Metastability and discontinuous condensation in zero-range processes

Transcription

1 Metastability and discontinuous condensation in zero-range processes Stefan Großkinsky Warwick in collaboration with G.M. Schütz February 17, 2009

2 The zero-range process Lattice: Λ L = {1,..., L} with pbc State space: X L = {0, 1,..} Λ L η = (η x ) x ΛL Jump rates: g : {0, 1,..} [0, ) g(k) = 0 k = 0 1 p g 3 p [Spitzer (1970), Andjel (1982)] Introduction Results Conclusion ZRP Condensation Applications

3 The zero-range process Lattice: Λ L = {1,..., L} with pbc State space: X L = {0, 1,..} Λ L η = (η x ) x ΛL Jump rates: g : {0, 1,..} [0, ) g(k) = 0 k = 0 1 p g 3 p Conservation law: x Λ L η x (t) N, ρ = N/L [Spitzer (1970), Andjel (1982)] Introduction Results Conclusion ZRP Condensation Applications

4 Condensation g(k) = k independent particles (Poisson) g(k) = g server queues (geometric) Introduction Results Conclusion ZRP Condensation Applications

5 Condensation g(k) = k independent particles (Poisson) g(k) = g server queues (geometric) g x (k) = g x condensation on the slowest site [Evans (1996), Krug, Ferrari (1996), Benjamini, Ferrari, Landim (1996)] g(k) condensation on a random site [Evans (2000)] g(k) 1 + b k σ, σ (0, 1) ; σ = 1, b > 2 Introduction Results Conclusion ZRP Condensation Applications

6 Condensation g(k) = k independent particles (Poisson) g(k) = g server queues (geometric) g x (k) = g x condensation on the slowest site [Evans (1996), Krug, Ferrari (1996), Benjamini, Ferrari, Landim (1996)] g(k) condensation on a random site [Evans (2000)] g(k) 1 + b k σ, σ (0, 1) ; σ = 1, b > 2 ρ c < (ρ ρ c)l fluid phase ρ ρ c condensed phase ρ > ρ c Introduction Results Conclusion ZRP Condensation Applications

7 Condensation g(k) condensation on a random site [Evans (2000)] g(k) 1 + b k σ, σ (0, 1); σ = 1, b > 2 Continuous phase transition (e.g. σ = 1) 4 3 Ρ 2 fluid phase Ρ bulk Ρ Ρ c b condensed phase Ρ bulk Ρ c Introduction Results Conclusion ZRP Condensation Applications

8 Condensation g(k) condensation on a random site [Evans (2000)] g(k) 1 + b k σ, σ (0, 1); σ = 1, b > 2 Continuous phase transition (e.g. σ = 1) 4 Ρ c b fix b > 2 3 Ρ 2 1 fluid phase Ρ bulk Ρ condensed phase Ρ bulk Ρ c bulk density Ρ c fluid condensed Ρ c Introduction Results Conclusion ZRP Condensation Applications

9 Applications Condensation phenomena [Evans, Hanney (2005), Evans (2000)] bus route model, ant trails epitaxial growth (step flow model) network dynamics: rewiring (directed) networks glassy dynamics due to entropic barriers (backgammon model) network of server queues (M/M/1) Introduction Results Conclusion ZRP Condensation Applications

10 Applications Condensation phenomena [Evans, Hanney (2005), Evans (2000)] bus route model, ant trails epitaxial growth (step flow model) network dynamics: rewiring (directed) networks glassy dynamics due to entropic barriers (backgammon model) network of server queues (M/M/1) Phase separation in one-dimensional exclusion models [Kafri, Levine, Mukamel, Schütz, Török (2002)] ideal Bose gas Introduction Results Conclusion ZRP Condensation Applications

11 Applications Clustering in granular gases [van der Meer, van der Weele, Lohse, Mikkelsen, Versluis ( )] stilton.tnw.utwente.nl/people/rene/clustering.html Introduction Results Conclusion ZRP Condensation Applications

12 Applications Clustering in granular gases [van der Meer, van der Weele, Lohse, Mikkelsen, Versluis ( )] stilton.tnw.utwente.nl/people/rene/clustering.html Introduction Results Conclusion ZRP Condensation Applications

