THEORETICAL MODEL. δ =0Øpotential dynamics

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1 THEOETICL MODEL t + [ g + g ] t + [ g + g ] t + [ g + g ] g +µ δ j, x j t r r δfbh r + δ δ (,, ) r r f( ) î δ 0Øpotential dynamics rδf r r δ BH dr r f ( ) + c.c. 0 δ nonrelaxational potential flow δ 0 otherwise Ø nonpotential

2 ZEO-DIMENSIONL SYSTEMS ZEO-DIMENSIONL SYSTEMS j 0,,,, ) ( ) ( θ j e t t i j j j, ] ) ( ) ( [ ) ( ] ) ( ) ( [ ) ( ] ) ( ) ( [ ) ( + µ δ + µ + δ + µ δ + µ + δ + µ δ + µ + δ t t t & & & θ θ θ & & & Equations for the real variables {,, } instead of complex variables {,, } Similar sets of equations proposed to study population competition dynamics

3 Stationary Solutions Null: (,, ) (0,0,0) hombus : (,, ) µ + δ µ ( µ + ) δ olls : (,, ) {(,0,0),...} Hexagon H: (,, ) /(+µ), µ δ µ ( µ + ) δ, 0, K Stable somewhere in (µ,δ) parameter space Küppers-Lortz instability (,0,0) (0,,0) (0,0,)

4 The Case µ 0 Orthogonality condition holds (δ µ 0) fl nonrelaxational potential flow & j V + δ f j, j,, elaxational part j esidual dynamics V ( + + ) + ( + + ) Equations of motion x( t) ( t) + ( t) + ( t) t e + ( ) fter t O() fl x(t) º (asymptotic dynamics) x 0

5 Eliminating for instance : δh & δ δ Hamiltonian dynamics H (, ) ( ) E ( energy ) δh & δ δ E (0) (0) (0) [ (0) + (0) + (0)] (only depends on initial conditions) Explicit solutions for the period of the orbits and the amplitudes T ( E) K( q) log E ( + O( E)) δ b( a c) δ E Ø 0

6 µ 0, δ.

7 The case µ t 0 Time-dependent energy ( ) de E t 4µ f ( t) E, f ( ) ( + + ) dt t > Motion occurs near the plane + + [after a transient time of order ] 0 µ 0., δ.

8 Period of the orbits is function of time E( t) E(0) exp( 4µ T T ( E( t)) long times (small energies) t t 0 f ( t ) dt ) E( t log E( t) δ 0 ) e T 4µ ( t t 0 0 ) 6µ + t δ ó δ., µ0.0 é δ, µ0.0 ñ δ., µ0. í δ, µ0. lines with slope 6µ/δ

9 Busse-Heikes Model with Noise Period increasing with time is unphysical. Solution: addition of fluctuations & & & [ [ [ ( + µ + δ) ( + µ + δ) ( + µ + δ) ( + µ δ) ( + µ δ) ( + µ δ) ] ] ] + ξ (t) + ξ (t) + ξ (t) White noise processes: ξ i (t)ξ j *(t ) εδ(t-t )δ ij Noise prevents E(t) from decaying to zero. Fluctuating period is established (but periodic on average) E(t) E T(t) T T( E )

10 ó δ., µ0.0 é δ, µ0.0 ñ δ., µ0. í δ, µ0. µ 0., δ., ε 0 6 ε 0 0 6

11 P E verage energy st ( st µ 0) exp[ V ( µ 0) / ε] P ( µ ) exp[ Φ ( µ ) / ε] exp[ V ( µ ) / ε] µ small d d d E exp( V / ε) d d d exp( V / ε) Saddle-point type integration: E (ε/µ), ε 0, µ small ó δ., µ0.0 é δ, µ0.0 ñ δ., µ0. í δ, µ0. T T ( E ) log( µ / ε) δ ε 0 0 6

12 ONE-DIMENSIONL SYSTEMS Three competing nonconserved real order parameters with short-range interactions & & & xx xx xx + [ + [ + [ ( + µ + δ) ( + µ + δ) ( + µ + δ) ( + µ δ) ( + µ δ) ( + µ δ) ] ] ] Prothotypical nonpotential problem to study domain growth and dynamical scaling We focus on the region below the KL instability: coexistence of three stable homogeneous states under nonpotential dynamics (δ 0)

13 Dynamics of of an an Isolated Kink Isolated kink moves due to nonpotential effects v(µ,δ)h(µ) δ + O(δ ) v Simulation Perturbative approach

14 Multikink Configurations Kink motion ô annihilation ô domain growth Front motion due to: attractive interaction forces + nonpotential effects t L(t) ± v(µ,δ) γ exp( µ / L(t)) L(t) ~ t L(t) ~ log t

15 Domain Growth and Dynamical Scaling L(t) ~ log t + crossover + t crossover L(t) ~ log t L(t) ~ t η.5, δ0

16 TWO-DIMENSIONL SYSTEMS We take real variables, isotropic diffusion terms and we focus on the region below the KL instability point Normal front velocity v n (r, t ;µ, δ) κ(r, t) + v p (µ, δ) v p (µ, δ) ~ δ + O(δ ) bsence of coarsening for system sizes large enough nonpotential dynamics + formation of vertex points POTENTIL µ, δ 0 NONPOTENTIL µ, δ

17 Spiral dynamics Spiral dynamics otation angular velocity w(µ,δ) v p / k 0 / v p small k 0 ~ v p (d) w~v p (d) ~d

18 Critical distance for vertex annihilation d 0 ~ δ - Consequence: coarsening will occur for system sizes S d d c

19 Vertex Motion For long times vertices diffuse through the system Vertex dynamics affected by the boundary conditions Periodic BC: - Even number of vertices: half and half - Vertices disappear by pairs of opposite sence of rotation - Correlated motions are observed - No restrictions about the number of vertices Null BC: - Vertices may disappear through the egdes of the system - No correlated motions observed

20 Null BC Periodic BC

21 Domain Growth and Dynamical Scaling δ 0 (potential limit) L(t) t / + dynamical scaling ( fields) δ 0 (nonpotential limit) L(t) L t / + dynamical scaling ( fields)

22 Spatial-dependent Terms lternative explanation for period stabilization Two kinds of differential operators j, j,, (ê j ) ISOTOPIC NISOTOPIC ( NWS, GOS terms) We focus on the region beyond the KL instability point Dynamics depends on the type of spatial derivatives and on the size of µ

23 µ small Similar morphology of domains lternating period dominated by the KL instability Intrinsic KL period stabilizes to a statistically constant value with both kinds of terms (ê j )

24 µ large Different morphology of domains Intrinsic KL period Different alternating periods : diverges with time (ê j ) : saturates to a constant value (ê j )