[PPT] - Bose-Einstein condensation and a two-dimensional walk model Farhad PowerPoint Presentation

SLIDE 1

Bose-Einstein condensation and a two-dimensional walk model

Farhad H. Jafarpour

Bu-Ali Sina University (BASU) Physics Department Hamedan Iran

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 1 / 19

Collaborator: S. Zeraati

SLIDE 2

Motivation

If we consider: 1) Driven-Diffusive Systems 2) Zero-Range Processes 3) Lattice Path Models are these three systems related? Sharing a common partition function, observable ... What is the role of the Matrix Product Ansatz?

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 2 / 19

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Outline

I) Open boundary conditions

PASEP Steady-state of PASEP as a superposition of single-shock measures Steady-state of PASEP as a superposition of multiple-shock measures Mapping onto a lattice path model

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 3 / 19

SLIDE 4

Outline

I) Open boundary conditions

PASEP Steady-state of PASEP as a superposition of single-shock measures Steady-state of PASEP as a superposition of multiple-shock measures Mapping onto a lattice path model

II) Periodic boundary conditions

A simple driven-diffusive system → Mapping onto a zero-range process→ Mapping onto a lattice path model A generalized driven-diffusive system → Mapping onto a zero-range process→ Mapping onto a lattice path model

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 3 / 19

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PASEP

Partially Asymmetric Simple Exclusion Process with open boundaries:

1 1 x x

1 2 Α Γ Β ∆ N 1 N

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 4 / 19

SLIDE 6

PASEP

Partially Asymmetric Simple Exclusion Process with open boundaries:

1 1 x x

1 2 Α Γ Β ∆ N 1 N

1 1

C A B

x1−d = κ+(β, δ)κ+(α, γ), d = 1, 2, 3, · · · κ+(u, v) = −u+v+1+√

(u−v−1)2+4uv 2u

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 4 / 19

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Steady-state of PASEP

For d = 2 the steady-state can be written as a linear superposition of single-shock measures with random walk dynamics: |k = 1 − ρ1 ρ1 ⊗k ⊗ 1 − ρ2 ρ2 ⊗N−k A shock:

k 1 N ρ ρ 1 2

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 5 / 19

SLIDE 8

Steady-state of PASEP

For d = 2 the steady-state can be written as a linear superposition of single-shock measures with random walk dynamics: |k = 1 − ρ1 ρ1 ⊗k ⊗ 1 − ρ2 ρ2 ⊗N−k A shock:

k 1 N ρ ρ 1 2

The steady-state can also be obtained using the matrix product method: |P∗ = 1 ZN

N

k=0

ck|k

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 5 / 19

SLIDE 9

Steady-state of PASEP

For d = 2 the steady-state can be written as a linear superposition of single-shock measures with random walk dynamics: |k = 1 − ρ1 ρ1 ⊗k ⊗ 1 − ρ2 ρ2 ⊗N−k A shock:

k 1 N ρ ρ 1 2

The steady-state can also be obtained using the matrix product method: |P∗ = 1 ZN

N

k=0

ck|k = 1 ZN W | E D ⊗N |V

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 5 / 19

SLIDE 10

Steady-state of PASEP

The partition function of the system: ZN =

N

k=0

ck = W |(D + E)N|V

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 6 / 19

SLIDE 11

Steady-state of PASEP

The partition function of the system: ZN =

N

k=0

ck = W |(D + E)N|V The matrix representation for d = 2: E = (1 − ρ1) d0

δl δr (1 − ρ2)

, D =

ρ1 −d0

δl δr ρ2

k

1 N ρ ρ 1 2 δ δ l r

Question!

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 6 / 19

SLIDE 12

PASEP mapped onto a lattice path model

The lattice path model is defined on a rotated square lattice as follows: Assign the weight δr

δl to each upward step

Assign the weight 1 to each downward step except those steps which end on the horizontal axis.

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 7 / 19

SLIDE 13

PASEP mapped onto a lattice path model

The lattice path model is defined on a rotated square lattice as follows: Assign the weight δr

δl to each upward step

Assign the weight 1 to each downward step except those steps which end on the horizontal axis. ZN = L|T N|R

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 7 / 19

SLIDE 14

PASEP mapped onto a lattice path model

The lattice path model is defined on a rotated square lattice as follows: Assign the weight δr

δl to each upward step

Assign the weight 1 to each downward step except those steps which end on the horizontal axis. ZN = L|T N|R ZN = W |(D + E)N|V

F. H. Jafarpour and S. Zeraati, PRE 81 (2010).
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 7 / 19

SLIDE 15

PASEP mapped onto a lattice path model

For an arbitrary d: x1−d = κ+(β, δ)κ+(α, γ)

ne can define a multiple-shock measure:

· · ·

1 N i1 i2 id−1 ρ1 ρ2 . . . ρd

which shares a common partition function with a multiple-transit lattice path:

❅❅ ❅ ❅❅❅ ❅❅ ❅❅

(0, 0) (2N, 0)

F. H. Jafarpour and S. Zeraati, PRE 82 (2010).
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 8 / 19

SLIDE 16

A simple disordered DDS

A disordered Driven Diffusive System (DDS) defined on a lattice of length N consisting

f M − 1 first-class particles in

the presence of a second-class particle.