13 Applications Clustering in granular gases [van der Meer, van der Weele, Lohse, Mikkelsen, Versluis ( )] stilton.tnw.utwente.nl/people/rene/clustering.html Model: ZRP with g(k) = k N e 1/(T 0+ (1 k/n)) [Lipowski, Droz (2002), Coppex, Droz, Lipowski (2002)] [Török (2005), van der Meer, Reimann, Lohse (2007)] Introduction Results Conclusion ZRP Condensation Applications

14 ZRP with size-dependent jump rates { c0, k < R g R (k) =, k R c 1 where c 0 > c 1, L, N, R such that R/L a and N/L ρ c 0 g R k c 1 0 R [G., Schütz (2008)]

15 ZRP with size-dependent jump rates { c0, k < R g R (k) =, k R c 1 where c 0 > c 1, L, N, R such that R/L a and N/L ρ C F Ρ trans a F C Ρ c a Ρ F Ρ c F E [G., Schütz (2008)] 0 a

16 ZRP with size-dependent jump rates { c0, k < R g R (k) =, k R c 1 where c 0 > c 1, L, N, R such that R/L a and N/L ρ bulk density Ρ c F E F F C C F 0 Ρ c Ρ c a Ρ Ρ trans [G., Schütz (2008)]

17 Stationary measures Stationary weights factorize wr(η) L := w R (η x ) with w R (k) γ k g R (i) 1 x Λ L i k with γ > 0 arbitrary (choice of timescale)

18 Stationary measures Stationary weights factorize wr(η) L := w R (η x ) with w R (k) γ k g R (i) 1 x Λ L i k with γ > 0 arbitrary (choice of timescale) here we choose γ = c 0 and get { 1, k R w R (k) = (c 0 /c 1 ) k R, k > R

19 Stationary measures { 1, k R stationary weights w R (k) = (c 0 /c 1 ) k R, k > R canonical measures π L,N (η) = 1 Z L,N ( w R (η x ) δ Σx η x, N ) x Λ L

20 Stationary measures { 1, k R stationary weights w R (k) = (c 0 /c 1 ) k R, k > R canonical measures π L,N (η) = 1 Z L,N ( w R (η x ) δ Σx η x, N ) x Λ L grand-canonical measures fugacity φ 0 ν L φ,r (η) = 1 z R (φ) L x Λ L w R (η x ) φ ηx

21 Stationary measures { 1, k R stationary weights w R (k) = (c 0 /c 1 ) k R, k > R canonical measures π L,N (η) = 1 Z L,N ( w R (η x ) δ Σx η x, N ) x Λ L grand-canonical measures fugacity φ 0 ν L φ,r (η) = 1 z R (φ) L x Λ L w R (η x ) φ ηx z R (φ) = k=0 w R(k) φ k < for φ < c 1 /c 0

22 Stationary measures z R (φ) = 1 1 φ + φr( φ 1 + φ + φ ), φ < c 1 /c 0 c 1 /c 0 φ log z R Φ log z R L 2,4,8 0 c 1 c 0

23 Stationary measures z R (φ) = 1 1 φ + φr( φ 1 + φ + φ ), φ < c 1 /c 0 c 1 /c 0 φ log z R Φ log z R L 2,4,8 0 c 1 c 0 density ρ R (φ) := η x νφ,r = φ φ log z R (φ) ρ R (φ) as φ c 1 /c 0 φ R (ρ)

24 Equivalence of ensembles Specific relative entropy h ( π L,N, νφ,r L ) 1 := L π L,N (η) log π L,N(η) ν L η X L,N φ (η)

25 Equivalence of ensembles Specific relative entropy h ( π L,N, νφ,r L ) 1 := L π L,N (η) log π L,N(η) ν L η X L,N φ (η) = log z R (φ) N L log φ 1 L log Z L,N

26 Equivalence of ensembles Specific relative entropy h ( π L,N, νφ,r L ) 1 := L π L,N (η) log π L,N(η) ν L η X L,N φ (η) = log z R (φ) N L log φ 1 L log Z L,N s gcan (ρ) s can (ρ) as L, N, N/L ρ, provided φ = φ R (ρ)

27 Equivalence of ensembles Specific relative entropy h ( π L,N, νφ,r L ) 1 := L π L,N (η) log π L,N(η) ν L η X L,N φ (η) = log z R (φ) N L log φ 1 L log Z L,N s gcan (ρ) s can (ρ) as L, N, N/L ρ, provided φ = φ R (ρ) entropy s gcan (ρ) := sup φ<c 1 /c 0 ( ρ log φ p(φ) ) pressure p(φ) := lim L log z R(φ) = log(1 φ), φ < c 1 /c 0

