The asymmetric KMP model

Transcription

1 June 14, 2017

2 Joint work with G. Carinci (Delft) C. Giardinà (Modena) T. Sasamoto (Tokyo)

3 Outline 1) Symmetric case: how KMP is connected to other processes SIP, BEP (via so-called thermalization), and how this leads immediately to a one-parameter family KMP(α). 2) Duality of KMP (and all other dualities between the models of this family) follows from self-duality of SIP(α). 3) Self-duality of SIP(α) in turn follows from its algebraic structure and consequent symmetries (commuting operators): the generator of SIP is the co-product of the Casimir in U (SU(1, 1)) (in a discrete representation).

4 4) To find the correct asymmetric SIP(α) (ASIP(q, α)), this algebraic construction has now to be performed in U q (SU(1, 1)). Built in the construction are symmetries and self-duality (comparable to Schütz self-duality of ASEP). 5) From ASIP(q, α) with weak asymmetry q = 1 σ N, we find a diffusion limit (many particle limit) called ABEP(q, α). 6) This ABEP(q, α) process then yields AKMP(σ, α) via thermalization. The AKMP(σ, α) has the same dual as KMP(α), all the asymmetry is put into the duality function.

5 The KMP process The KMP (Kipnis, Marchioro, Presutti, J. Stat. Phys. 1982) process on a (finite) graph (S, E) is a Markov process {X (t), t 0} on [0, ) S (energies associated to vertices) described as follows 1. Every edge is selected with rate 1 (independently for different edges) 2. If the edge e = (ij), i, j S is selected, then the energies x i, x j associated to the vertices of the edge are replaced by ɛ(x i + x j ), (1 ɛ)(x i + x j ) with ɛ uniformly distributed on [0, 1] (every time of updating independently chosen).

6 The discrete KMP process The dkmp process on a (finite) graph (S, E) is a Markov process {η(t), t 0} on [0, ) S (particle numbers associated to vertices) described as follows 1. Every edge is selected with rate 1 (independently for different edges) 2. If the edge e = (ij), i, j S is selected, then the particle numbers η i, η j associated to the vertices of the edge are replaced by k e, η i + η j k e where k e is (discrete) uniformly distributed on {0, 1, 2,..., η i + η j } (every time of updating independently chosen).

7 Duality of KMP and dkmp Putting D(η, x) = i S x η i i η i! We have the duality E dkmp η Which implies e.g. D(η(t), x) = E KMP x D(η, X (t)) E KMP x (X i (t)) = j p t (i, j)x j where p t (i, j) is the transition probability for continuous-time rate 1 simple random walk on (S, E).

8 Self-duality of dkmp The dkmp is self-dual: putting D(ξ, η) = η i!γ(1) (η i ξ i )!Γ(1 + ξ i ) = i S i ( ηi ξ i ) We have E dkmp ξ D(ξ(t), η) = E dkmp η D(ξ, η(t))

9 Further relations between dkmp and KMP The KMP is the many particle limit of dkmp. Taking η i = x i N in dkmp and denoting ηt N its time-evolution under dkmp, we have, when N η N (t) N X (t) with X i (0) = x i The duality between dkmp and KMP can thus be derived from the self-duality of dkmp via lim N 1 N ξ i ( ) xi N = x ξ i i ξ i! ξ i

10 A one-parameter family of KMP models Given α > 0 we define KMP(α) as the Markov process {X (t), t 0} on [0, ) S (energies associated to vertices) described as follows 1. Every edge is selected with rate 1 (independently for different edges) 2. If the edge e = (ij), i, j S is selected, then the energies x i, x j associated to the vertices of the edge are replaced by ɛ(x i + x j ), (1 ɛ)(x i + x j ) with ɛ Beta(α, α) distributed on [0, 1] (every time of updating independently chosen).

11 The discrete KMP process The dkmp(α) process on a (finite) graph (S, E) is a Markov process {η(t), t 0} on [0, ) S (particle numbers associated to vertices) described as follows 1. Every edge is selected with rate 1 (independently for different edges) 2. If the edge e = (ij), i, j S is selected, then the particle numbers η i, η j associated to the vertices of the edge are replaced by k e, η i + η j k e where k e is (discrete) Beta(α, α) binomial distributed on {0, 1, 2,..., η i + η j } (every time of updating independently chosen). Beta Binomial is defined via ( ) ηi + η j P(k e = n) = E(p n (1 p) η i +η j n ) n where E is w.r.t. p according to Beta(α, α) distribution.

