Feedback Couplings in Chemical Reactions

Transcription

1 Feedback Coulings in Chemical Reacions Knud Zabocki, Seffen Time DPG Fühjahsagung Regensbug

2 Conen Inoducion Moivaion Geneal model Reacion limied models Diffusion wih memoy Oen Quesion and Summay DPG Fühjahsagung Regensbug

3 Inoducion Gowing inees in non-makovian effecs in dynamical sysems Desciion of memoy in evoluion models 1Physics, i.e. Econohysics Biology, i.e. Poulaion dynamics 3Chemisy Inclusion of memoy hough nonlinea feedback, couling of aes DPG Fühjahsagung Regensbug

4 Moivaion Densiy-densiy coelaion in undecooled liquids Moi-Zwanzig ojeco fomalism Coninuous Time Random Walks Tanso in heeogeneous oous media viscoelasic media Non-Fickian anso, i.e. Caanneo s law j,, j,, DPG Fühjahsagung Regensbug

5 Time delays in oulaion dynamics and hei influences on diffusion and gowh Nicholson s blowfly lucilia cuina 1954,1957 Desciion wih discee delay logisic equaion by May 1974 dn N N 1 d N Egg-o-adul develomen ime 8.5 days N numbe of adul flies N 75 N max 89 N min osc osc osc N av DPG Fühjahsagung Regensbug

6 Model Invesigaion of evoluion equaions Relevan vaiable,, M[, ] d d K,, L[, ] