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
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
Ouline Densiy ucuaions in non-equilibrium sysems Curren ucuaions Open quesions Non seady sae siuaions Deerminisic dynamics
Non equilibrium seady saes: curren of paricles
Non equilibrium seady saes: curren of paricles ρ a Sysem ρ b
Non equilibrium seady saes: curren of hea T a Sysem T b
Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) L
Q Ta T b Equilibrium (T a = T b ) Seady sae (T a T b ) L P(Q) = P( Q)
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 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
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
Q Ta T b L Equilibrium (T a = T b ) Non-equilibrium seady sae (T a T b )
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
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
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
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
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.
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 10 23 J/K
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
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)}) ]
Exclusion processes SSEP (Symmeric simple exclusion process) α γ 1 1 1 1 1 L ρ a = α α + γ, ρ b = δ β + δ β δ
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 ]
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 2001-2002 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
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
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
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
TWO APPROACHES SSEP (Symmeric simple exclusion process) 1 1 1 1 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)
Marix ansaz Seady sae of he SSEP α 1 1 1 1 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
Marix ansaz Seady sae of he SSEP α 1 1 1 1 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
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
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
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
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, )
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( )
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?
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, ) )
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?
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)
Macroscopic ucuaion heory Berini De Sole Gabrielli Jona-Lasinio Landim 2001-2002 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)
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
Curren ucuaions in non-equilibrium seady saes T a T b
Curren ucuaions in non-equilibrium seady saes T a T b Curren ucuaions on a ring T
AVERAGE CURRENT OF HEAT Q Ta L T b Q = F (T a, T b, L, conacs)
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
AVERAGE CURRENT OF PARTICLES Q ρa L ρ b Q = F (ρ a, ρ b, L, conacs)
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
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 )
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,...
DIFFUSIVE SYSTEMS: average curren Symmeric exclusion α 1 1 1 1 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
SSEP (Symmeric simple exclusion process) D. Douço Roche 2004 α γ 1 1 1 1 1 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 ]
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
DIFFUSIVE SYSTEMS: higher cumulans Q ρa ρ b Bodineau D. 2004 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
ADDITIVITY PRINCIPLE Pro ( Q = j, ρ a, ρ b ) exp[ F L+L (j, ρ a, ρ b )] Bodineau D. 2004 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 )]
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. 2004 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 1994... Kurchan 1998 Lebowiz Spohn 1999
DIFFUSIVE SYSTEMS: all he cumulans Bodineau D. 2004 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 I1 2 14 I 1I 2 I 3 + 9 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
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 0 1 0 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. 2005-2007 he opimal ρ (x) sars o become ime dependen
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
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
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
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
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
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
HARD PARTICLE GAS: he second cumulan T a T a
HARD PARTICLE GAS: he second cumulan T a T a 2000 Q 2 ring Q 2 ring 0.45 0.4 Q 2 ring / Q 2 ring / 1500 0.35 1000 0.3 500 0.25 0.2 a) 0 2000 4000 6000 Q 2 b) 0 2 10 4 4 10 4 1/ has a limi
HARD PARTICLE GAS: Size dependence of he cumulans OPEN SYSTEM lim Qn c/ 10 2 10 1 10 0 Q 4 open c Q 3 open c Q 2 open c Q open lim Qn c/ 10 2 10 1 10 0 Q 4 open c Q 2 open c 10 1 10 1 Ta >Tb Ta = Tb a) 10 2 10 3 N b) 10 2 10 3 N RING 10 2 Q 4 ring c Q 2 ring c lim Qn c/ 10 1 10 0 10 1 10 2 10 3 N
HARD PARTICLE GAS: Size dependence of he cumulans OPEN SYSTEM lim Qn c/ 10 2 10 1 10 0 Q 4 open c Q 3 open c Q 2 open c Q open lim Qn c/ 10 2 10 1 10 0 Q 4 open c Q 2 open c 10 1 10 1 Ta >Tb Ta = Tb a) 10 2 10 3 N b) 10 2 10 3 N RING 10 2 Q 4 ring c Q 2 ring c lim Qn c/ 10 1 10 0 ANOMALOUS FOURIER'S LAW 10 1 10 2 10 3 N
Correlaions on a mesoscopic scale T a T a L Cor(x,.25) 1 0 2 4 100 200 400 6 0 0.2 0.4 0.6 0.8 1 x Delni, Lepri, Livi, Mejia-Monaserio, Polii 2010 Gerschenfeld, D., Lebowiz 2010
Sep iniial condiion ρ a Q 1 0 1 2 Q ρ b ASEP SSEP G. Schüz 98. Prähofer Spohn 2000-2002. Tracy Widom 2008 D Gerschenfeld 2009 ρ a ρ b e λq =?
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 ]
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 / ]
Densiy proles condionned on he curren Densiy 0.8 0.6 0 /2 0.4 0.2 0 space
Densiy proles condionned on he curren Densiy 0.8 0.6 0 /2 0.4 0.2 0 space
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