Large deviations of the density and of the current in non equilibrium steady states
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- Ήφαιστος Ζερβός
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1 Large deviaions of he densiy and of he curren in non equilibrium seady saes Bernard DERRIDA Universié Pierre e Marie Curie and Ecole Normale Supérieure, Paris June 2012 Lyon
2 COLLABORATORS J.L. Lebowiz, E.R. Speer C. Enaud B. Douço, P.-E. Roche T. Bodineau C. Apper, V. Lecome, F. van Wijland A. Gerschenfeld, E. Brune
3 Ouline Densiy ucuaions in non-equilibrium sysems Curren ucuaions Open quesions Non seady sae siuaions Deerminisic dynamics
4 Non equilibrium seady saes: curren of paricles
5 Non equilibrium seady saes: curren of paricles ρ a Sysem ρ b
6 Non equilibrium seady saes: curren of hea T a Sysem T b
7 Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) L
8 Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) L P(Q) = P( Q)
9 Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) P(Q) = P( Q) L P(Q) P( Q) exp [ Q ( 1 1 )] kt b kt a Flucuaion Theorem
10 Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) P(Q) = P( Q) L P(Q) P( Q) exp [ Q ( 1 1 )] kt b kt a Flucuaion Theorem Q 2 1 L Q A(T a, T b ) L Fourier's law
11 Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) P(Q) = P( Q) L P(Q) P( Q) exp [ Q ( 1 1 )] kt b kt a Flucuaion Theorem Q 2 1 L P(Q)? Q A(T a, T b ) L Fourier's law
12 Q Ta T b L Equilibrium (T a = T b ) Non-equilibrium seady sae (T a T b )
13 Q Ta T b L Equilibrium (T a = T b ) P(C) exp [ E(C) ] kt Non-equilibrium seady sae (T a T b ) P(C) =? Densiy ucuaions
14 Q Ta T b L Equilibrium (T a = T b ) P(C) exp [ E(C) ] kt Non-equilibrium seady sae (T a T b ) P(C) =? Densiy ucuaions Shor range correlaions Long range correlaions
15 Q Ta T b L Equilibrium (T a = T b ) P(C) exp [ E(C) ] kt Non-equilibrium seady sae (T a T b ) P(C) =? Densiy ucuaions Shor range correlaions Time symmery of ucuaions Long range correlaions Time asymmery of ucuaions
16 Densiy ucuaions a equilibrium Saring from S = k log Ω Einsein n N paricles n paricles in volume v v N = Vρ N n paricles in v V
17 Densiy ucuaions a equilibrium Saring from S = k log Ω Einsein n N paricles n paricles in volume v v N = Vρ n 2 n 2 = kt vρ κ V N n paricles in v κ is he compressibiliy, ρ he densiy in he reservoir and T he emperaure.
18 Densiy ucuaions a equilibrium Saring from S = k log Ω Einsein n N paricles n paricles in volume v v N = Vρ n 2 n 2 = kt vρ κ V N n paricles in v κ is he compressibiliy, ρ he densiy in he reservoir and T he emperaure. Variance of he ucuaion = k Response coecien Bolzmann Consan k J/K
19 Large deviaions of he densiy n v N paricles N = Vρ ( n ) Pro v = r exp[ v a(r)] N n paricles in v V a(r) a(r) is he large deviaion funcion ρ r A equilibrium a(r) is he free energy
20 Large deviaion funcional ρ 1 ρ 2 ρ k L Pro(ρ 1,...ρ k ) exp [ L d F(ρ 1,...ρ k ) ] Large number k of boxes r = L x Pro({ρ( x)}) exp [ L d F({ρ( x)}) ]
21 Exclusion processes SSEP (Symmeric simple exclusion process) α γ L ρ a = α α + γ, ρ b = δ β + δ β δ
22 Large deviaion funcion for he SSEP Pro({ρ(x)}) exp[ LF({ρ(x)})] Equilibrium ρ a = ρ b = F F({ρ(x)}) = 1 0 dx [ (1 ρ(x)) log 1 ρ(x) 1 F + ρ(x) log ρ(x) F ]
