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Coarsening versus selection of a lenghtscale. Chaouqi Misbah , LIPHy (Laboratoire Interdisciplinaire de Physique) Univ . J. Fourier, Grenoble and CNRS, France. with P. Politi , Florence , Italy. 2 general classes of evolution. 1) Length scale selection. Time.
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Coarsening versus selection of a lenghtscale ChaouqiMisbah, LIPHy (Laboratoire Interdisciplinaire de Physique) Univ. J. Fourier, Grenoble and CNRS, France with P. Politi, Florence, Italy Errachidia 2011
2 general classes of evolution 1) Length scale selection Time Errachidia 2011
2 general classes of evolution 1) Length scale selection Time 2) Coarsening Time Errachidia 2011
Questions • Can one say if coarseningtakes place in advance? • Whatis the main idea? • How canthisbeexploited? • Can one saysomething about coarseningexponent? • Is this possible beyond one dimension? • How general are the results? Bray, Adv. Phys. 1994: necessity for vartiaionaleqs. Non variationaleqs. are the rule in nonequilibriumsystems P. Politi et C.M. PRL (2004), PRE(2006,2007,2009) Errachidia 2011
Someexamples of coarsening Errachidia 2011
Andreotti et al. Nature, 457 (2009) Errachidia 2011
That’s not me! Errachidia 2011
Myriadof pattern formingsystems 1) Finite wavenumber bifurcation Lengthscale (no room for complex dynamics, generically ) Amplitude equation (one or two modes) Errachidia 2011
2) Zero wavenumber bifurcation Errachidia 2011
2) Zero wavenumber bifurcation Far from threshold Complex dynamics expected Errachidia 2011
Can one say in advance if coarseningtakes place ? Yes, analytically, for a certain class of equations and more generally ……. Errachidia 2011
Whatis the main idea? Coarseningis due to phase instability (wavelength fluctuations) Phase modes are the relevant ones! Eckhaus stable Errachidia 2011 unstable
General class of equations (step flow, sand ripples….) Arbitrary functions Errachidia 2011
How canthisbeexploited? Errachidia 2011
Example:GeneralizedLandau-Ginzburgequation (trivial solution is supposed unstable) Example of LG eq.: or Unstable if Errachidia 2011
steady solution Patricle subjected to a force B Example Errachidia 2011
Coarsening U=1 U=-1 Kink-Antikink anihilation time +1 -1 Errachidia 2011
Stability vs phase fluctuations? :slow phase : Fast phase Local wavenumber: Errachidia 2011
Full branch unstable vs phase fluctuations :slow phase : Fast phase Local wavenumber: Errachidia 2011
Full branch unstable vs phase fluctuations :slow phase : Fast phase Local wavenumber: Sovability condition: Derivation possible for anynonlinearequation Errachidia 2011
Full branch unstable vs phase fluctuations :slow phase : Fast phase Local wavenumber: Sovability condition: Errachidia 2011
Particle with mass unity in time Subject to a force Errachidia 2011
Particle with mass unity in time Subject to a force is the action Errachidia 2011
Particle with mass unity in time Subject to a force is the action But remind that :energy Errachidia 2011
Particle with mass unity in time Subject to a force is the action But remind that :energy Errachidia 2011
has sign of A: amplitude : wavelength Errachidia 2011
wavelength No coarsening amplitude Errachidia 2011
wavelength No coarsening coarsening amplitude Errachidia 2011
wavelength No coarsening coarsening amplitude Interrupted coarsening Errachidia 2011
wavelength P. Politi, C.M., Phys. Rev. Lett. (2004) No coarsening Coarsening amplitude Interrupted coarsening Coarsening C.M., O. Pierre-Louis, Y. Saito, Review of Modern Physics(sous press) Errachidia 2011
General class of equations (step flow, sand ripples….) Arbitrary functions Errachidia 2011
Sand Ripples, Csahok, Misbah, Rioual,ValanceEPJE (1999). Errachidia 2011
Example: meandering of steps on vicinal surfaces Wavelength frozen branch stops amplitude O. Pierre-Louis et al. Phys. Rev. Lett. 80, 4221 (1998) and manyother examples , See :C.M., O. Pierre-Louis, Y. Saito, Review of Modern Physics(sous press) Errachidia 2011
Andreotti et al. Nature, 457 (2009) Errachidia 2011
Dunes (Andreotti et al. Nature, 457 (2009)) Errachidia 2011
Can one saysomething about coarseningexponent? P. Politi, C.M., Phys. Rev. E (2006) Errachidia 2011
Coarsening exponent LG GL and CH in 1d Other types of equations Errachidia 2011
Some illustrations If non conserved: remove If non conserved Use of Errachidia 2011
Coarsening time U=1 U=-1 Finite (order 1) Errachidia 2011
Remark: what really matters is the behaviour of V close to maximum; if it is quadratic, then ln(t) Conserved: Nonconserved Errachidia 2011
Other scenarios (which arise in MBE) B(u) (the force) vanishes at infinity only Conserved Non conserved Benlahsen, Guedda (Univ. Picardie, Amiens) Errachidia 2011
General class of equations (step flow, sand ripples….) Arbitrary functions Errachidia 2011
Transition from coarsening to selection of a length scale Golovin et al. Phys. Rev. Lett. 86, 1550 (2001). Cahn-Hilliard equation coarsening Kuramoto-Sivashinsky After rescaling no coarsening For a critical Fold singularity of the steady branch Wavelength Errachidia 2011 Amplitude
New class of eqs: new criterion ; P. Politi and C.M., PRE (2007) KS equation If Steady-state periodic solutions exist only if G is odd If not stability depends on sign of Errachidia 2011
Extension to higherdimension possible C.M., and P. Politi, Phys. Rev. E (2009) Analogywithmechanicsisnot possible Phase diffusion equationcanbederived A linkbetweensignof D and slope of a certain quantity (not the amplitude itselflike in 1D) The exploitation of allows extraction of coarsening exponent Errachidia 2011
Summary Phase diffusion eq. providesthe key for coarsening, D is a function of steady-state solutions (e.g. fluctuations-dissipation theorem). 1) 2) D has sign of for a certain class of eqs 3) Which type of criterionholds for other classes of equations? But D canbecomputed in any case. 4)Coarseningexponentcanbeextracted for anyequation and at any dimension fromsteadyconsiderations, using Errachidia 2011
