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Stochastic energy-exchange models of non-equilibrium. Cristian - - PowerPoint PPT Presentation

Nov 27, 2023 355 likes •1.45k views

Stochastic energy-exchange models of non-equilibrium. Cristian Giardina Joint work with J. Kurchan (Paris), F . Redig (Delft) K.Vafayi (Eindhoven), G. Carinci, C. Giberti (Modena). Cristian Giardin` a (UniMoRe) Fourier law J = T -

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Stochastic energy-exchange models of non-equilibrium.

Cristian Giardina’ Joint work with J. Kurchan (Paris), F . Redig (Delft) K.Vafayi (Eindhoven), G. Carinci, C. Giberti (Modena).

Cristian Giardin` a (UniMoRe)

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Fourier law J = κ∇T

  • 1D Hamiltonian models:

Oscillators chains (Lebowitz, Lieb, Rieder, 1967): κ ∼ N. Non-linear oscillators chains (Lepri, Livi, Politi, Phys. Rep. 2003): κ ∼ Nα, 0 < α < 1 Non-linear fluctuating hydrodynamics (van Beijeren 2012, Spohn 2013)

Cristian Giardin` a (UniMoRe)

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Stochastic energy exchange models Kipnis, Marchioro, Presutti (1982): Observables: Energies at every site z = (z1, . . . , zN) ∈ RN

+

Dynamics: Select a bond at random and uniformly redistribute the energy under the constraint of conserving the total energy. LKMPf(z) =

N

  • i=1

1 dp

  • f(z1, . . . , p(zi + zi+1), (1 − p)(zi + zi+1), . . . , zN) − f(z)
  • Cristian Giardin`

a (UniMoRe)

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Outline

1

From Hamiltonian to stochastics: a simple model.

2

Duality Theory:

Brownian Momentum Process (BMP). Symmetric Inclusion Process (SIP).

3

Self-duality (SIP).

4

Boundary driven systems.

5

A larger picture & “redistribution” models.

Cristian Giardin` a (UniMoRe)

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From Hamiltonian to stochastics

Cristian Giardin` a (UniMoRe)

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A simple Hamiltonian model (G., Kurchan) H(q, p) =

N

  • i=1

1 2

  • pi − Ai

2 A = (A1(q), . . . , AN(q)) “vector potential” in RN. dqi dt = vi dvi dt =

N

  • j=1

Bijvj where Bij(q) = ∂Ai(q) ∂qj − ∂Aj(q) ∂qi antisymmetric matrix containing the “magnetic fields”

Cristian Giardin` a (UniMoRe)

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Conservation laws Conservation of Energy: Even if the forces depend on velocities and positions, the model conserves the total (kinetic) energy d dt

  • i

1 2v2

i

  • =
  • i,j

Bijvivj = 0 Conservation of Momentum: If we choose the Ai(x) such that they are left invariant by the simultaneous translations xi → xi + δ, then the quantity

i pi is

conserved.

Cristian Giardin` a (UniMoRe)

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Example: discrete time dynamics with “magnetic kicks” q(t + 1) = q(t) + v(t) v(t + 1) = R(t + 1) · v(t) with R(t) a rotation matrix R(t + 1) = cos(B(q(t + 1))) sin(B(q(t + 1))) − sin(B(q(t + 1))) cos(B(q(t + 1)))

  • Cristian Giardin`

a (UniMoRe)

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Chaoticity properties of the map on T2

Figure : Poincare section with plane q(2) = 0 of the map

     q(1)

t+1 =

q(1)

t

+ v cos(βt) q(2)

t+1 =

q(2)

t

+ v sin(βt) βt+1 = βt + B(q(1)

t

, q(2)

t

) with v =

  • v2

1 + v2 2 ,

β = arctan(v2/v1), B(q(1), q(2)) = q(1) + q(2) − 2π .