28 Pressure and entropy { log(1 φ), φ < c1 /c pressure p(φ) = 0, φ c 1 /c 0 p Φ log z R Φ log z R L 2,4,8 log 1 Φ p Φ 0 c 1 c 0

29 Pressure and entropy { log(1 φ), φ < c1 /c pressure p(φ) = 0, φ c 1 /c 0 p Φ s gcan log z R Φ log z R L 2,4,8 log 1 Φ entropy s Ρ s Ρ c s fluid p Φ 0 c 1 c 0 0 Ρ c where s fluid (ρ) = (1+ρ) log(1+ρ) ρ log ρ, ρ c = 1 c 0 /c 1 1

30 Canonical entropy h ( π L,N, νφ L ) R (ρ),r s gcan (ρ) s can (ρ) }{{} 1 lim L L log Z L,N

31 Canonical entropy h ( π L,N, νφ L ) R (ρ),r s gcan (ρ) s can (ρ) }{{} 1 lim L L log Z L,N Main Result where s can (ρ) = { sfluid (ρ), ρ ρ trans (a) s cond (ρ, ρ c ), ρ > ρ trans (a) 1 s cond (ρ, ρ c ) = s fluid (ρ c ) + lim L L log w ( R (ρ ρc )L )

32 Canonical entropy h ( π L,N, νφ L ) R (ρ),r s gcan (ρ) s can (ρ) }{{} 1 lim L L log Z L,N Main Result where s can (ρ) = { sfluid (ρ), ρ ρ trans (a) s cond (ρ, ρ c ), ρ > ρ trans (a) 1 s cond (ρ, ρ c ) = s fluid (ρ c ) + lim L L log w ( R (ρ ρc )L ) = s fluid (ρ c ) + (ρ ρ c a) log c 0 /c 1.

33 Canonical entropy Recursion relation Z L,N = Z 1,N = N w R (k) Z L 1,N k... k=0 N w R (k) k=0

34 Canonical entropy 3.0 c 0 1,c 1 0.5,a 1 s gcan Ρ 2.5 s fluid Ρ 2.0 s can Ρ 1.5 s cond Ρ,Ρ c Ρ c Ρ c a 3 Ρ trans 4

35 Phase diagram 3.0 c 0 1,c 1 0.5,a 1 s gcan Ρ 2.5 s fluid Ρ 2.0 s can Ρ 1.5 s cond Ρ,Ρ c Ρ c Ρ c a 3 Ρ trans 4 Ρ Ρ c 0 C F Ρ trans a a F C F F E Ρ c a

36 Phase diagram 3.0 c 0 1,c 1 0.5,a 1 s gcan Ρ 2.5 s fluid Ρ 2.0 s can Ρ 1.5 s cond Ρ,Ρ c Ρ c Ρ c a 3 Ρ trans 4 Ρ Ρ c 0 C F Ρ trans a a F C F F E Ρ c a a = ρ trans ρ c ( s fluid (ρ trans ) s fluid (ρ c ) )/ log c 0 c 1

37 Phase diagram bulk density Ρ c F E F F C C F Ρ C F Ρ trans a F C F Ρ c a Ρ c F E 0 Ρ c Ρ c a Ρ Ρ trans 0 a a = ρ trans ρ c ( s fluid (ρ trans ) s fluid (ρ c ) )/ log c 0 c 1

38 Metastability Number of particles in the background (bulk): Σ bg L (η) := Σ xη x max x Λ L η x, ρ bulk (η) = 1 L Σbg L (η)

39 Metastability Number of particles in the background (bulk): Σ bg L (η) := Σ xη x max x Λ L η x, ρ bulk (η) = 1 L Σbg L (η) Second result 1 L log π ( L,N {Σ bg L = S L} ) I ρ (ρ bg ) [0, ] as L, N/L ρ, S L /L ρ bg,

40 Metastability Number of particles in the background (bulk): Σ bg L (η) := Σ xη x max x Λ L η x, ρ bulk (η) = 1 L Σbg L (η) Second result 1 L log π ( L,N {Σ bg L = S L} ) I ρ (ρ bg ) [0, ] as L, N/L ρ, S L /L ρ bg, where for ρ bg ρ I ρ (ρ bg ) = s can (ρ) s fluid (ρ bg ) { (ρ a ρbg ) log c 0 c 1, ρ bg ρ a 0, ρ bg ρ a

