The 1D KPZ equation: exact solutions and universality T. Sasamoto 5 Nov 2015 @ IHP 1
Plan 1. Introduction 2. The KPZ equation 3. Exact solutions (”Stochastic integrability”) Height distribution Stationary space-time two point correlation function 4. Universality (”KPZ is everywhere.”) Experiments KPZ in Hamiltonian dynamics 2
1. Non-linearity and fluctuations for far-from-equilibrium systems • Non-eq systems: various interesting phenomena • Dissipative structure ： Benard convection T T + ∆ T • Experimental developments: colloids, single electron counting, cold atom... (can measure even fluctuations) • Fundamental principle is unknown (cf Kubo for linear regime) • Studying simple model systems play important roles. 3
Nonlinearity for non-eq systems: Fermi-Pasta-Ulam A first numerical simulation of Hamiltonian dynamics for studying ergodic properties. • Harmonic chain is easy, but no dissipation. • Unharmonic chain (nonlinearlity). Hamiltonian N N − 1 p 2 j ∑ ∑ H = 2 + V ( x j +1 − x j ) j =1 j =1 where V ( x ) = 1 2 x 2 + α 3 x 3 + β 4 x 4 • No relaxation. Recurrence. • It remains difficult to study various properties of this model. 4
Soliton equations, Toda lattice, Integrable systems • KdV equation u t + 6 uu x + u xxx = 0 Soliton solutions • Toda lattice V ( x ) = e − x • Nonlinear Schr¨ odinger equation iu t + u xx + 2 | u | 2 u = 0 • Classical integrable systems Inverse scattering. Linearization. → Quantum integrable systems (Quantization of above, Heisenberg chain, XXZ chain, etc.) 5
Hydrodynamics: non-linear but no noise • Navier-Stokes equation • Kuramoto-Shivashinsky equation u t + uu x + u xx + u xxxx = 0 • Burgers equation u t = u xx + uu x Solvable by the Cole-Hopf transformation φ = e u ⇒ φ t = φ xx • One can add noise to study fluctuations ⇒ Nonlinear SPDE (stochastic partial differential equation) 6
2. Basics of the KPZ equation: Surface growth • Paper combustion, bacteria colony, crystal growth, etc • A typical non-equilibrium phenomenon • Recent developments due to the connections to integrable systems, representation theory, etc 7
Simulation models Ex: ballistic deposition Height fluctuation O ( t β ) , β = 1 / 3 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 Flat 8
KPZ equation h ( x, t ) : height at position x ∈ R and at time t ≥ 0 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 mean 0 and covariance ⟨ η ( x, t ) η ( x ′ , t ′ ) ⟩ = δ ( x − x ′ ) δ ( t − t ′ ) • Dynamical RG analysis: → β = 1 / 3 (KPZ class) • A simplest nonequilibrium model with nonlinearity, noise and ∞ -degrees of freedom ( u = ∂ x h satisfies the Burgers equation with noise.) • Ill-posed as its is. (Bertini-Giacomin, Hairer) • By a simple scaling we set ν = 1 2 , λ = D = 1 . 9
A discrete model: ASEP 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 ⇔ height for surface growth • In a certain weakly asymmetric limit ASEP ⇒ KPZ equation 10
3. Exact solutions: 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 . h(x,t) 2 λ t/ δ x δ → 0 c δ e −| x | /δ The polymer from the origin: Z ( x, 0) = δ ( x ) = lim corresponds to narrow wedge for KPZ. 11
Exact solution for the height distribution Thm (2010 TS Spohn, Amir Corwin Quastel) For the initial condition Z ( x, 0) = δ ( x ) (narrow wedge for KPZ) h ( x, t ) = − x 2 / 2 t − 12 γ 3 1 t + γ t ξ t where γ t = ( t/ 2) 1 / 3 . The distribution function of ξ t is ∫ ∞ − 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 −∞ 12
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 • In the large t limit, F t tends to the GUE Tracy-Widom distribution F 2 from random matrix theory. 13