Cristian Giardin` a (UniMoRe)

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Numerical result

10

−3

10

−2

10

−1

10 10

1

10

2

10

3

10

4

T

0.10 1.00

k

N=128 N=512 N=2048

Thermal conductivity

Cristian Giardin` a (UniMoRe)

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Duality theory

Cristian Giardin` a (UniMoRe)

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Duality Definition (ηt)t≥0 Markov process on Ω with generator L, (ξt)t≥0 Markov process on Ωdual with generator Ldual

Cristian Giardin` a (UniMoRe)

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Duality Definition (ηt)t≥0 Markov process on Ω with generator L, (ξt)t≥0 Markov process on Ωdual with generator Ldual ξt is dual to ηt with duality function D : Ω × Ωdual → R if ∀t ≥ 0 Eη(D(ηt, ξ)) = Eξ(D(η, ξt)) ∀(η, ξ) ∈ Ω × Ωdual

Cristian Giardin` a (UniMoRe)

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Duality Definition (ηt)t≥0 Markov process on Ω with generator L, (ξt)t≥0 Markov process on Ωdual with generator Ldual ξt is dual to ηt with duality function D : Ω × Ωdual → R if ∀t ≥ 0 Eη(D(ηt, ξ)) = Eξ(D(η, ξt)) ∀(η, ξ) ∈ Ω × Ωdual Equivalently LD(·, ξ)(η) = LdualD(η, ·)(ξ)

Cristian Giardin` a (UniMoRe)

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How to find a dual process?

1

Write the generator in abstract form , i.e. as an element of a Lie algebra, using creation and annihilation operators.

2

Duality is related to a change of representation, i.e. new

  • perators that satisfy the same algebra.

3

Self-duality is associated to symmetries, i.e. conserved quantities.

Cristian Giardin` a (UniMoRe)

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The method at work Brownian momentum process

↓

SU(1,1) algebra

↓

Inclusion process

Cristian Giardin` a (UniMoRe)

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Brownian momentum process (BMP) on two sites Given (xi, xj) ≡ velocities of the couple (i, j) LBMP

i,j

f(xi, xj) =

  • xi

∂ ∂xj − xj ∂ ∂xi 2 f(xi, xj) polar coordinates LBMP

i,j

= ∂2 ∂θ2

ij

Brownian motion for angle θi,j = arctan(xj/xi) total kinetic energy conserved: r 2

i,j = x2 i + x2 j

Cristian Giardin` a (UniMoRe)

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Brownian momentum process (BMP) For a graph G = (V, E) let Ω = ⊗i∈VΩi = R|V|. Configuration x = (x1, . . . , x|V|) ∈ Ω Generator BMP LBMP =

  • (i,j)∈E

LBMP

i,j

=

  • (i,j)∈E
  • xi

∂ ∂xj − xj ∂ ∂xi 2

Cristian Giardin` a (UniMoRe)

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Brownian momentum process (BMP) For a graph G = (V, E) let Ω = ⊗i∈VΩi = R|V|. Configuration x = (x1, . . . , x|V|) ∈ Ω Generator BMP LBMP =

  • (i,j)∈E

LBMP

i,j

=

  • (i,j)∈E
  • xi

∂ ∂xj − xj ∂ ∂xi 2

Cristian Giardin` a (UniMoRe)

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Brownian momentum process (BMP) For a graph G = (V, E) let Ω = ⊗i∈VΩi = R|V|. Configuration x = (x1, . . . , x|V|) ∈ Ω Generator BMP LBMP =

  • (i,j)∈E

LBMP

i,j

=

  • (i,j)∈E
  • xi

∂ ∂xj − xj ∂ ∂xi 2 Stationary measures: Gaussian product measures dµ(x) =

|V|

  • i=1

e−

x2 i 2T

√ 2πT dxi

Cristian Giardin` a (UniMoRe)

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Symmetric Inclusion Process (SIP) Ωdual = ⊗i∈VΩdual

i

= {0, 1, 2, . . .}|V| Configuration ξ = (ξ1, . . . , ξ|V|) ∈ Ωdual

Cristian Giardin` a (UniMoRe)

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Symmetric Inclusion Process (SIP) Ωdual = ⊗i∈VΩdual

i

= {0, 1, 2, . . .}|V| Configuration ξ = (ξ1, . . . , ξ|V|) ∈ Ωdual Generator SIP LSIPf(ξ) =

  • (i,j)∈E

LSIP

i,j f(ξ)

=

  • (i,j)∈E

ξi

  • ξj + 1

2

  • [f(ξi,j) − f(ξ)] +
  • ξi + 1

2

  • ξj [f(ξj,i) − f(ξ)]

Cristian Giardin` a (UniMoRe)

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Symmetric Inclusion Process (SIP) Ωdual = ⊗i∈VΩdual

i

= {0, 1, 2, . . .}|V| Configuration ξ = (ξ1, . . . , ξ|V|) ∈ Ωdual Generator SIP LSIPf(ξ) =

  • (i,j)∈E

LSIP

i,j f(ξ)

=

  • (i,j)∈E

ξi

  • ξj + 1

2

  • [f(ξi,j) − f(ξ)] +
  • ξi + 1

2

  • ξj [f(ξj,i) − f(ξ)]

Stationary measures: product of Negative Binomial(r, p) with r = 2 Pr(ξ1 = n1, . . . , ξ|V| = n|V|) =

|V|

  • i=1

pni(1 − p)r ni! Γ(r + ni) Γ(r)

Cristian Giardin` a (UniMoRe)

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Duality between BMP and SIP Theorem 1 The process {x(t)}t≥0 with generator L = LBMP and the process {ξ(t)}t≥0 with generator Ldual = LSIP are dual on D(x, ξ) =

  • i∈V

x2ξi

i

(2ξi − 1)!!

Cristian Giardin` a (UniMoRe)

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Duality between BMP and SIP Theorem 1 The process {x(t)}t≥0 with generator L = LBMP and the process {ξ(t)}t≥0 with generator Ldual = LSIP are dual on D(x, ξ) =

  • i∈V

x2ξi

i

(2ξi − 1)!! Proof: An explicit computation gives LBMPD(·, ξ)(x) = LSIPD(x, ·)(ξ)

Cristian Giardin` a (UniMoRe)

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Duality explained SU(1, 1) ferromagnetic quantum spin chain Abstract operator L =

  • (i,j)∈E
  • K +

i K − j

+ K −

i K + j

− 2K o

i K o j + 1

8

  • with {K +

i , K − i , K o i }i∈V satisfying SU(1, 1) commutation relations:

[K o

i , K ± j ] = ±δi,jK ± i

[K −

i , K + j ] = 2δi,jK o i

Cristian Giardin` a (UniMoRe)

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Duality explained SU(1, 1) ferromagnetic quantum spin chain Abstract operator L =

  • (i,j)∈E
  • K +

i K − j

+ K −

i K + j

− 2K o

i K o j + 1

8

  • with {K +

i , K − i , K o i }i∈V satisfying SU(1, 1) commutation relations:

[K o

i , K ± j ] = ±δi,jK ± i

[K −

i , K + j ] = 2δi,jK o i

Duality between LBMP e LSIP corresponds to two different representations of the operator L . Duality fct is the intertwiner.

Cristian Giardin` a (UniMoRe)

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SU(1, 1) structure Continuous representation K +

i

= 1 2x2

i

K −

i

= 1 2 ∂2 ∂x2

i

K o

i = 1

4

  • xi

∂ ∂xi + ∂ ∂xi xi

  • satisfy commutation relations of the SU(1, 1) Lie algebra

[K o

i , K ± i ] = ±K ± i

[K −

i , K + i ] = 2K o i

Cristian Giardin` a (UniMoRe)

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SU(1, 1) structure Continuous representation K +

i

= 1 2x2

i

K −

i

= 1 2 ∂2 ∂x2

i

K o

i = 1

4

  • xi

∂ ∂xi + ∂ ∂xi xi

  • satisfy commutation relations of the SU(1, 1) Lie algebra

[K o

i , K ± i ] = ±K ± i

[K −

i , K + i ] = 2K o i

In this representation L = LBMP =

  • (i,j)∈E
  • xi

∂ ∂xj − xj ∂ ∂xi 2

Cristian Giardin` a (UniMoRe)

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SU(1, 1) structure Discrete representation K+

i |ξi =

  • ξi + 1

2

  • |ξi + 1

K−

i |ξi = ξi|ξi − 1

Ko

i |ξi =

  • ξi + 1

4

  • |ξi

Cristian Giardin` a (UniMoRe)

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SU(1, 1) structure Discrete representation K+

i |ξi =

  • ξi + 1

2

  • |ξi + 1

K−

i |ξi = ξi|ξi − 1

Ko

i |ξi =

  • ξi + 1

4

  • |ξi

In a canonical base

K+

i

=           

1 2

...

3 2

... ... ...            K−

i

=            1 ... 2 ... ... ...            K0

i =

           

1 4 5 4

...

9 4

... ...             Cristian Giardin` a (UniMoRe)

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SU(1, 1) structure Discete representation K+

i |ξi =

  • ξi + 1

2

  • |ξi + 1

K−

i |ξi = ξi|ξi − 1

Ko

i |ξi =

  • ξi + 1

4

  • |ξi

In this representation L f(ξ) = LSIPf(ξ) =

  • (i,j)∈E

ξi

  • ξj + 1

2

  • [f(ξi,j) − f(ξ)] +
  • ξi + 1

2

  • ξj[f(ξj,i) − f(ξ)]

Cristian Giardin` a (UniMoRe)

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SU(1, 1) structure Intertwiner K +

i Di(·, ξi)(xi) = K+ i Di(xi, ·)(ξi)

K −

i Di(·, ξi)(xi) = K− i Di(xi, ·)(ξi)

K o

i Di(·, ξi)(xi) = Ko i Di(xi, ·)(ξi)

From the creation operators x2

i

2 Di(xi, ξi) =

  • ξi + 1

2

  • D(x, ξi + 1)

Therefore Di(xi, ξi) = x2ξi

i

(2ξi − 1)!!Di(xi, 0)

Cristian Giardin` a (UniMoRe)

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Self-duality

Cristian Giardin` a (UniMoRe)

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Markov chain with finite state space

Cristian Giardin` a (UniMoRe)

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Markov chain with finite state space

  • 1. Matrix formulation of self-duality (Ldual = L)

LD = DLT

Cristian Giardin` a (UniMoRe)

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Markov chain with finite state space

  • 1. Matrix formulation of self-duality (Ldual = L)

LD = DLT Indeed

Cristian Giardin` a (UniMoRe)

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Markov chain with finite state space

  • 1. Matrix formulation of self-duality (Ldual = L)

LD = DLT Indeed LD(·, ξ)(η) = LD(η, ·)(ξ)

Cristian Giardin` a (UniMoRe)

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Markov chain with finite state space

  • 1. Matrix formulation of self-duality (Ldual = L)

LD = DLT Indeed

  • η′

L(η, η′)D(η′, ξ) = LD(·, ξ)(η) = LD(η, ·)(ξ)

Cristian Giardin` a (UniMoRe)

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Markov chain with finite state space

  • 1. Matrix formulation of self-duality (Ldual = L)

LD = DLT Indeed

  • η′

L(η, η′)D(η′, ξ) = LD(·, ξ)(η) = LD(η, ·)(ξ) =

  • ξ′

L(ξ, ξ′)D(η, ξ′)

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 2. trivial self-duality ⇐

⇒ reversible measure µ d(η, ξ) = 1 µ(η)δη,ξ

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 2. trivial self-duality ⇐

⇒ reversible measure µ d(η, ξ) = 1 µ(η)δη,ξ Indeed

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 2. trivial self-duality ⇐

⇒ reversible measure µ d(η, ξ) = 1 µ(η)δη,ξ Indeed Ld(η, ξ) = dLT(η, ξ)

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 2. trivial self-duality ⇐

⇒ reversible measure µ d(η, ξ) = 1 µ(η)δη,ξ Indeed L(η, ξ) µ(ξ) = Ld(η, ξ) = dLT(η, ξ)

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 2. trivial self-duality ⇐

⇒ reversible measure µ d(η, ξ) = 1 µ(η)δη,ξ Indeed L(η, ξ) µ(ξ) = Ld(η, ξ) = dLT(η, ξ) = L(ξ, η) µ(η)

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function.

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed LD

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdLT

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdLT = DLT

Cristian Giardin` a (UniMoRe)

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Self-Duality

  • 3. S: symmetry of the generator, i.e. [L, S] = 0,

d: trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdLT = DLT Self-duality is related to the action of a symmetry.

Cristian Giardin` a (UniMoRe)

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Self-duality of the SIP process Theorem 2 The process with generator LSIP is self-dual on functions D(η, ξ) =

  • i∈V

ηi! (ηi − ξi)! Γ 1

2

  • Γ

1

2 + ξi

  • Cristian Giardin`

a (UniMoRe)

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Self-duality of the SIP process Theorem 2 The process with generator LSIP is self-dual on functions D(η, ξ) =

  • i∈V

ηi! (ηi − ξi)! Γ 1

2

  • Γ

1

2 + ξi

  • Proof:

[LSIP,

  • i

K o

i ] = [LSIP,

  • i

K +

i ] = [LSIP,

  • i

K −

i ] = 0

Self-duality fct related to the simmetry S = e

  • i K +

i Cristian Giardin` a (UniMoRe)

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Boundary driven systems.

Cristian Giardin` a (UniMoRe)

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1 1 2 1 2

x x x T L

L res L

     

N N N R res R

x x x T L      

2 2

BMP

L

2 i

x

2 j

x

Brownian Momentum Process with reservoirs

Cristian Giardin` a (UniMoRe)

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i j

 

 

 

 

    f f f Labs  

, 1 1 1

2

abs N

L

SIP

L

Inclusion Process with absorbing reservoirs

Cristian Giardin` a (UniMoRe)

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Duality between BMP with reservoirs and SIP with absorbing boundaries Configurations ¯ ξ = (ξ0, ξ1, . . . , ξN, ξN+1) ∈ Ωdual = NN+2

Cristian Giardin` a (UniMoRe)

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Duality between BMP with reservoirs and SIP with absorbing boundaries Configurations ¯ ξ = (ξ0, ξ1, . . . , ξN, ξN+1) ∈ Ωdual = NN+2 Theorem 3 The process {x(t)}t≥0 with generator LBMP,res is dual to the process {¯ ξ(t)}t≥0 with generator LSIP,abs on D(x, ¯ ξ ) = T ξ0

L

N

  • i=1

x2ξi

i

(2ξi − 1)!!

  • T ξN+1

R

Cristian Giardin` a (UniMoRe)

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CONSEQUENCES OF DUALITY From continuous to discrete: Interacting diffusions (BMP) studied via particle systems (SIP). From many to few: n-points correlation functions of N particles using n dual walkers Remark: n ≪ N From reservoirs to absorbing boundaries: Stationary state of dual process described by absorption probabilities at the boundaries

Cristian Giardin` a (UniMoRe)

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Proposition Let P¯

ξ(a, b) = P(ξ0(∞) = a, ξN+1(∞) = b | ξ(0) = ¯

ξ). Then E(D(x, ¯ ξ )) =

  • a,b

T a

L T b R P¯ ξ(a, b)

Cristian Giardin` a (UniMoRe)

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Proposition Let P¯

ξ(a, b) = P(ξ0(∞) = a, ξN+1(∞) = b | ξ(0) = ¯

ξ). Then E(D(x, ¯ ξ )) =

  • a,b

T a

L T b R P¯ ξ(a, b)

Proof: E(D(x, ¯ ξ )) = lim

t→∞ Ex0(D(xt, ¯

ξ ))

Cristian Giardin` a (UniMoRe)

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Proposition Let P¯

ξ(a, b) = P(ξ0(∞) = a, ξN+1(∞) = b | ξ(0) = ¯

ξ). Then E(D(x, ¯ ξ )) =

  • a,b

T a

L T b R P¯ ξ(a, b)

Proof: E(D(x, ¯ ξ )) = lim

t→∞ Ex0(D(xt, ¯

ξ )) = lim

t→∞ E¯ ξ(D(x0, ¯

ξt ))

Cristian Giardin` a (UniMoRe)

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SLIDE 65

Proposition Let P¯

ξ(a, b) = P(ξ0(∞) = a, ξN+1(∞) = b | ξ(0) = ¯

ξ). Then E(D(x, ¯ ξ )) =

  • a,b

T a

L T b R P¯ ξ(a, b)

Proof: E(D(x, ¯ ξ )) = lim

t→∞ Ex0(D(xt, ¯

ξ )) = lim

t→∞ E¯ ξ(D(x0, ¯

ξt )) using D(x, ¯ ξ) = T ξ0

L

N

  • i=1

x2ξi

i

(2ξi − 1)!!

  • T ξN+1

R

Cristian Giardin` a (UniMoRe)

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SLIDE 66

Proposition Let P¯

ξ(a, b) = P(ξ0(∞) = a, ξN+1(∞) = b | ξ(0) = ¯

ξ). Then E(D(x, ¯ ξ )) =

  • a,b

T a

L T b R P¯ ξ(a, b)

Proof: E(D(x, ¯ ξ )) = lim

t→∞ Ex0(D(xt, ¯

ξ )) = lim

t→∞ E¯ ξ(D(x0, ¯

ξt )) using D(x, ¯ ξ) = T ξ0

L

N

  • i=1

x2ξi

i

(2ξi − 1)!!

  • T ξN+1

R

= E¯

ξ(T ξ0(∞) L

T ξN+1(∞)

R

)

Cristian Giardin` a (UniMoRe)

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SLIDE 67

Temperature profile

  • ξ = (0, . . . , 0, 1, 0, . . . , 0) ⇒ D(x,

ξ ) = x2

i

⇒ 1 SIP walker (Xt)t≥0 with X0 = i site i ր

Cristian Giardin` a (UniMoRe)

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SLIDE 68

Temperature profile

  • ξ = (0, . . . , 0, 1, 0, . . . , 0) ⇒ D(x,

ξ ) = x2

i

⇒ 1 SIP walker (Xt)t≥0 with X0 = i site i ր E

  • x2

i

  • = TL Pi(X∞ = 0) + TR Pi(X∞ = N + 1)

Cristian Giardin` a (UniMoRe)

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SLIDE 69

Temperature profile

  • ξ = (0, . . . , 0, 1, 0, . . . , 0) ⇒ D(x,

ξ ) = x2

i

⇒ 1 SIP walker (Xt)t≥0 with X0 = i site i ր E

  • x2

i

  • = TL Pi(X∞ = 0) + TR Pi(X∞ = N + 1)

E(x2

i ) = TL +

TR − TL N + 1

  • i

J = E(x2

i+1) − E(x2 i ) = TR − TL

N + 1 Fourier ′s law

Cristian Giardin` a (UniMoRe)

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SLIDE 70

Energy covariance If ξ = (0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0) ⇒ D(x, ξ ) = x2

i x2 j

site i ր site j ր

Cristian Giardin` a (UniMoRe)

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SLIDE 71

Energy covariance If ξ = (0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0) ⇒ D(x, ξ ) = x2

i x2 j

site i ր site j ր In the dual process we initialize two SIP walkers (Xt, Yt)t≥0 with (X0, Y0) = (i, j)

Cristian Giardin` a (UniMoRe)

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SLIDE 72

i j

 

 

 

 

    f f f Labs  

, 1 1 1

2

abs N

L

SIP

L

Inclusion Process with absorbing reservoirs

Cristian Giardin` a (UniMoRe)

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SLIDE 73

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 74

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 75

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 76

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 77

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 78

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 79

i j

Cristian Giardin` a (UniMoRe)

slide-80
SLIDE 80

i j

Cristian Giardin` a (UniMoRe)

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SLIDE 81

i j

 

         

P P P P

2 2 2 2

   

R L R L j i

T T T T x x E

Cristian Giardin` a (UniMoRe)

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SLIDE 82

Energy covariance E

  • x2

i x2 j

  • − E
  • x2

i

  • E
  • x2

j

  • =

2i(N + 1 − j) (N + 3)(N + 1)2 (TR − TL)2 ≥ 0 Remark: up to a sign, covariance is the same in the boundary driven Exclusion Process.

Cristian Giardin` a (UniMoRe)

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SLIDE 83

A larger picture & redistribution models (i). Brownian Energy Process BEP(m) (ii). Instantaneous thermalization (iii). Symmetric exclusion (SEP(n))

Cristian Giardin` a (UniMoRe)

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SLIDE 84

(i) Brownian Energy Process: BEP The energies of the Brownian Momentum Process zi(t) = x2

i (t)

Cristian Giardin` a (UniMoRe)

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SLIDE 85

(i) Brownian Energy Process: BEP The energies of the Brownian Momentum Process zi(t) = x2

i (t)

evolve with Generator LBEP =

  • (i,j)∈E

zizj ∂ ∂zi − ∂ ∂zj 2 − 1 2(zi − zj) ∂ ∂zi − ∂ ∂zj

  • Cristian Giardin`

a (UniMoRe)

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SLIDE 86

Generalized Brownian Energy Process: BEP(m) LBMP(m) =

  • (i,j)∈E

m

  • α,β=1
  • xi,α

∂ ∂xj,β − xj,β ∂ ∂xi,α 2

Cristian Giardin` a (UniMoRe)

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SLIDE 87

Generalized Brownian Energy Process: BEP(m) LBMP(m) =

  • (i,j)∈E

m

  • α,β=1
  • xi,α

∂ ∂xj,β − xj,β ∂ ∂xi,α 2 The energies zi(t) = m

α=1 x2 i,α(t)

evolve with Generator LBEP(m) =

  • (i,j)∈E

zizj ∂ ∂zi − ∂ ∂zj 2 − m 2 (zi − zj) ∂ ∂zi − ∂ ∂zj

  • Stationary measures: product Gamma( m

2 , θ)

Cristian Giardin` a (UniMoRe)

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SLIDE 88

Adding-up SU(1, 1) spins L(m) =

  • (i,j)∈E
  • K+

i K− j + K− i K+ j − 2Ko i Ko j + m2

8

  • K+

i , K− i , Ko i

  • i∈V

satisfy SU(1, 1)

Cristian Giardin` a (UniMoRe)

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SLIDE 89

Adding-up SU(1, 1) spins L(m) =

  • (i,j)∈E
  • K+

i K− j + K− i K+ j − 2Ko i Ko j + m2

8

  • K+

i , K− i , Ko i

  • i∈V

satisfy SU(1, 1)            K+

i = zi

K−

i = zi ∂2 zi + m 2 ∂zi

K0

i = zi ∂zi + m 4

           K+

i |ξi =

  • ξi + m

2

  • |ξi + 1

K−

i |ξi = ξi|ξi − 1

Ko

i |ξi = (ξi + m) |ξi

Cristian Giardin` a (UniMoRe)

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SLIDE 90

Generalized Symmetric Inclusion Process: SIP(m) Generator LSIP(m)f(ξ) =

  • (i,j)∈E

ξi

  • ξj + m

2

  • [f(ξi,j) − f(ξ)] + ξj
  • ξi + m

2

  • [f(ξj,i) − f(ξ)]

Cristian Giardin` a (UniMoRe)

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SLIDE 91

Duality between BEP(m) and SIP(m) Theorem 4 The process {z(t)}t≥0 with generator LBEP(m) and the process {ξ(t)}t≥0 with generator LSIP(m) are dual on D(z, ξ) =

  • i∈V

zξi

i

Γ( m

2 )

Γ( m

2 + ξi)

Cristian Giardin` a (UniMoRe)

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SLIDE 92

(ii) Redistribution models Generator LKMPf(z) =

  • i

1 dp[f(z1, . . . , p(zi + zi+1), (1 − p)(zi + zi+1), . . . , zN) − f(z)] KMP model is an instantaneous thermalization limit of BEP(2).

Cristian Giardin` a (UniMoRe)

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SLIDE 93

Instantaneous thermalization limit LIT

i,j f(zi, zj)

Cristian Giardin` a (UniMoRe)

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SLIDE 94

Instantaneous thermalization limit LIT

i,j f(zi, zj) := lim t→∞

  • etLBEP(m)

i,j

− 1

  • f(zi, zj)

Cristian Giardin` a (UniMoRe)

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SLIDE 95

Instantaneous thermalization limit LIT

i,j f(zi, zj) := lim t→∞

  • etLBEP(m)

i,j

− 1

  • f(zi, zj)

=

  • dz′

i dz′ j ρ(m)(z′ i , z′ j | z′ i + z′ j = zi + zj)[f(z′ i , z′ j ) − f(zi, zj)]

Cristian Giardin` a (UniMoRe)

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SLIDE 96

Instantaneous thermalization limit LIT

i,j f(zi, zj) := lim t→∞

  • etLBEP(m)

i,j

− 1

  • f(zi, zj)

=

  • dz′

i dz′ j ρ(m)(z′ i , z′ j | z′ i + z′ j = zi + zj)[f(z′ i , z′ j ) − f(zi, zj)]

= 1 dp ν(m)(p) [f(p(zi + zj), (1 − p)(zi + zj)) − f(zi, zj)]

Cristian Giardin` a (UniMoRe)

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SLIDE 97

Instantaneous thermalization limit LIT

i,j f(zi, zj) := lim t→∞

  • etLBEP(m)

i,j

− 1

  • f(zi, zj)

=

  • dz′

i dz′ j ρ(m)(z′ i , z′ j | z′ i + z′ j = zi + zj)[f(z′ i , z′ j ) − f(zi, zj)]

= 1 dp ν(m)(p) [f(p(zi + zj), (1 − p)(zi + zj)) − f(zi, zj)] X, Y ∼ Gamma m 2 , θ

  • i.i.d.

= ⇒ P = X X + Y ∼ Beta m 2 , m 2

  • For m = 2: uniform redistribution

Cristian Giardin` a (UniMoRe)

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SLIDE 98

(iii) Generalized Symmetric Exclusion Process, SEP(n) [Sch¨ utz] Configuration ξ = (ξ1, . . . , ξ|V|) ∈ {0, 1, 2, . . . , n}|V|

Cristian Giardin` a (UniMoRe)

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SLIDE 99

(iii) Generalized Symmetric Exclusion Process, SEP(n) [Sch¨ utz] Configuration ξ = (ξ1, . . . , ξ|V|) ∈ {0, 1, 2, . . . , n}|V| LSEP(n)f(ξ) =

