dµ _t (γ)

=

X ∞ n=0

1 n!

Z

(R ^d ) ⁿ

G(x ₁ , . . . , x _n )k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n ) dx ₁ . . . dx _n

= Z

Γ 0

G(η)k _t (η) dλ(η), where

Γ ₀ :=

G ∞ n=0

{γ ∈ Γ : |γ| = n}

| {z }

{x ₁ ,...,x _n }

, λ :=

X ∞ n=0

1

n! dx ^⊗n

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Summing over n:

Z

Γ

X ∞ n =0

X

{x ₁ ,...,x n }⊂γ

G(x ₁ , . . . , x _n )

| {z }

:=(KG)(γ)

dµ _t (γ)

=

X ∞ n=0

1 n!

Z

(R ^d ) ⁿ

G(x ₁ , . . . , x _n )k _t ⁽ⁿ⁾ (x ₁ , . . . , x _n ) dx ₁ . . . dx _n

= Z

Γ 0

G(η)k _t (η) dλ(η).

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

d dt

Z

Γ

F (γ) dµ t (γ) = Z

Γ

(LF )(γ) dµ t (γ)

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

d dt

Z

Γ

F (γ)

| {z }

(KG)

dµ _t (γ) = Z

Γ

(LF )(γ) dµ t (γ)

⇓ d

dt Z

Γ

(KG)(γ) dµ t (γ) = Z

Γ

(KK ⁻¹ LKG)(γ) dµ t (γ)

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

d dt

Z

Γ

F (γ)

| {z }

(KG)

dµ _t (γ) = Z

Γ

(LF )(γ) dµ t (γ)

⇓ d

dt Z

Γ

(KG)(γ) dµ t (γ)

| {z }

Z

Γ ₀

G(η)k t (η) λ(η)

= Z

Γ

(KK ⁻¹ LKG)(γ)dµ t (γ)

| {z }

Z

Γ ₀

(K ⁻¹ LKG)(η)k t (η) λ(η)

Z

Γ

(KG)(γ) dµ t (γ) = Z

Γ ₀

G(η)k t (η) λ(η)

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Consequences of d

dt Z

Γ

(KG)(γ) dµ t (γ)

| {z }

Z

Γ ₀

G(η)k t (η) λ(η)

= Z

Γ

(KK ⁻¹ LKG)(γ)dµ t (γ)

| {z }

Z

Γ ₀

(K ⁻¹ LKG)(η)k t (η) λ(η) For ˆ L := K ⁻¹ LK :

✔ Correlation functions _∂t ^∂ k _t = ˆ L ^∗ k _t

Stochastic Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Consequences of d

dt Z

Γ

(KG)(γ) dµ t (γ)

| {z }

Z

Γ ₀

G(η)k t (η) λ(η)

= Z

Γ

(KK ⁻¹ LKG)(γ)dµ t (γ)

| {z }

Z

Γ ₀

(K ⁻¹ LKG)(η)k t (η) λ(η) For ˆ L := K ⁻¹ LK :

✔ Correlation functions _∂t ^∂ k _t = ˆ L ^∗ k _t

✔ Bogoliubov functionals (G(η) = Q

x∈η θ(x) =: e λ (θ, η))

∂

∂t B _t (θ) = Z

Γ 0

( ˆ Le _λ (θ))(η)k _t (η) dλ(η) =: ( ˜ LB _t )(θ)

Glauber Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

∂

∂t B _t = ˜ LB _t ( ˜ LB )(θ) =−

Z

R ^d

dx θ(x) δB(θ)

δθ(x) − zB((1 + θ) e ^{−φ(x−·)} − 1

+ θ)



Glauber Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

∂

∂t B _t = ˜ LB _t ( ˜ LB )(θ) =−

Z

R ^d

dx θ(x) δB(θ)

δθ(x) − zB((1 + θ) e ^{−φ(x−·)} − 1

+ θ)



Natural Banach spaces: E α , α > 0:= the space of all entire functionals B on L ¹ (dx) s.t.

kBk _α := sup

θ∈L ¹

 |B(θ)| e ^{− α 1 kθk L1} 

< ∞.

Glauber Dynamics

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

∂

∂t B _t = ˜ LB _t ( ˜ LB )(θ) =−

Z

R ^d

dx θ(x) δB(θ)

δθ(x) − zB((1 + θ) e ^{−φ(x−·)} − 1

+ θ)



Natural Banach spaces: E α , α > 0:= the space of all entire functionals B on L ¹ (dx) s.t.

kBk _α := sup

θ∈L ¹

 |B(θ)| e ^{− α 1 kθk L1} 

< ∞.

′

Gel’fand & Shilov’ 58 – Ovsjannikov’ 65

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Scale of Banach spaces: A family of Banach spaces {B s , k · k s : 0 < s ≤ s ₀ } s.t.

B _s ⊆ B _s ^′ , k · k _s ^′ ≤ k · k s

for any pair s, s ^′ such that 0 < s ^′ < s ≤ s ₀ .

Gel’fand & Shilov’ 58 – Ovsjannikov’ 65

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Scale of Banach spaces: A family of Banach spaces {B s , k · k s : 0 < s ≤ s ₀ } s.t.

B _s ⊆ B _s ^′ , k · k _s ^′ ≤ k · k s

for any pair s, s ^′ such that 0 < s ^′ < s ≤ s ₀ .

Example: {E _α : 0 < α ≤ α ₀ }, α ₀ > 0.

General Theorem

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

On a scale of Banach spaces {B _s : 0 < s ≤ s ₀ } consider

du(t)

dt = F u(t), u(0) = u ₀ ∈ B s

(1)

where, for each pair s ^′ < s, F : B _s → B _s ^′ is a linear mapping.

General Theorem

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

On a scale of Banach spaces {B _s : 0 < s ≤ s ₀ } consider

du(t)

dt = F u(t), u(0) = u ₀ ∈ B s

(1)

where, for each pair s ^′ < s, F : B _s → B _s ^′ is a linear mapping.

✔ For each s ∈ (0, s ₀ ) fixed, there is a constant M > 0 s.t.

kF uk _s

≤ M

s ^′′ − s ^′ kuk _s

, ∀ s ≤ s ^′ < s ^′′ ≤ s ₀ , ∀ u ∈ B _s

(M is independent of s ^′ , s ^′′ and u; it might depend on s, s ₀ )

General Theorem

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

On a scale of Banach spaces {B _s : 0 < s ≤ s ₀ } consider

du(t)

dt = F u(t), u(0) = u ₀ ∈ B s

(1)

where, for each pair s ^′ < s, F : B _s → B _s ^′ is a linear mapping.

✔ For each s ∈ (0, s ₀ ) fixed, there is a constant M > 0 s.t.

kF uk _s

≤ M

s ^′′ − s ^′ kuk _s

, ∀ s ≤ s ^′ < s ^′′ ≤ s ₀ , ∀ u ∈ B _s

(M is independent of s ^′ , s ^′′ and u; it might depend on s, s ₀ ) Theorem. For each s ∈ (0, s ₀ ), there is a constant δ > 0

(δ = (eM ) ⁻¹ ) s.t. there is a unique continuously differentiable function u : [0, δ(s ₀ − s)[ → B s which solves (1) in the

time-interval 0 ≤ t < δ(s ₀ − s).

In terms of the Glauber dynamics...

General Framework Analiticity

Stochastic Dynamics (cont.)

Glauber Dynamics General Result Glauber Dynamics (cont.)

Theorem. Given an entire GF B ₀ on L ¹ s.t. for some C, α ₀ > 0,

|B ₀ (θ)| ≤ Ce ^{α0 1 kθk L1} for all θ ∈ L ¹ ( =⇒ B ₀ ∈ E α ₀ ) , there is a local (in time) solution ([0, T )) of the initial value problem

∂B _t

∂t = ˜ LB _t , B _t|t=0 = B ₀

which, for each time t ∈ [0, T ), is an entire functional on L ¹ .