KPZ 普遍性の新たな展開 笹本智弘（東工大） 10 Mar 2014 @ 学習院 1
1. KPZ for surface growth • Paper combustion, bacteria colony, crystal growth, liquid crystal turbulence • Non-equilibrium statistical mechanics • Connections to integrable systems 2
Simulation models Ex: ballistic deposition A B ↓ ↓ ↓ 100 "ht10.dat" "ht50.dat" "ht100.dat" 80 60 A ′ 40 B ′ 20 0 0 10 20 30 40 50 60 70 80 90 100 3
Scaling h ( x, t ) : surface height at position x and at time t Scaling ( L : system size) h W ( L, t ) = ⟨ ( h ( x, t ) − ⟨ h ( x, t ) ⟩ ) 2 ⟩ 1 / 2 x = L α Ψ( t/L z ) W ( L, t ) ∼ L α For t → ∞ W ( L, t ) ∼ t β For t ∼ 0 where α = βz In many models, α = 1 / 2 , β = 1 / 3 Dynamical exponent z = 3 / 2 : Anisotropic scaling 4
KPZ equation 1986 Kardar Parisi Zhang √ 2 λ ( ∂ x h ( x, t )) 2 + ν∂ 2 ∂ t h ( x, t ) = 1 x h ( x, t ) + Dη ( x, t ) where η is the Gaussian noise with covariance ⟨ η ( x, t ) η ( x ′ , t ′ ) ⟩ = δ ( x − x ′ ) δ ( t − t ′ ) √ 1 + ( ∂ x h ) 2 ∂ t h = v ≃ v + ( v/ 2)( ∂ x h ) 2 + . . . • Dynamical RG analysis: → α = 1 / 2 , β = 1 / 3 (KPZ class ) • New analytic and experimental developments 5
2: Limiting height distribution ASEP = asymmetric simple exclusion process q p q p q ⇒ ⇒ · · · ⇐ ⇐ ⇐ · · · -3 -2 -1 0 1 2 3 • TASEP(Totally ASEP, p = 0 or q = 0 ) • N ( x, t ) : Integrated current at ( x, x + 1) upto time t • Bernoulli (each site is independently occupied with probability ρ ) is stationary 6
Mapping to surface growth 2 initial conditions besides stationary Flat Droplet ↕ ↕ ↕ ↕ Wedge Step Alternating Integrated current N ( x, t ) in ASEP ⇔ Height h ( x, t ) in surface growth 7
TASEP with step i.c. 2000 Johansson As t → ∞ N (0 , t ) ≃ 1 4 t − 2 − 4 / 3 t 1 / 3 ξ 2 Here N ( x = 0 , t ) is the integrated current of TASEP at the origin and ξ 2 obeys the GUE Tracy-Widom distribution; F 2 ( s ) = P [ ξ 2 ≤ s ] = det(1 − P s K Ai P s ) 0.5 where P s : projection onto the interval [ s, ∞ ) 0.4 0.3 and K Ai is the Airy kernel 0.2 ∫ ∞ 0.1 0.0 K Ai ( x, y ) = d λ Ai( x + λ )Ai( y + λ ) � 6 � 4 � 2 0 2 s 0 Random universality in KPZ universality 8
Tracy-Widom distributions Random matrix theory, Gaussian ensembles H : N × N matrix 1 e − β 2 Tr H 2 P ( H ) dH = Z Nβ GOE (real symmetric, β = 1 ), GUE (hermitian, β = 2 ). Joint eigenvalue distribution N 1 e − β 2 x 2 ∏ ( x i − x j ) β ∏ P Nβ ( x 1 , x 2 , . . . , x N ) = i Z Nβ i =1 1 ≤ i<j ≤ N • Average density … Wigner semi-circle 9
Largest eigenvalue distribution Largest eigenvalue distribution of Gaussian ensembles 1 ∫ e − β 2 x 2 ∏ ( x i − x j ) β ∏ i dx 1 · · · dx P Nβ [ x max ≤ s ] = Z Nβ ( −∞ ,s ] N i<j i Scaling limit (expected to be universal) √ √ [ 2 N 1 / 6 < s ] lim ( x max − 2 N ) = F β ( s ) N →∞ P Nβ GUE (GOE) Tracy-Widom distribution 10