41 Metastability Ρ 1.2 Ρ 2 IΡ Ρ bg Ρ Ρ c a Ρ c Ρ c a 2 Ρ trans Ρ bg

42 Metastability Ρ 2 Ρ 3 IΡ Ρ bg Ρ Ρ trans Ρ c Ρ c a 2 Ρ trans Ρ bg

43 Life-times Σ bg ( ) L η(t) is a simple random walk (non-markovian) with stationary large deviation function I ρ

44 Life-times Σ bg L ( η(t) ) is a simple random walk (non-markovian) with stationary large deviation function I ρ Life-times τ L exp(lξ) (Arrhenius law) Ρ 2 Ρ 3 IΡ Ρ bg Ρ Ρ trans Ρ c Ρ c a 2 Ρ trans Ρ bg

45 Life-times Σ bg ( ) L η(t) is a simple random walk (non-markovian) with stationary large deviation function I ρ Life-times τ L exp(lξ) (Arrhenius law) ξ fluid (ρ) = s fluid (ρ) s fluid (ρ a) ξ cond (ρ) = s fluid (ρ c ) s fluid (ρ a) + (ρ a ρ c ) log c 0 c 1

46 Life-times Ξ 0.4 Ξ fluid Ρ Ξ cond Ρ Ρ c a 2 Ρ trans Ρ

47 Life-times Τ L Ρtrans condensed fluid L P Τ L Τ L t Exp t 0.30 L t ρ = ρ trans

48 Summary Results first order transition for size-dep. jump rates metastable phases, ergodicity breaking iidrv s with distribution depending on number of rv s Introduction Results Conclusion

49 Summary Results first order transition for size-dep. jump rates metastable phases, ergodicity breaking iidrv s with distribution depending on number of rv s Work in progress use current matching heuristics for more general systems relevant for granular clustering, including coarsening prove Arrhenius law? Introduction Results Conclusion

50 Dependence on the number of particles g ( η x, Σ L (η) ) = { c0, η x aσ L (η) c 1, η x > aσ L (η), a [0, 1) Introduction Results Conclusion

51 Dependence on the number of particles g ( η x, Σ L (η) ) = { c0, η x aσ L (η) c 1, η x > aσ L (η), a [0, 1) Ρ C F Ρ trans a Ρ c a F C Ρ meta a 2 F E Ρ c a Introduction Results Conclusion

52 Dependence on the number of particles g ( η x, Σ L (η) ) = { c0, η x aσ L (η) c 1, η x > aσ L (η), a [0, 1) 0.7 c 0 2, c 1 1, a s gcan Ρ s can 0.4 s flluid Ρ c s cond Ρ s fluid Ρ Ρ meta Ρ c a 2 Ρ trans 3 4 Ρ Introduction Results Conclusion

53 Heuristics matching currents: j fluid (ρ bg ) = φ(ρ bg ) = c 0 ρ bg 1 + ρ bg Introduction Results Conclusion

54 Heuristics ρ bg matching currents: j fluid (ρ bg ) = φ(ρ bg ) = c ρ bg j cond (ρ, ρ bg ) := lim g ( R (ρ ρbg ) L ) L Introduction Results Conclusion

55 Heuristics ρ bg matching currents: j fluid (ρ bg ) = φ(ρ bg ) = c ρ bg j cond (ρ, ρ bg ) := lim g ( R (ρ ρbg ) L ) L c 0 Φ cond Ρ,Ρ bg Φ c 1 Φ Ρ bg 0 Ρ c Ρ a Ρ Ρ bg Introduction Results Conclusion

56 Another example g L (k) = (k/l) + 1 { 1, as k 2, as L Introduction Results Conclusion

57 Another example g L (k) = (k/l) + 1 { 1, as k 2, as L Φ 1.0 Φ cond Ρ meta,ρ bg Φ cond 5,Ρ bg 0.5 Φ Ρ bg Ρ bg 5 Ρ bg Ρ crit Ρ meta 5 Ρ bg Introduction Results Conclusion

58 Another example g L (k) = (k/l) + 1 { 1, as k 2, as L 1.0 s gcan Ρ 0.5 s fluid Ρ s 0.0 can s flluid Ρ bg s cond Ρ,Ρ bg Ρ c 2 3 Ρ trans Ρ meta Ρ Introduction Results Conclusion