23 Large deviaion funcion for he SSEP Pro({ρ(x)}) exp[ LF({ρ(x)})] Equilibrium ρ a = ρ b = F F({ρ(x)}) = 1 0 dx [ (1 ρ(x)) log 1 ρ(x) 1 F + ρ(x) log ρ(x) F ] Non-equilibrium (ρ a ρ b ) D Lebowiz Speer Berini De Sole Gabrielli Jona-Lasinio Landim 2002 F = sup F (x) [ dx (1 ρ(x)) log 1 ρ(x) ρ(x) + ρ(x) log 1 F (x) F (x) + log F ] (x) ρ b ρ a wih F (x) monoone, F (0) = ρ a and F (1) = ρ b
24 Non-equilibrium ρ a ρ b F = sup F (x) [ dx (1 ρ(x)) log 1 ρ(x) ρ(x) + ρ(x) log 1 F (x) F (x) + log F ] (x) ρ b ρ a F is non-local: for example for small ρ a ρ b F({ρ(x)}) = Z 1 + (ρ a ρ b ) 2 (ρ a ρ 2 a) 2 + O(ρ a ρ b ) 3 dx (1 ρ(x)) log 1 ρ(x) 1 ρ (x) 0 Z 1 Z 1 dx 0 x where ρ (x) = ρ(x) =(1 x)ρ a + xρ b Long-range correlaions Spohn 82 + ρ(x) log ρ(x) ρ (x) dy x(1 y)`ρ(x) ρ (x) `ρ(y) ρ (y) ρ(x)ρ(y) ρ(x) ρ(y) 1 G(x, y) = (ρ a ρ b ) 2 x(1 y) L L
25 Large deviaion funcional Pro({ρ(x)}) exp[ Acion] = exp [ LF({ρ(x)})] Equilibrium 1. F local 2. F = T 1 d x f (ρ( x)) f is he free energy per uni volume 3. Shor range correlaions
26 Large deviaion funcional Pro({ρ(x)}) exp[ Acion] = exp [ LF({ρ(x)})] Equilibrium 1. F local 2. F = T 1 d x f (ρ( x)) f is he free energy per uni volume 3. Shor range correlaions Non-equilibrium 1. F non local 2. Weak long range correlaions (SSEP for x < y) Spohn 1982 (ρ(x) ρ (x))(ρ(y) ρ (y)) 1 L G(x, y) = (ρ a ρ b ) 2 x(1 y) L 3. Higher correlaions ρ(x 1 )ρ(x 2 ))...ρ(x n ) c L 1 n
27 TWO APPROACHES SSEP (Symmeric simple exclusion process) α β γ 1 L δ Microscopic Behe ansaz, Perurbaion heory, Compuer,... Macroscopic i = Lx, = L 2 τ Pro({ρ(x, τ), j(x, τ)}) exp [ L T /L 2 0 d 1 0 ] dx [j + ρ ] 2 4ρ(1 ρ)
28 Marix ansaz Seady sae of he SSEP α Fadeev 1980,..., D Evans Hakim Pasquier 1993 β γ 1 L δ P(τ 1,..., τ L ) = W X 1... X L V W (D + E) L V where X i = { D E if sie i occupied if sie i empy
29 Marix ansaz Seady sae of he SSEP α Fadeev 1980,..., D Evans Hakim Pasquier 1993 β γ 1 L δ P(τ 1,..., τ L ) = W X 1... X L V W (D + E) L V where X i = { D E if sie i occupied if sie i empy W (αe γd) = W DE ED = D + E (βd δe) V = V
30 Large diusive sysems ρ a ρ b A diusive sysem For ρ a ρ b small: Q = D(ρ) (ρ a ρ b ) L Fick's law ρ a = ρ b = ρ : Q 2 = σ(ρ) L
31 Large diusive sysems ρ a ρ b A diusive sysem For ρ a ρ b small: Q = D(ρ) (ρ a ρ b ) L Fick's law ρ a = ρ b = ρ : Q 2 = σ(ρ) L Noe ha σ(ρ) = 2kT ρ 2 κ(ρ)d(ρ) where κ is he compressibiliy
32 Macroscopic ucuaion heory Onsager Machlup heory for non equilibrium diusive sysems Kipnis Olla Varadhan 89 Spohn 91. Berini De Sole Gabrielli Jona-Lasinio Landim 2001 Evoluion of a prole ρ(x, ) for 0 T Pro({ρ(x, ), j(x, )}) [ exp L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) dρ d = dj dx (conservaion law) ρ(0, ) = ρ a ; ρ(1, ) = ρ b
33 Macroscopic ucuaion heory Pro({ρ(x, ), j(x, )}) [ exp L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) j(x, ) = ρ (x, )D(ρ(x, )) + 1 L η(x, )
34 Macroscopic ucuaion heory Pro({ρ(x, ), j(x, )}) [ exp L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) j(x, ) = ρ (x, )D(ρ(x, )) + 1 L η(x, ) wih he whie noise η(x, )η(x, ) = 2σ(ρ(x, ))δ(x x )δ( )
35 dρ d = dj dx ; ρ(0, ) = ρ a ; ρ(1, ) = ρ b Pro({ρ(x, ), j(x, )}) exp [ Acion] = exp [ L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) How does a ucuaion ρ(x, 0) relax?