Tracy-Widom distributions GUE Tracy-Widom distribution F 2 ( s ) = det(1 − P s K 2 P s ) where P s : projection onto [ s, ∞ ) and K 2 is the Airy kernel ∫ ∞ K 2 ( x, y ) = d λ Ai( x + λ )Ai( y + λ ) 0 Painlev´ e II representation ∫ ∞ [ ] ( x − s ) u ( x ) 2 dx − F 2 ( s ) = exp s where u ( x ) is the solution of the Painlev´ e II equation ∂ 2 ∂x 2 u = 2 u 3 + xu, u ( x ) ∼ Ai ( x ) x → ∞ 11
GOE Tracy-Widom distribution ∫ ∞ − 1 [ ] ( F 2 ( s )) 1 / 2 F 1 ( s ) = exp u ( x ) dx 2 s GSE Tracy-Widom distribution ∫ ∞ − 1 [ ] ( F 2 ( s )) 1 / 2 F 4 ( s ) = cosh u ( x ) dx 2 s Figures for Tracy-Widom distributions 12
Step TASEP and random matrix • Generalize to discrete TASEP with parallel update. A waiting time is geometrically distributed. . . ✻ j . ( N, N ) w ij on ( i, j ) : geometrically distributed waiting time of i th hop of j th particle · · · (1 , 1) ✲ i • Time at which N th particle arrives at the origin ∑ = max w i,j (= G ( N, N )) up-right paths from ( i,j ) on a path (1 , 1) to ( N,N ) Zero temperature directed polymer 13
LUE formula for TASEP • By RSK algorithm a matrix of size N × N with non-negative integer entries is mapped to a pair of semi-standard Young tableau with the same shape λ with entries from { 1 , 2 , . . . , N } , with G ( N, N ) = λ 1 . √ • When the j th particle does i th hop with parameter a i b j , the measure on λ is given by the Schur measure 1 Z s λ ( a ) s λ ( b ) • Using a determinant formula of the Schur function and taking the continuous time limit, one gets 1 ∫ ∏ ( x i − x j ) 2 ∏ e − x i dx 1 · · · dx N P [ N ( t ) ≥ N ] = Z N [0 ,t ] N i<j i 14
Generalizations Current Fluctuations of TASEP with flat initial conditions: GOE TW distribution More generalizations: stationary case: F 0 distribution, multi-point fluctuations: Airy process, etc Experimental relevance? What about the KPZ equation itself? 15
Takeuchi-Sano experiments 16
See Takeuchi Sano Sasamoto Spohn, Sci. Rep. 1,34(2011) 17
3. Exact solution for the KPZ equation Remember the KPZ equation √ 2 λ ( ∂ x h ( x, t )) 2 + ν∂ 2 ∂ t h ( x, t ) = 1 x h ( x, t ) + Dη ( x, t ) 2010 Sasamoto Spohn, Amir Corwin Quastel • Narrow wedge initial condition • Based on (i) the fact that the weakly ASEP is KPZ equation (1997 Bertini Giacomin) and (ii) a formula for step ASEP by 2009 Tracy Widom • The explicit distribution function for finite t 18
Narrow wedge initial condition Scalings h → λ x → α 2 x, t → 2 να 4 t, 2 ν h where α = (2 ν ) − 3 / 2 λD 1 / 2 . We can and will do set ν = 1 2 , λ = D = 1 . We consider the droplet growth with macroscopic shape − x 2 / 2 t for | x | ≤ t/δ , h ( x, t ) = (1 / 2 δ 2 ) t − | x | /δ for | x | > t/δ which corresponds to taking the following narrow wedge initial conditions: h ( x, 0) = −| x | /δ , δ ≪ 1 19
h(x,t) 2 λ t/ δ x 20