  • (i,j)∈E

ξi(n − ξj)[f(ξi,j) − f(ξ)] + (n − ξi)ξj[f(ξj,i) − f(ξ)]

Cristian Giardin` a (UniMoRe)

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SLIDE 100

(iii) Generalized Symmetric Exclusion Process, SEP(n) [Sch¨ utz] Configuration ξ = (ξ1, . . . , ξ|V|) ∈ {0, 1, 2, . . . , n}|V| LSEP(n)f(ξ) =

  • (i,j)∈E

ξi(n − ξj)[f(ξi,j) − f(ξ)] + (n − ξi)ξj[f(ξj,i) − f(ξ)] Stationary measures: product with marginals Binomial(n,p)

Cristian Giardin` a (UniMoRe)

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SLIDE 101

Generalized Symmetric Exclusion Process: SEP(n) L(n) =

  • (i,j)∈E
  • J+

i J− j + J− i J+ j + 2Jo i Jo j − n2

2

  • J+

i , J− i , Jo i

  • satisfy SU(2) commutation relations

[Jo

i , J± j ] = ±δi,jJ± i

[J−

i , J+ j ] = −2δi,jJo i

Cristian Giardin` a (UniMoRe)

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SLIDE 102

Generalized Symmetric Exclusion Process: SEP(n) L(n) =

  • (i,j)∈E
  • J+

i J− j + J− i J+ j + 2Jo i Jo j − n2

2

  • J+

i , J− i , Jo i

  • satisfy SU(2) commutation relations

[Jo

i , J± j ] = ±δi,jJ± i

[J−

i , J+ j ] = −2δi,jJo i

           J+

i |ξi = (n − ξi) |ξi + 1

J−

i |ξi = ξi|ξi − 1

Jo

i |ξi =

  • ξi − n

2

  • |ξi

Cristian Giardin` a (UniMoRe)

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SLIDE 103

Self-duality of the SEP(n) process Theorem 5 The process with generator LSEP(n) is self-dual on functions D(η, ξ) =

  • i∈V

ηi! (ηi − ξi)! Γ (n + 1 − ξi) Γ (n + 1)

Cristian Giardin` a (UniMoRe)

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SLIDE 104

Self-duality of the SEP(n) process Theorem 5 The process with generator LSEP(n) is self-dual on functions D(η, ξ) =

  • i∈V

ηi! (ηi − ξi)! Γ (n + 1 − ξi) Γ (n + 1) Proof: [LSEP(n),

  • i

Jo

i ] = [LSEP(n),

  • i

J+

i ] = [LSEP(n),

  • i

J−

i ] = 0

Self-duality corresponds to the action of the symmetry S = e

  • i J+

i Cristian Giardin` a (UniMoRe)

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SLIDE 105

Summary of Self-duality Theorem 6 The INCLUSION process is self-dual on D(η, ξ) =

  • i

ηi! (ηi − ξi)! Γ m

2

  • Γ

m

2 + ξi

  • Cristian Giardin`

a (UniMoRe)

slide-106
SLIDE 106

Summary of Self-duality Theorem 6 The INCLUSION process is self-dual on D(η, ξ) =

  • i

ηi! (ηi − ξi)! Γ m

2

  • Γ

m

2 + ξi

  • The INDEPENDENT WALKERS process is self-dual on

D(η, ξ) =

  • i

ηi! (ηi − ξi)!

Cristian Giardin` a (UniMoRe)

slide-107
SLIDE 107

Summary of Self-duality Theorem 6 The INCLUSION process is self-dual on D(η, ξ) =

  • i

ηi! (ηi − ξi)! Γ m

2

  • Γ

m

2 + ξi

  • The INDEPENDENT WALKERS process is self-dual on

D(η, ξ) =

  • i

ηi! (ηi − ξi)! The EXCLUSION process is self-dual on D(η, ξ) =

  • i

ηi! (ηi − ξi)! Γ (n + 1 − ξi) Γ (n + 1)

Cristian Giardin` a (UniMoRe)

slide-108
SLIDE 108

Perspectives Energy/particle redistribution models [JSP (2013)] Instantaneous thermalization limit of inclusion process, independent walkers, exclusion process Mathematical population genetics: [arXiv:1302.3206] e.g. Multi-type Wright Fisher diffusion, Moran model Bulk-driven models: [work in progress] Asymmetric processes and q-deformed algebras

Cristian Giardin` a (UniMoRe)