36 dρ d = dj dx ; ρ(0, ) = ρ a ; ρ(1, ) = ρ b Pro({ρ(x, ), j(x, )}) exp [ Acion] = exp [ L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) How does a ucuaion ρ(x, 0) relax? Acion = 0 dρ d = (D(ρ(x, ))ρ(x, ) )
37 dρ d = dj dx ; ρ(0, ) = ρ a ; ρ(1, ) = ρ b Pro({ρ(x, ), j(x, )}) exp [ Acion] = exp [ L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) How does a ucuaion ρ(x, 0) relax? Acion = 0 dρ d = (D(ρ(x, ))ρ(x, ) ) How is a ucuaion ρ(x, 0) produced?
38 dρ d = dj dx ; ρ(0, ) = ρ a ; ρ(1, ) = ρ b Pro({ρ(x, ), j(x, )}) exp [ Acion] = exp [ L T /L 2 0 d 1 0 ] dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) How does a ucuaion ρ(x, 0) relax? Acion = 0 dρ d = (D(ρ(x, ))ρ(x, ) ) How is a ucuaion ρ(x, 0) produced? Minimize he Acion wih ρ(x, ) = ρ seady sae (x)
39 Macroscopic ucuaion heory Berini De Sole Gabrielli Jona-Lasinio Landim ρ a ρ(x,) ρ(x) ρ*(x) ρ b 0 x 1 Z 0 F({ρ(x)}) = min d ρ(x,) Z 1 0 dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) wih ρ(x, 0) = ρ(x) ; ρ(x, ) = ρ (x)
40 SSEP How does a ucuaion ρ(x, 0) relax? dρ(x, ) d = d 2 ρ(x, ) dx 2 How is a ucuaion ρ(x, 0) produced? dρ(x, s) ds = d 2 ρ(x, s) 2 (ρ a ρ b ) dρ(x, s) (1 2ρ(x, s)) +O ( (ρ dx 2 a ρ b ) 2) ρ a (1 ρ a ) dx
41 Curren ucuaions in non-equilibrium seady saes T a T b
42 Curren ucuaions in non-equilibrium seady saes T a T b Curren ucuaions on a ring T
43 AVERAGE CURRENT OF HEAT Q Ta L T b Q = F (T a, T b, L, conacs)
44 AVERAGE CURRENT OF HEAT Q Ta L T b Q = F (T a, T b, L, conacs) FOURIER's law: large L Q G(T a, T b ) L large L and T a T b small Q D(T ) T a T b D(T ) T L
45 AVERAGE CURRENT OF PARTICLES Q ρa L ρ b Q = F (ρ a, ρ b, L, conacs)
46 AVERAGE CURRENT OF PARTICLES Q ρa L ρ b Q = F (ρ a, ρ b, L, conacs) FICK's law: large L Q G(ρ a, ρ b ) L large L and ρ a ρ b small Q D(ρ) ρ a ρ b D(ρ) ρ L
47 ONE DIMENSIONAL MECHANICAL SYSTEMS Ideal gas No Fourier's law Harmonic chain E = p 2 i 2m + i g(x i+1 x i ) 2 Q G(T a, T b )
48 ONE DIMENSIONAL MECHANICAL SYSTEMS Ideal gas No Fourier's law Harmonic chain E = p 2 i 2m + i g(x i+1 x i ) 2 Q G(T a, T b ) Hard paricle gas Anomalous Fourier's law T a Anharmonic chain T a Q G(T a, T b ) L 1 α.3 α.5 Delni Lepri Livi Polii Livi, Spohn, Grassberger,...