1

2 1 3 N 1 1 p

1 ∅ → ∅ 1 with rate 1 2 ∅ → ∅ 2 with rate p

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 9 / 19

SLIDE 17

A simple disordered DDS

A disordered Driven Diffusive System (DDS) defined on a lattice of length N consisting

f M − 1 first-class particles in

the presence of a second-class particle.

1

2 1 3 N 1 1 p

1 ∅ → ∅ 1 with rate 1 2 ∅ → ∅ 2 with rate p A Zero Range Process (ZRP) defined on a lattice of length M consisting of N − M

particles. The particles in the

first box leave it with the rate p.

1 2 3 4 5 p 1 1 1

M. R. Evans EPL (1996).
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 9 / 19

SLIDE 18

A simple disordered DDS

Steady-state as a matrix product state

2 ∅ ∅ ∅

n1

1 ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

n2

1 · · · 1 ∅ ∅ ∅ ∅ ∅

nM

1 P({n1, n2, · · · , nM}) = 1 ZN,M Tr(D′E n1DE n2 · · · DE nM) D′ → 2 , D → 1 , E → ∅

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 10 / 19

SLIDE 19

A simple disordered DDS

Steady-state as a matrix product state

2 ∅ ∅ ∅

n1

1 ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

n2

1 · · · 1 ∅ ∅ ∅ ∅ ∅

nM

1 P({n1, n2, · · · , nM}) = 1 ZN,M Tr(D′E n1DE n2 · · · DE nM) D′ → 2 , D → 1 , E → ∅ These operators satisfy a quadratic algebra: pD′E = D′, DE = D. The matrix representation of this algebra is: D′ =

∞

i=0

p−i|0i| , D =

∞

i=0

|0i| , E =

∞

i=0

|i + 1i|

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 10 / 19

SLIDE 20

A simple disordered DDS

Canonical partition function

ZN,M(p) =

{ni}

δ(

M

i=1

ni − N + M)Tr(D′E n1DE n2 · · · DE nM) =

N−M

i=0

N − i − 2 M − 2

p−i
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 11 / 19

SLIDE 21

A simple disordered DDS

Canonical partition function

ZN,M(p) =

{ni}

δ(

M

i=1

ni − N + M)Tr(D′E n1DE n2 · · · DE nM) =

N−M

i=0

N − i − 2 M − 2

p−i

Two phases: Bose-Einstein condensation!

Depending on the values of p and ρ = M

N the system has two phases:

n1 ≃    O(N) for p < 1 − ρ, O(1) for p > 1 − ρ.

M. R. Evans EPL (1996).
M. R. Evans and T. Hanney JPA (2005).
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 11 / 19

SLIDE 22

A simple disordered DDS

Question!

Is there an equivalent lattice path model?

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 12 / 19

SLIDE 23

A simple disordered DDS

Question!

Is there an equivalent lattice path model?

An equivalent lattice path model

A lattice path model defined on Z2

+ = {(i, j) : i, j ≥ 0 are integers}.

From (i, j) to (i + 1, j + 1) with a weight 1

p.

From (i, j) to (i + 1, 0) with a weight zpj.

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 12 / 19

SLIDE 24

A simple disordered DDS

Question!

Is there an equivalent lattice path model?

An equivalent lattice path model

A lattice path model defined on Z2

+ = {(i, j) : i, j ≥ 0 are integers}.

From (i, j) to (i + 1, j + 1) with a weight 1

p.

From (i, j) to (i + 1, 0) with a weight zpj.

1 2 3 4 5 i 1 2 3 4 j

A transfer matrix T can be defined: T|j = zpj|0 + 1

p|j + 1

T N−1|0 = N−2

j=0 z(z+1)N−j−2 pj

|j +

1 pN−1 |N − 1

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 12 / 19

SLIDE 25

A simple disordered DDS

The grand-canonical partition function of the model:

ZN(p, z) =

∞

j=0

j|T N−1|0 =

N−1

i=1

N−i−1

j=0

N − j − 2 i − 1

p−jzi +

1 pN−1

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 13 / 19

SLIDE 26

A simple disordered DDS

The grand-canonical partition function of the model:

ZN(p, z) =

∞

j=0

j|T N−1|0 =

N−1

i=1

N−i−1

j=0

N − j − 2 i − 1

p−jzi +

1 pN−1 The canonical partition function of the model in which the total number of upward steps is exactly N − M. This is given by the coefficient zM−1.

The canonical partition function of the model

ZN,M(p) =

N−M

i=0

N − i − 2 M − 2

p−i

PN,M(j) = 1 ZN,M(p) N − j − 2 M − 2

p−j.
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 13 / 19

SLIDE 27

A simple disordered DDS

The mean height after N − 1 steps (ρ = M

N ):

h =

N−M

j=0

j PN,M(j) ≃      N(1 −

ρ 1−p)

for p < 1 − ρ,

1−ρ p−1+ρ

for p > 1 − ρ.

