Spectral shock waves in dynamical random matrix models Maciej A. Nowak (in collaboration with Jean-Paul Blaizot and Piotr Warcho� l) Mark Kac Complex Systems Research Center, Marian Smoluchowski Institute of Physics, Jagiellonian University, Krak´ ow, Poland 8-10 October 2012 -Telecom ParisTech - Paris 13 e , France Supported in part by the MAESTRO grant DEC-2011/02/A/ST1/00119 of National Centre of Science.
Real Burgers equation GUE Chiral GUE Catastrophes Outline Trivia on real Burgers equation Large N limit of RMT and complex, inviscid Burgers Where are the shocks? 1 What plays the role of spectral viscosity? 2 Finite N as the inverse of viscosity in the spectral flow - Airy, Pearcey, Bessel and Bessoid functions as heralds of the shocks Optical analogies and applications Summary Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Real Burgers equation ∂ t f ( x , t ) + f ( x , t ) ∂ x f ( x , t ) = µ∂ xx f ( x , t ) f ( x , t ) is the velocity field at time t and position x of the fluid with viscosity µ . One-dimensional toy model for turbulence [Burgers 1939] But, equation turned out to be exactly integrable [Hopf 1950],[Cole 1951] If f ( x , t ) = − 2 µ∂ x ln d ( x , t ), then ∂ t d ( x , t ) = µ∂ xx d ( x , t ) (diffusion equation), so general solution comes from Cole-Hopf transformation where ′ ′ )2 � x � + ∞ ′′ , 0) dx ′′ −∞ e − ( x − x − 1 d ( x 1 ′ d ( x , t ) = dx √ 4 πµ t 4 µ t 2 µ 0 Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Inviscid real Burgers equation ∂ t f ( x , t ) + f ( x , t ) ∂ x f ( x , t ) = 0 where f ( x , 0) = f 0 ( x ). Solution by the method of characteristics: If x ( t ) is the solution of ODE dx ( t ) / dt = f ( x ( t ) , t ), then F ( t ) ≡ f ( x ( t ) , t ) is constant in time along characteristic curve on the ( x , t ) plane Then dx / dt = F and dF / dt = 0 lead to x ( t ) = x (0) + tF (0) and F ( t ) = F (0) Defining ξ ≡ x (0) we get f ( x , t ) = f ( ξ, 0) = f 0 ( ξ ) = f 0 ( x − tf ( x , t )), i.e. implicit relation determining the solution of the Burgers equation. When d ξ/ dx = ∞ , we get the shock wave. Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Inviscid real Burgers equation In the case of inviscid Burgers equation, characteristics are straight lines, but with different slopes (velocity depends on the position) Characteristics method fails when lines cross (shock wave) Finite viscosity (or diffusive constant) smoothens the shock Inviscid limit of viscid Burgers equation is highly non-trivial Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Inviscid complex Burgers equation After considerable and fruitless efforts to develop a Newtonian theory of ensembles, we discovered that the correct procedure is quite different and much simpler...... from F.J. Dyson, J. Math. Phys. 3 (1962) 1192 H ij → H ij + δ H ij with < δ H ij = 0 > and < ( δ H ij ) 2 > = (1 + δ ij ) δ t For eigenvalues x i , random walk undergoes in the ”electric � � 1 field” (Dyson) < δ x i > ≡ E ( x i ) δ t = � δ t and i � = j x j − x i < ( δ x i ) 2 > = δ t Resulting SFP equation for the resolvent in the limit N = ∞ and τ = Nt reads ∂ τ G ( z , τ ) + G ( z , τ ) ∂ z G ( z , τ ) = 0 where � � G ( z , τ ) = 1 1 is the resolvent tr N z − H ( τ ) Non-linear, inviscid complex Burgers (Hopf, Voiculescu) equation Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Inviscid complex Burgers equation - details SFP eq: ∂ t P ( { x j } , t ) = 1 i ∂ 2 � ii P ( { x j } , t ) − � i ∂ i ( E ( x i ) P ( { x j } , t )) 2 Integrating, normalizing densities to 1 and rescaling the time τ = Nt we get dy ρ ( y ) � ∂ τ ρ ( x ) + ∂ x ρ ( x ) P . V . x − y = dy ρ c ( x , y ) 2 N ∂ 2 1 � xx ρ ( x ) + P . V . x − y r.h.s. tends to zero in the large N limit x ± i ǫ = P . V . 1 1 x ∓ i πδ ( x ) Taking Hilbert transform of the above equation and using above Sochocki formula converts pair of singular integral-differential equations onto complex inviscid Burgers equation. Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Dolphins wisdom - surfing the shock wave Tracing the singularities of the flow allows to understand the pattern of the evolution of the complex system without explicit solutions of the complicated hydrodynamic equations... UK Daily Mail, July 11th 2007 Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Complex inviscid Burgers Equation Complex Burgers equation ∂ τ G + G ∂ z G = 0 Complex characteristics, trivial initial conditions G 0 ( z ) = G ( τ = 0 , z ) = 1 G ( z , τ ) = G 0 ( ξ [ z , τ )]) z ξ = z − G 0 ( ξ ) τ ( ξ = x − vt ), so solution reads G ( z , τ ) = G 0 ( z − τ G ( z , τ )) Shock wave when d ξ dz = ∞ Equivalently, dz / d ξ = 0, then ξ c = ±√ τ , so z c = ξ c + G 0 ( ξ c ) τ = ± 2 √ τ Since explicit solution easily reads √ √ z 2 − 4 τ ), i.e. ρ ( x , τ ) = 1 1 4 τ − x 2 , G ( z , τ ) = 2 πτ ( z − 2 πτ we see that shock waves appear at the edges of the spectrum ( x = ± 2 √ τ ). Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Where is the viscosity? Let us define D N ( z , τ ) ≡ � det( z − H ( τ )) � Opening the determinant with the help of auxilliary Grassmann variables and perfoming the averaging one gets easily �� � � η i z η i − τ � D N ( z , τ ) = exp i ¯ � i < j ¯ η i η i ¯ η j η j l , r d ¯ η l d η r N Differentiating and using the properties of the Grassmann variables one gets that D N obeys complex equation ∂ τ D N ( z , τ ) = − 1 2 N ∂ zz D ( z , τ ). Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Where is the viscosity? - cont. ∂ τ D N ( z , τ ) = − 1 2 N ∂ zz D ( z , τ ). Then complex Cole Hopf transformation f N ( z , τ ) = 1 N ∂ z ln D N ( z , τ ) leads to exact for any N , viscid complex Burgers equation 1 ∂ τ f N + f N ∂ z f N = − µ∂ zz f N µ = 2 N Positive viscosity ”smoothens” the shocks, negative is ”roughening” them, triggering violent oscillations � � Note than G ( z , τ ) = 1 1 = Tr N z − H ( τ ) � 1 � 1 � � ∂ z N Tr ln( z − H ( τ )) = ∂ z N ln det( z − H ( τ )) so f N and G coincide only when N = ∞ (cumulant expansion). � 1 � N = ∞ 1 N ln det( z − H ( τ )) = N ln � det( z − H ( τ )) � , Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Airy function as the herald of the shock Shock wave corresponds to square root singularities Number of eigenvalues in the narrow strip of width s around strip λ 1 / 2 d λ = Ns 3 / 2 , so the � branch point scales like n = N spacing between the eigenvalues ( n = 1) scales like N − 2 / 3 ∼ µ 2 / 3 Then ± x = 2 √ τ + µ 2 / 3 s and f N ( x , τ ) ∼ ± 1 √ τ + µ 1 / 3 ξ N ( s , τ ) Solving viscid Burgers equation with above parametrization yields, in the large N , limit Riccati equation, with solution s ξ N ∼ ∂ s ln Ai ( 2 √ τ ) Herald of ”soft edge” universality Note that despite we know in this case the exact finite viscosity solution (monic, time-dependent Hermite polynomial), we do not need its form to infer the large N asymptotics at the end-points. Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Non-trivial boundary conditions Complex characteristics, nontrivial initial conditions � � G 0 ( z ) = 1 1 1 G ( z , τ ) = G 0 ( ξ [ z , τ )]) z − 1 + 2 z +1 ξ = z − G 0 ( ξ ) τ , so solution reads again G ( z , τ ) = G 0 ( z − τ G ( z , τ )) but now is given by the cubic (Cardano) equation Shock wave when d ξ dz = ∞ Novel phenomenon happens at τ = τ ∗ = 1, where square root branch points collide forming cubic root branch point (inflexion point, i.e. the change of curvature of the colliding shock waves). This triggers different scaling in viscosity ( N ), yielding the Pearcey function as the solution of the viscid Burgers at the collision of the shocks. Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Non-trivial boundaty conditions - visualization [Collage of Hokusai woodcut] � ∞ � � t 3 Two Airy heralds Ai ( x ) = 2 π −∞ exp i 3 + xt collide forming � ∞ � � t 4 4 + x t 2 Pearcey (Turrittin) herald P ( x , y ) = −∞ exp i 2 + yt Maciej A. Nowak Spectral shock waves
Real Burgers equation GUE Chiral GUE Catastrophes Chiral GUE Temporal dynamics of the matrix W ( τ ) � K † ( τ ) � 0 W ( τ ) = K ( τ ) 0 where K is a M × N complex matrix ( M > N ), whose elements are undergoing complex Brownian walk. We define ”zero modes number ” ν = M − N and ”rectangularity number” r = N / M . We define det( w 2 − K † K ) D ν N ( z , τ ) = � det( w − W ( τ )) � = w ν � � N ( z , τ ), where w 2 = z . ≡ z ν/ 2 R ν Using Grassmannian tricks we derive exact for any finite M , N evolution equations. Maciej A. Nowak Spectral shock waves
Recommend
More recommend