49 DIFFUSIVE SYSTEMS: average curren Symmeric exclusion α γ 1 L β δ Fourier's law General laice gas Random walkers KMP model Kipnis Marchioro Presui Q G(T a, T b ) L
50 SSEP (Symmeric simple exclusion process) D. Douço Roche 2004 α γ L ρ a = α α + γ, ρ b = δ β + δ β δ Q() lim 1 L [ρ a ρ b ] Fick s law Q 2 () c lim 1 L Q 3 () c lim 1 L (ρ a ρ b ) [ ρ a + ρ b 2(ρ2 a + ρ a ρ b + ρ 2) b 3 [ ] 1 2(ρ a + ρ b ) + 16ρ2 a + 28ρ a ρ b + 16ρ 2 b 15 ]
51 CURRENT FLUCTUATIONS AND LARGE DEVIATIONS Q T a T b Q Energy ransferred during ime F(j) Pro ( Q ) = j exp[ F (j)] j * j Expansion of F (j) near j gives all cumulans of Q
52 DIFFUSIVE SYSTEMS: higher cumulans Q ρa ρ b Bodineau D L For ρ a ρ b small: Q = D(ρ)(ρ a ρ b ) L ρ a = ρ b = ρ : Then one can calculae Q 2 = σ(ρ) L All cumulans of Q for arbirary ρ a and ρ b
53 ADDITIVITY PRINCIPLE Pro ( Q = j, ρ a, ρ b ) exp[ F L+L (j, ρ a, ρ b )] Bodineau D Q ρ a ρ b L L Addiiviy P L+L (Q, ρ a, ρ b ) max ρ [P L(Q, ρ a, ρ) P L (Q, ρ, ρ b )] F L+L (j, ρ a, ρ b ) = min ρ [F L (j, ρ a, ρ) + F L (j, ρ, ρ b )]
54 VARIATIONAL PRINCIPLE F L (j, ρ a, ρ b ) = 1 L min ρ(x) Z 1 0 dx [j L + ρ (x) D(ρ(x))] 2 2σ(ρ(x)) Bodineau D ρ a ρ (x) ρ*(x) ρ b 0 x 1 Saises he ucuaion heorem F (j) F ( j) = j ρb ρ a D(ρ) σ(ρ) dρ Gallavoi Cohen 1995 Evans Searls Kurchan 1998 Lebowiz Spohn 1999
55 DIFFUSIVE SYSTEMS: all he cumulans Bodineau D Q c Q 2 c Q 3 c Q 4 c = 1 L I 1 = 1 L = 1 L = 1 L I 2 I 1 3 ( I 3 I 1 I2 2 ) I1 3 3 ( 5 I 4 I I 1I 2 I I2 3 ) I1 5 where I n = ρa ρ b D(ρ)σ(ρ) n 1 dρ For he SSEP D(ρ) = 1 and σ(ρ) = 2ρ(1 ρ) For he KMP D(ρ) = 1 and σ(ρ) = 4ρ 2 For he random walkers D(ρ) = 1 and σ(ρ) = 2ρ
56 TRUE VARIATIONAL PRINCIPLE Berini De Sole Gabrielli Jona-Lasinio Landim 2005 F (j) = 1 L lim min T ρ(x,),j(x,) 1 T d T dx [j(x, ) + ρ (x, )D(ρ(x, ))] 2 2σ(ρ(x, )) wih dρ d = dj dx (conservaion), ρ (0) = ρ a, ρ (1) = ρ b and j T = T 0 j (x)d Sucien condiion for he opimal prole o be ime independen Dynamical phase ransiion Bodineau D he opimal ρ (x) sars o become ime dependen
57 SSEP ON A RING Apper D Lecome Van Wijland 2008 N paricles L sies ρ = N L 1 1 Q ux hrough a bond during ime
58 CUMULANTS OF THE CURRENT FOR THE SSEP ON A RING paricles and L sies σ(ρ) = 2ρ(1 ρ) = 2N(L N) L 2 N Q 2 c = σ L 1 Q 4 c = σ2 2(L 1) 2 Q 6 c = (L2 L+2)σ 3 2(L 1)σ 2 4(L 1) 3 (L 2) Q 8 c = (10L4 2L 3 +27L 2 15L+18)σ 4 4(L 1)(11L 2 L+12)σ 3 +48(L 1) 2 σ 2 24(L 1) 4 (L 2)(L 3) Q 2 c = σ L Gaussian + Fick's law Q 2n c σn L 2 for n 2