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 14 / 19

SLIDE 28

A simple disordered DDS

The mean height after N − 1 steps (ρ = M

N ):

h =

N−M

j=0

j PN,M(j) ≃      N(1 −

ρ 1−p)

for p < 1 − ρ,

1−ρ p−1+ρ

for p > 1 − ρ.

Grand-canonical partition function of the lattice path model

ZN(p, z) =

∞

j=0

j|T N−1|0

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 14 / 19

SLIDE 29

A simple disordered DDS

The mean height after N − 1 steps (ρ = M

N ):

h =

N−M

j=0

j PN,M(j) ≃      N(1 −

ρ 1−p)

for p < 1 − ρ,

1−ρ p−1+ρ

for p > 1 − ρ.

Grand-canonical partition function of the lattice path model

ZN(p, z) =

∞

j=0

j|T N−1|0

Grand-canonical partition function of the disordered DDS

ZN(p, z) = Tr(D′(E + zD)N−1) =

∞

j=0

j|C N−1|0

F. H. Jafarpour PRE (2011).
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 14 / 19

SLIDE 30

A generalized disordered DDS

A generalized disordered DDS defined on a lattice of length N consisting of M − 1 first-class particles in the presence of a second-class particle.

1

2 3 N u11 u13 u12 u12 u23

µ 0 · · · 0

nµ

µ′ uµ(nµ) − − − − → 0 µ 0 · · · 0

nµ−1

µ′

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 15 / 19

SLIDE 31

A generalized disordered DDS

A generalized disordered DDS defined on a lattice of length N consisting of M − 1 first-class particles in the presence of a second-class particle.

1

2 3 N u11 u13 u12 u12 u23

µ 0 · · · 0

nµ

µ′ uµ(nµ) − − − − → 0 µ 0 · · · 0

nµ−1

µ′ A generalized ZRP defined on a lattice of length M consisting of N − M particles. The particles in the first box leave it with the rate p.

1 2 3 4 5 u23 u12 u12 u11 u13

M. R. Evans and T. Hanney, JPA (2005).
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 15 / 19

SLIDE 32

A generalized disordered DDS

Steady-state as a matrix product state

2 ∅ ∅ ∅

n1

1 ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

n2

1 · · · 1 ∅ ∅ ∅ ∅ ∅

nM

1 P({n1, n2, · · · , nM}) = 1 ZN,M Tr(D2E n1D1E n2 · · · D1E nM) D2 → 2 , D1 → 1 , E → ∅

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 16 / 19

SLIDE 33

A generalized disordered DDS

Steady-state as a matrix product state

2 ∅ ∅ ∅

n1

1 ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

n2

1 · · · 1 ∅ ∅ ∅ ∅ ∅

nM

1 P({n1, n2, · · · , nM}) = 1 ZN,M Tr(D2E n1D1E n2 · · · D1E nM) D2 → 2 , D1 → 1 , E → ∅ These operators satisfy an algebra: DµE nµDµ′ = fµ(nµ)Dµ′ for µ, µ′ = 1, 2 where fµ(nµ) =

nµ

m=1

1 uµ(m) The matrix representation of this algebra is: Dµ =

∞

i=0

fµ(i)|0i| (for µ = 1, 2), E =

∞

i=0

|i + 1i|

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 16 / 19

SLIDE 34

A generalized disordered DDS

The grand-canonical partition function

ZN(z) = Tr(D2C N−1) =

N−1

i=0

f2(i)i|C N−1|0 where C = E + zD1 and that we have used: C|i = zf1(i)|0 + |i + 1

Question!

Is there an equivalent lattice path model?

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 17 / 19

SLIDE 35

A generalized disordered DDS

A generalized WM

A lattice path model defined on Z2

+ = {(i, j) : i, j ≥ 0 are integers}.

From (i, j) to (i + 1, j + 1) with a weight 1. From (i, j) to (i + 1, 0) with a weight zf1(j). A weight f2(j) is assigned to the point (N − 1, j).

F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 18 / 19

SLIDE 36

A generalized disordered DDS

A generalized WM

A lattice path model defined on Z2

+ = {(i, j) : i, j ≥ 0 are integers}.

From (i, j) to (i + 1, j + 1) with a weight 1. From (i, j) to (i + 1, 0) with a weight zf1(j). A weight f2(j) is assigned to the point (N − 1, j).

1 2 3 4 5 i 1 2 3 4 j

The grand-canonical partition function: ZN(z) =

N−1

j=0

f2(j)j|T N−1|0 where T = E+zD1 D1 = ∞

i=0 f1(i)|0i| , E = ∞ i=0 |i + 1i|

S. R. Masharian and F. H. Jafarpour, arXiv:1106.0174
F. H. Jafarpour (BASU)

MPIPKS, LAFNES11 18 / 19