12 Self-duality of dkmp(α) The dkmp(α) is also self-dual: putting then we have E dkmp(α) ξ D(ξ, η) = i S η i!γ(α) (η i ξ i )!Γ(α + ξ i ) D(ξ(t), η) = E dkmp(α) D(ξ, η(t)) from this we can derive, as before, duality of KMP(α) and dkmp(α) with D(η, x) = x η i i Γ(α) Γ(α + η i ) i η

13 Thermalization For a process on X S (X = [0, ) or X = N) with generator of type L = e E L e we define its thermalization as T (L) := L = e E L e with L e f = lim t (e tle I )f Notice that this is a kind of projection, i.e., T (T (L)) = T (L)

14 Relation between SIP(α) and dkmp(α) In the SIP(α) only one particle jumps at a time and a particle hops from i to j (if ij E) at rate r(η i, η j ) = η i (α + η j ) So the generator reads L SIP(α) = [ r(ηi, η j )(f (η ij ) f (η)) + r(η j, η i )(f (η ji ) f (η)) ] e=ij E We then have L dkmp(α) = T (L SIP(α) ) i.e., dkmp(α) is the thermalization of SIP(α).

15 Self-duality of SIP(α) then we have D(ξ, η) = i S E SIP(α) ξ η i!γ(α) (η i ξ i )!Γ(α + ξ i ) D(ξ(t), η) = E SIP(α) D(ξ, η(t)) This self-duality is the source duality from which all the others follow (by taking many particle limits or thermalizations) η

16 Brownian energy process BEP(α) If one takes the many particle limit η i = Nx i in the SIP(α) we obtain a process of diffusion type with generator L BEP(α) = [ xi x j ( i j ) 2 2α(x i x j )( i j ) ] ij=e E From self-duality of SIP(α), one infers duality of this process with SIP(α) with D(η, x) = x η i i Γ(α) Γ(α + η i ) i S Moreover, the thermalization of this process is the process KMP(α).

17 Self-duality and symmetries The self-duality of SIP(α) follows from its algebraic structure. The self-duality of a process with generator L can (in most cases) be summarized via L left D(ξ, η) = L right D(ξ, η) We denote this by L D L In the finite state space case this relation reads in matrix form LD = DL T

18 The following fact connects symmetries with self-duality functions: if S commutes with L, i.e., if then implies [S, L] = SL LS = 0 L D L L S leftd L i.e., from a given self-duality function and a symmetry one can produce a new self-duality function.

19 A cheap self-duality function is given by D cheap (ξ, η) = 1 µ(ξ) δ ξ,η where µ is a reversible measure. Other, more useful self-dualities can then be made by acting with symmetries on this one (provided we have symmetries). In this sense, self-duality can be viewed as a generalization of reversibility (from diagonal to non-diagonal D).

20 Symmetries of the SIP generator The single edge generator of SIP(α) is [ r(ηi, η j )(f (η ij ) f (η)) + r(η j, η i )(f (η ji ) f (η)) ] where we remind r(k, n) = k(α + n). In order to discover its symmetries, we have to go to its algebraic structure

21 We introduce the following operators working on functions f : N R. K + f (n) = (α + n)f (n + 1) These operators K +, K, K 0 satisfy K f (n) = nf (n 1) K 0 f (n) = ( α 2 + n) f (n) (1) [K ±, K 0 ] = ±K ±, [K +, K ] = 2K 0 (2) These are the commutation relations of the dual algebra of U (SU(1, 1)) (the commutation relations of U (SU(1, 1)) being the same with opposite signs, i.e. [K 0, K ± ] = ±K ±, [K, K + ] = 2K 0 ).

22 In terms of these operators the single edge generator of SIP(α) reads L 12 = K 1 + K 2 + K 1 K 2 + 2K 1 0 K2 0 + α2 (3) 2 This operator L 12 is naturally related to a distinguished central element of U (SU(1, 1), C = (K 0 ) (K + K + K K + ) (4) the so-called Casimir element. This is the reason that this operator has many symmetries.

23 First we define the co-product on the generating elements: for u {+,, 0} (K u ) = K u I + I K u = K u 1 + K u 2 (5) and extend to a homomorphism between the algebras A and A A. : A A A is then called coproduct. It has the property (co-associativity) ( I ) = (I ) which allows to consider iterated coproducts, e.g., 2 : A A A A 2 (K u ) = ( I ) (K u ) = K u 1 + K u 2 + K u 3, u {, +, 0}

24 We have ( C) = (K + 1 K 2 + K + 2 K 1 ) 2K 0 1 K 0 2 C 1 C 2 (6) As a consequence, the generator L 12 commutes with (A) for every algebra-element (because C is central and preserves commutators). In particular L 12 commutes with K u 1 + K u 2, u {0, +, } These symmetries are responsible for the self-duality of SIP(α): D = e K + 1 +K + 2 Dcheap Taking the exponential is natural because we want factorized (over vertices) self-dualities.

25 Summary so far The generator (on two edges) of the SIP(α) is the coproduct applied to the Casimir operator (in the discrete representation). As a consequence, the generator (on two edges) of the SIP(α) has many commuting elements (symmetries). The self-duality of SIP(α) follows immediately from the application of a symmetry (e K + 1 +K + 2 ) on a trivial self-duality function coming from the reversible product measure. All dualities and self-dualities of processes related to SIP(α) (BEP(α), dkmp(α), KMP(α)) follow from this self-duality of SIP(α), and taking limits and or thermalizations.