59 UNIVERSAL CUMULANTS OF THE CURRENT Q 2 c = σ L Gaussian Q 4 c σ2, Q 6 c 2L 2 σ3, Q 8 c 4L 2 5σ4 12L 2 Universal
60 UNIVERSAL CUMULANTS OF THE CURRENT Q 2 c = σ L Gaussian Q 4 c σ2, Q 6 c 2L 2 σ3, Q 8 c 4L 2 5σ4 12L 2 Universal Macroscopic ucuaion heory
61 UNIVERSAL CUMULANTS OF THE CURRENT Q 2 c = σ L Gaussian Q 4 c σ2, Q 6 c 2L 2 σ3, Q 8 c 4L 2 5σ4 12L 2 Universal Macroscopic ucuaion heory + ucuaions around a a prole
62 DIFFUSIVE SYSTEMS: summary Open sysem T a T b Q n c / L 1 Ring T Q 2 c / L 1 Q 2n c / L 2 for n 2
63 HARD PARTICLE GAS: he second cumulan T a T a
64 HARD PARTICLE GAS: he second cumulan T a T a 2000 Q 2 ring Q 2 ring Q 2 ring / Q 2 ring / a) Q 2 b) / has a limi
65 HARD PARTICLE GAS: Size dependence of he cumulans OPEN SYSTEM lim Qn c/ Q 4 open c Q 3 open c Q 2 open c Q open lim Qn c/ Q 4 open c Q 2 open c Ta >Tb Ta = Tb a) N b) N RING 10 2 Q 4 ring c Q 2 ring c lim Qn c/ N
66 HARD PARTICLE GAS: Size dependence of he cumulans OPEN SYSTEM lim Qn c/ Q 4 open c Q 3 open c Q 2 open c Q open lim Qn c/ Q 4 open c Q 2 open c Ta >Tb Ta = Tb a) N b) N RING 10 2 Q 4 ring c Q 2 ring c lim Qn c/ ANOMALOUS FOURIER'S LAW N
67 Correlaions on a mesoscopic scale T a T a L Cor(x,.25) x Delni, Lepri, Livi, Mejia-Monaserio, Polii 2010 Gerschenfeld, D., Lebowiz 2010
68 Sep iniial condiion ρ a Q Q ρ b ASEP SSEP G. Schüz 98. Prähofer Spohn Tracy Widom 2008 D Gerschenfeld 2009 ρ a ρ b e λq =?
69 Sep iniial condiion on he innie line ρ a Q ρ b SSEP D Gerschenfeld 2009 e λq exp[ H(ω)] ] wih H(ω) = 1 [1 π dk log + ωe k2 and ω = 1 [1 (e λ 1)ρ a ][1 (1 e λ )ρ b ]
70 Sep iniial condiion on he innie line ρ a Q ρ b SSEP D Gerschenfeld 2009 e λq exp[ H(ω)] ] wih H(ω) = 1 [1 π dk log + ωe k2 and ω = 1 [1 (e λ 1)ρ a ][1 (1 e λ )ρ b ] For large Q : Pro(Q ) exp [ π2 12 Q3 / ]
71 Densiy proles condionned on he curren Densiy / space
72 Densiy proles condionned on he curren Densiy / space
73 OPEN QUESTIONS Large deviaion funcion of he densiy for a general diusive sysem Non seady sae siuaions Theory for mechanical sysems which conserved impulsion Mean eld heory for large deviaion funcions
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