Sergei Nechaev LPTMS, Orsay Thanks to: K. Khanin (Toronto) G. Oshanin (Jussieu) A. Sobolevski (Poncelet Lab, Moscow) O. Vasilyev (Stuttgardt) Shocks in “Tetris”

clusters “crack”

The questions of interest 1. Number of clusters surviving till height h 2. Mean-square displacement of cracks 3. Cluster’s mass distribution

Deposition rules i i+1i i i-1 rewrite this as a single “Langevin-type” equation

The equation with the multiplicative noise describes the growth in absence of surface relaxation and differs from the equation with the additive noise Define “thin” and “thick” discrete delta-functions and

Consider first the trivial dynamics described by equation. It can be rewritten as Since, one obviously gets So, the stochastic equations with multiplicative and additive noises are respectively

Discrete dynamic equation is a solution of a “variational problem” of finding a trajectory that maximizes the action for boundary conditions. “Physically” maximization of means that we find a trajectory that terminates in and passes through the maximal number of dropping events under the condition: unless is the dropping in adjacent column,.

Direct maximization of the action is a difficult problem (the solution depends on the whole future history). However, knowing we can easily reconstruct minimizing trajectory in a reverse time. Algorithmic implementation is as follows. Solve for given initial conditions and obtain set of values, then restore the path back in time to a given point. This procedure is known in optimization as dynamic programming and recursive equation for is a Bellman equation.

What we learn from all that? 1.There are discrete trajectories (“maximizers”) which maximize the discrete action. 2.Two maximizers meet at dropping event giving birth of a discrete shock (“crack”). The shock is located between two dropping events in adjacent columns (if they belong to different clusters).

Directed polymer (DP) in a random environment and KPZ growth Lagrangian of (1+1)D DP in a potential  (x,t) reads The partition function Z(x,t) satisfies the equation By using Cole-Hopf transform one can arrive at KPZ equation for h(x,t)

KPZ growth and Burgers turbulence By setting one passes from KPZ to Burgers equation with random forcing where u(x,t) is the velocity field, which is by definition For =0 and F(x,t)=0, the field u(x,t) gives rise to characteristics (“minimizers”).

The full derivative reads From inviscid Burgers equation it follows that Thus, the characteristics are straight lines defined by boundary conditions and can intersect, creating conflicts. These conflicts are called shocks.

we get The minimum is reached on minimizers x(t), with where and. By Legendre transform Why characteristics are minimizers? For =0 the height h(x,t) solves the Hamilton-Jacobi equation

For minimizers are not srtaight lines anymore. Shocks in inviscid Burgers Turbulence correspond to We conjecture the following dictionary For small there are no exact shocks anymore, but there are trajectories with Ballistic DepositionBurgers Turbulence Bellman equation ↔ Hamilton-Jacobi equation discrete maximizers (clusters) ↔ forest of minimizers discrete shocks (cracks) ↔ shocks

Burgers Turbulence and Ballistic Deposition (a) Aggregates growing by sequential deposition and highlighted cracks; (b) Growing aggregate in (2+1)D spacetime; (c) Density plot of 2nd local difference (discrete analog of 2nd derivative) of the height, which highlights the discontinuities corresponding to shocks.

Interface growth for Ballistic Deposition Let n=t/L be average number of particles per column. For KPZ universality class In (1+1)D one has. Ballistic Deposition belongs to KPZ universality class with scaling relations

Note that This allows one to rewrite dynamic equation as Continuous limit and relation to KPZ (Kardar-Parisi-Zhang equation)

Scaling of shocks in Burgers Turbulence Let d(t) be the horizontal size of the cluster at the time t 1. There is a “driving force” promoting the “smearing” of the cluster due to velocity fluctuations Increments are uncorrelated Hence, 2. There is “smearing” of a cluster due to random adding of new particles in vicinity of the cluster's boundary. The effects 1 and 2 lead to the growth of d(t): Thus,

Density of clusters and transversal fluctuations of cracks Let d be the average horizontal size of clusters, then the density of clusters is c ~1/d, and since t~h, one has The transversal fluctuations of cracks (the mean square displacement of the cluster’s boundary),

Mass distribution of clusters “Mass”, m(h), of a cluster alive at a height h is Since, by eliminating h we get The clusters are ranked (ordered) according to their masses and c is the rank in the corresponding Zipf's law. Let be the mass distribution. We have for “integral” distribution:. From definition of Zipf’s law, we have:. So

Silver electrodeposition in quasi-2d geometry C. Horowitz, M. Pasquale, E. Albano, A. Ariva, Phys. Rev. B, 70, (2004). We have  =1.4

“In”– and “out”– clusters correlations

Geometric interpretation of the standard RSK (Robinson-Schensted-Knuth) algorithm for the Young Tableau construction Plancherel measure Is here any interpretation of cracks?

Three-dimensional view of (1+1)-dimensional Asymmetric Ballistic Deposition Cracks in vertex models?