Distribution h ( x, t ) = − x 2 / 2 t − 12 γ 3 1 t + γ t ξ t where γ t = (2 t ) 1 / 3 . The distribution function of ξ t ∫ ∞ − e γ t ( s − u ) ] [ F t ( s ) = P [ ξ t ≤ s ] = 1 − exp −∞ ( ) × det(1 − P u ( B t − P Ai ) P u ) − det(1 − P u B t P u ) d u where P Ai ( x, y ) = Ai( x )Ai( y ) , P u is the projection onto [ u, ∞ ) and the kernel B t is ∫ ∞ d λ Ai( x + λ )Ai( y + λ ) B t ( x, y ) = e γ t λ − 1 −∞ • In the large t limit, F t tends to F 2 . 21
Finite time KPZ distribution and TW 0.5 0.4 0.3 0.2 0.1 0.0 � 6 � 4 � 2 0 2 s : exact KPZ density F ′ t ( s ) at γ t = 0 . 94 −− : Tracy-Widom density ( t → ∞ limit) • : ASEP Monte Carlo at q = 0 . 6 , t = 1024 MC steps 22
Cole-Hopf transformation If we set Z ( x, t ) = exp ( h ( x, t )) this quantity (formally ) satisfies ∂ 2 Z ( x, t ) ∂tZ ( x, t ) = 1 ∂ + η ( x, t ) Z ( x, t ) ∂x 2 2 This can be interpreted as a (random) partition function for a directed polymer in random environment η . 23
Replica method For a system with randomness, e.g. for random Ising model, ∑ H = J ij s i s j ⟨ ij ⟩ where i is site, s i = ± 1 is Ising spin, J ij is iid random variable(e.g. Bernoulli), we are often interested in the averaged s i = ± 1 e − H . free energy ⟨ log Z ⟩ , Z = ∑ In some cases, computing ⟨ log Z ⟩ seems hopeless but the calculation of N th replica partition function ⟨ Z N ⟩ is easier. In replica method, one often resorts to the following identity ⟨ Z N ⟩ − 1 ⟨ log Z ⟩ = lim . N N → 0 24
For KPZ: Feynmann-Kac and δ Bose gas Feynmann-Kac expression for the partition function, ∫ t ( ) 0 η ( b ( s ) ,t − s ) ds Z ( b ( t ) , 0) Z ( x, t ) = E x e Because η is a Gaussian variable, one can take the average over the noise η to see that the replica partition function can be written as (for pt-to-pt case) ⟨ Z N ( x, t ) ⟩ = ⟨ x | e − H N t | 0 ⟩ where H N is the Hamiltonian of the δ -Bose gas, N N ∂ 2 H N = − 1 − 1 ∑ ∑ δ ( x j − x k ) . ∂x 2 2 2 j j =1 j ̸ = k 25
Remark: More generally, the N point correlation function satisfies ⟨ N ⟨ N ⟩ ⟩ d ∏ ∏ Z ( x i , t ) = H N Z ( x i , t ) dt i =1 i =1 Remember h = log Z . We are interested not only in the average ⟨ h ⟩ but the full distribution of h . Here we compute the generating function G t ( s ) of the replica partition function, − e − γ t s ) N ∞ ( γ 3 t ∑ Z N (0 , t ) e N ⟨ ⟩ G t ( s ) = 12 N ! N =0 with γ t = ( t/ 2) 1 / 3 . This turns out to be written as a Fredholm determinant. We apply the inversion formula to recover the p.d.f for h . But for the KPZ, ⟨ Z N ⟩ ∼ e N 3 . 26
4. Stationary case 2012-2013 Imamura S • Narrow wedge is technically the simplest. • Flat case is a well-studied case in surface growth • Stationary case is important for stochastic process and nonequilibrium statistical mechanics – Two-point correlation function – Experiments: Scattering, direct observation – A lot of approximate methods (renormalization, mode-coupling, etc.) have been applied to this case. – Nonequilibrium steady state(NESS): No principle. Dynamics is even harder. 27
Recommend
More recommend