26 The asymmetric inclusion process Now we start from deformed algebra U q (SU(1, 1)) with commutation relations [K +, K ] = [2K 0 ] q, [K 0, K ± ] = ±K ± 0 < q < 1 is the parameter tuning the asymmetry. q-numbers are defined via [n] q = qn q n q q 1 The Casimir element of U q (SU(1, 1)) is given by C = [K 0 ] q [K 0 1] q K + K and the coproduct on the generating elements is given by iterated coproducts via (K ± ) = K ± q K 0 + q K 0 K ± (K 0 ) = K 0 I + I K 0 n = ( I )( n 1 )

27 We now have to start from the q-deformed version U q (SU(1, 1)), and apply the same strategy: Copy the coproduct of the Casimir along the edges (i, i + 1) of the finite graph {1, 2,..., L}. This gives an operator of the form L 1 H = i=1 h i,i+1 which is not yet a Markov generator, but of the form Hg = Lg ϕg i.e., a Markov generator minus a multiplication operator.

28 Turn the Hamiltonian operator thus obtained into a generator via a ground-state transformation : if He f = 0 (positive groundstate) then L g = e f H(e f g) is a Markov generator. The symmetries of H are in one-to-one correspondence with the symmetries of L. The analogue of the exponential symmetries e i K i u are a well-chosen q-deformed exponential of (L 1) (K u ). These symmetries then yield the self-dualities of the process with generator L

29 Explicitly, we have the generator of SIP(q, α) is given by L = i with L i,i+1 L i,i+1 f (η) = q η i η i+1 +(α 1) [η i ] q [α + η i+1 ] q (f (η i,i+1 ) f (η)) + q η i η i+1 (α 1) [η i+1 ] q [α + η i ] q (f (η i+1,i ) f (η)) This process is self-dual with self-duality functions n D(ξ l 1,...,l n, η) = q 2α m=1 n lm n2 q α q α (q 2N lm (η) q 2Nlm+1(η) ) m=1 where ξ l 1,...,l n denotes the configuration with particles at the n different location l 1,..., l m, and N i (η) = L j=i is the number of particles to the right of i. η j

30 The ABEP(σ, α) Now we take the limit q = 1 σ N (weak asymmetry), η i = Nx i (many particles) in the ASIP(q, α) and we find a diffusion (in limit N ) process called ABEP(σ, α) with generator L 1 L = i=1 L i,i+1 L i,i+1 = 1 ( ) ( 1 e 2σx i e 2σx i+1 4σ 2 1 ) ( i i+1 ) 2 1 (( ) ( 1 e 2σx i e 2σx i+1 1 ) + 2α(2 e 2σx i e 2σx i+1 ) ) 2σ ( i i+1 )

31 Duality of ABEP(σ, α) From the self-duality of ASIP(q, α) we obtain duality between ABEP(σ, α) and SIP(α) with duality functions D σ (ξ, x) = L i=1 ( Γ(α) e 2σEi+1(x) e 2σE i (x) Γ(α + ξ i ) 2σ ) ξi with E i (x) = L j=i x i is the total energy to the right of i. i.e., in the dual process, the asymmetry is disappearing, and the only trace of the asymmetry is in the duality function. So this is an example of a truly bulk-asymmetric process dual to a symmetric process. Example Ex ABEP(σ,α) (e 2σJ i (x(t)) ) = k p t (i, k)e 2σ(E k(x) E i (x))

32 The AKMP(σ, α) The AKMP(σ, α) is then defined as the thermalization of ABEP(σ, α) This gives the following process: the energies of every edge are (at rate 1) updated according to (x i, x i+1 ) (B (x i +x i+1 ) σ (x i + x i+1 ), (1 B (x i +x i+1 ) σ )(x i + x i+1 )) with B E σ a random variable on [0, 1] with probability density f B E σ = C 1 E,σ,α e2σew ((e 2σEw 1)(1 e 2σE(1 w) )) α 1 C E,σ,α = 1 0 e 2σEw ((e 2σEw 1)(1 e 2σE(1 w) )) α 1 dw which is the asymmetric analogue of the Beta(α, α) distribution in KMP(α).

33 This AKMP(σ, α) is dual to the dkmp(α) with the duality functions ( ) L D σ Γ(α) e 2σEi+1(x) e 2σE ξi i (x) (ξ, x) = Γ(α + ξ i ) 2σ i=1 Two open questions For SIP(α) we can characterize all self-duality functions among which there are also orthogonal polynomials (Franceschini, Giardinà; R., Sau). Can this be done also in the asymmetric case? Are there correct reservoirs for ABEP(σ, α) (or AKMP(σ, α)) such that the dual has absorbing boundaries?

34 Thanks for your attention!