Bose-Einstein condensation and a two-dimensional walk model Farhad H. Jafarpour Bu-Ali Sina University (BASU) Physics Department Hamedan Iran Collaborator: S. Zeraati F. H. Jafarpour (BASU) MPIPKS, LAFNES11 1 / 19
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
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
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
PASEP Partially Asymmetric Simple Exclusion Process with open boundaries: Α Β x 1 x 1 1 2 N � 1 N Γ ∆ F. H. Jafarpour (BASU) MPIPKS, LAFNES11 4 / 19
PASEP Partially Asymmetric Simple Exclusion Process with open boundaries: Α Β x 1 x 1 1 2 N � 1 N Γ ∆ x 1 − d = κ + ( β, δ ) κ + ( α, γ ), d = 1 , 2 , 3 , · · · A κ + ( u , v ) = − u + v +1+ √ 1 ( u − v − 1) 2 +4 uv C 2 u B 0 1 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 4 / 19
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: � 1 − ρ 1 � 1 − ρ 2 � ⊗ k � ⊗ N − k | k � = ⊗ ρ 1 ρ 2 A shock: ρ 2 ρ 1 1 k N F. H. Jafarpour (BASU) MPIPKS, LAFNES11 5 / 19
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: � 1 − ρ 1 � 1 − ρ 2 � ⊗ k � ⊗ N − k | k � = ⊗ ρ 1 ρ 2 A shock: ρ 2 ρ 1 1 k N The steady-state can also be obtained using the matrix product method: N | P ∗ � = 1 � c k | k � Z N k =0 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 5 / 19
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: � 1 − ρ 1 � 1 − ρ 2 � ⊗ k � ⊗ N − k | k � = ⊗ ρ 1 ρ 2 A shock: ρ 2 ρ 1 1 k N The steady-state can also be obtained using the matrix product method: � E N � ⊗ N | P ∗ � = 1 c k | k � = 1 � �� W | | V �� Z N Z N D k =0 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 5 / 19
Steady-state of PASEP The partition function of the system: N � c k = �� W | ( D + E ) N | V �� Z N = k =0 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 6 / 19
Steady-state of PASEP The partition function of the system: N � c k = �� W | ( D + E ) N | V �� Z N = k =0 The matrix representation for d = 2: � (1 − ρ 1 ) � ρ 1 � � − d 0 d 0 E = , D = δ l δ l δ r (1 − ρ 2 ) 0 0 δ r ρ 2 δ δ r l ρ 2 ρ 1 1 k N Question! F. H. Jafarpour (BASU) MPIPKS, LAFNES11 6 / 19
PASEP mapped onto a lattice path model The lattice path model is defined on a rotated square lattice as follows: Assign the weight δ r δ l 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
PASEP mapped onto a lattice path model The lattice path model is defined on a rotated square lattice as follows: Assign the weight δ r δ l to each upward step Assign the weight 1 to each downward step except those steps which end on the horizontal axis. Z N = � L | T N | R � F. H. Jafarpour (BASU) MPIPKS, LAFNES11 7 / 19
PASEP mapped onto a lattice path model The lattice path model is defined on a rotated square lattice as follows: Assign the weight δ r δ l to each upward step Assign the weight 1 to each downward step except those steps which end on the horizontal axis. Z N = � L | T N | R � Z N = �� W | ( D + E ) N | V �� • F. H. Jafarpour and S. Zeraati, PRE 81 (2010). F. H. Jafarpour (BASU) MPIPKS, LAFNES11 7 / 19
PASEP mapped onto a lattice path model For an arbitrary d : x 1 − d = κ + ( β, δ ) κ + ( α, γ ) one can define a multiple-shock measure: ρ d . . · · · . ρ 2 ρ 1 1 i 1 i 2 i d − 1 N which shares a common partition function with a multiple-transit lattice path: �❅�❅ �❅�❅�❅ �❅�❅ ❅� ❅�❅� (0 , 0) (2 N , 0) • F. H. Jafarpour and S. Zeraati, PRE 82 (2010). F. H. Jafarpour (BASU) MPIPKS, LAFNES11 8 / 19
A simple disordered DDS A disordered Driven Diffusive System (DDS) defined on a lattice of length N consisting of M − 1 first-class particles in the presence of a second-class particle. 1 � � � 1 � � � � N 1 � � � 1 2 � � 3 � � � � � p 1 ∅ → ∅ 1 with rate 1 2 ∅ → ∅ 2 with rate p F. H. Jafarpour (BASU) MPIPKS, LAFNES11 9 / 19
A simple disordered DDS A disordered Driven Diffusive A Zero Range Process (ZRP) System (DDS) defined on a defined on a lattice of length lattice of length N consisting M consisting of N − M of M − 1 first-class particles in particles. The particles in the the presence of a second-class first box leave it with the rate particle. p . 1 � 1 p � � 1 � � 1 � � N 1 1 � � � 1 2 � � 3 1 2 3 4 5 � � � � � p 1 ∅ → ∅ 1 with rate 1 2 ∅ → ∅ 2 with rate p • M. R. Evans EPL (1996). F. H. Jafarpour (BASU) MPIPKS, LAFNES11 9 / 19
A simple disordered DDS Steady-state as a matrix product state 2 ∅ ∅ ∅ 1 ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ 1 · · · 1 ∅ ∅ ∅ ∅ ∅ 1 ���� � �� � � �� � n 1 n 2 n M 1 Tr ( D ′ E n 1 DE n 2 · · · DE n M ) P ( { n 1 , n 2 , · · · , n M } ) = Z N , M D ′ → 2 , D → 1 , E → ∅ F. H. Jafarpour (BASU) MPIPKS, LAFNES11 10 / 19
A simple disordered DDS Steady-state as a matrix product state 2 ∅ ∅ ∅ 1 ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ 1 · · · 1 ∅ ∅ ∅ ∅ ∅ 1 ���� � �� � � �� � n 1 n 2 n M 1 Tr ( D ′ E n 1 DE n 2 · · · DE n M ) P ( { n 1 , n 2 , · · · , n M } ) = Z N , M D ′ → 2 , D → 1 , E → ∅ These operators satisfy a quadratic algebra: pD ′ E = D ′ , DE = D . The matrix representation of this algebra is: ∞ ∞ ∞ � � � D ′ = p − i | 0 �� i | , D = | 0 �� i | , E = | i + 1 �� i | i =0 i =0 i =0 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 10 / 19
A simple disordered DDS Canonical partition function M � � n i − N + M ) Tr ( D ′ E n 1 DE n 2 · · · DE n M ) Z N , M ( p ) = δ ( i =1 { n i } � N − i − 2 N − M � � p − i = M − 2 i =0 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 11 / 19
A simple disordered DDS Canonical partition function M � � n i − N + M ) Tr ( D ′ E n 1 DE n 2 · · · DE n M ) Z N , M ( p ) = δ ( i =1 { n i } � N − i − 2 N − M � � p − i = M − 2 i =0 Two phases: Bose-Einstein condensation! Depending on the values of p and ρ = M N the system has two phases: O ( N ) for p < 1 − ρ, � n 1 � ≃ 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
A simple disordered DDS Question! Is there an equivalent lattice path model? F. H. Jafarpour (BASU) MPIPKS, LAFNES11 12 / 19
A simple disordered DDS Question! Is there an equivalent lattice path model? An equivalent lattice path model A lattice path model defined on Z 2 + = { ( 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 zp j . F. H. Jafarpour (BASU) MPIPKS, LAFNES11 12 / 19
A simple disordered DDS Question! Is there an equivalent lattice path model? An equivalent lattice path model A lattice path model defined on Z 2 + = { ( 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 zp j . j A transfer matrix T can be defined: 4 3 T | j � = zp j | 0 � + 1 p | j + 1 � 2 1 z ( z +1) N − j − 2 T N − 1 | 0 � = � N − 2 1 | j � + p N − 1 | N − 1 � j =0 p j 5 i 1 2 3 4 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 12 / 19
A simple disordered DDS The grand-canonical partition function of the model: � N − j − 2 ∞ N − 1 N − i − 1 � 1 � � � p − j z i + � j | T N − 1 | 0 � = Z N ( p , z ) = i − 1 p N − 1 j =0 i =1 j =0 F. H. Jafarpour (BASU) MPIPKS, LAFNES11 13 / 19
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