Tracy-Widom distributions For GUE (Gaussian unitary ensemble) with density P ( H ) dH ∝ e − Tr H 2 dH for H : N × N hermitian matrix, the joint eigenvalue density is (with ∆( x ) Vandelmonde) 1 e − x 2 Z ∆( x ) 2 ∏ i i GUE Tracy-Widom distribution √ [ ] x max − 2 N = F 2 ( s ) = det(1 − P s K 2 P s ) lim 2 − 1 / 2 N − 1 / 6 < s N →∞ P 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 There is also GOE TW ( F 1 ) for GOE (Gaussian orthogonal ensemble, real symmetric matrices, for flat surface) 14
Probability densities of Tracy-Widom distributions F ′ 2 (GUE), F ′ 1 (GOE) 15
Replica approach Dotsenko, Le Doussal, Calabrese 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 narrow wedge case) ⟨ Z N ( x, t ) ⟩ = ⟨ x | e − H N t | 0 ⟩ where H N is the Hamiltonian of the (attractive) δ -Bose gas, N N ∂ 2 H N = − 1 − 1 ∑ ∑ δ ( x j − x k ) . ∂x 2 2 2 j j =1 j ̸ = k 16
We expand the quantity of our interest as − e − γ t s ) N ∞ ⟨ e − e h ( x,t )+ x 2 ( γ 3 2 t + t 24 − γts ⟩ = t ∑ Z N ( x, t ) e N ⟨ ⟩ 12 N ! N =0 The δ -Bose gas is the quantum version of the NLS equation and is quantum integrable. This allows us to get explicit expressions for the moment ⟨ Z N ⟩ and see that the generating function can be written as a Fredholm determinant. But for the KPZ, ⟨ Z N ⟩ ∼ e N 3 ! • For discrete models like ASEP, one can apply a rigorous version of the replica method. Note the ASEP is related to the XXZ spin chain. A semi-discrete finite temperature polymer model is related to quantum Toda lattice. (”Stochastic integrability”) 17
A finite temperature polymer model 2001 O’Connell Yor Semi-discrete directed polymer in random media B i , 1 ≤ i ≤ N : independent Brownian motions Energy of the polymer π E [ π ] = B 1 ( t 1 ) + B 2 ( t 1 , t 2 ) + · · · + B N ( t N − 1 , t ) with B j ( s, t ) = B j ( t ) − B j ( s ) , j = 2 , · · · , N for s < t Partition function ( β = 1 /k B T : inverse temperature ) ∫ e βE [ π ] dt 1 · · · dt N − 1 Z N ( t ) = 0 <t 1 < ··· <t N − 1 <t In a limit, this becomes the polymer related to KPZ equation. 18
Zero-temperature limit In the T → 0 (or β → ∞ ) limit f N ( t ) := lim β →∞ log Z N ( t ) /β = 0 <s 1 < ··· <s N − 1 <t E [ π ] max 2001 Baryshnikov Connection to random matrix theory N ∫ ∏ Prob ( f N (1) ≤ s ) = dx j · P GUE ( x 1 , · · · , x N ) , ( −∞ ,s ] N j =1 e − x 2 N j / 2 ∏ ∏ ( x k − x j ) 2 P GUE ( x 1 , · · · , x N ) = √ · j ! 2 π j =1 1 ≤ j<k ≤ N where P GUE ( x 1 , · · · , x N ) is the probability density function of the eigenvalues in the Gaussian Unitary Ensemble (GUE) 19
A generalization to finite β By using the connection to the quantum Toda lattice Thm (2015 TS Imamura) N ( ) − e − βuZN ( t ) ∫ ∏ β 2( N − 1) e = dx j f F ( x j − u ) · W ( x 1 , · · · , x N ; t ) E R N j =1 N 1 ( x k − x j ) · det ( ψ k − 1 ( x j ; t )) N ∏ ∏ W ( x 1 , · · · , x N ; t ) = j,k =1 j ! j =1 1 ≤ j<k ≤ N where f F ( x ) = 1 / ( e βx + 1) is the Fermi distribution function and ∫ ∞ ( iw ) k ψ k ( x ; t ) = 1 dwe − iwx − w 2 t/ 2 Γ (1 + iw/β ) N 2 π −∞ 20
Stationary 2pt correlation Not only the height/current distributions but correlation functions show universal behaviors. • For the KPZ equation, the Brownian motion is stationary. h ( x, 0) = B ( x ) where B ( x ) , x ∈ R is the two sided BM. • Two point correlation h t 2/3 t 1/3 ∂ x h ( x , t ) ∂ x h (0,0) x o 21
Figure from the formula Imamura TS (2012) ⟨ ∂ x h ( x, t ) ∂ x h (0 , 0) ⟩ = 1 2(2 t ) − 2 / 3 g ′′ t ( x/ (2 t ) 2 / 3 ) The figure can be drawn from the exact formula (which is a bit involved though). 2.0 γ t = 1 γ t = ∞ 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 y 1 t ( y ) for γ t := ( t Stationary 2pt correlation function g ′′ 3 = 1 . 2 ) The solid curve is the scaling limit g ′′ ( y ) . 22
4 Univerality 1: Expeirments by Takeuchi-Sano 23
Takeuchi Sano TS Spohn, Sci. Rep. 1,34(2011) 24
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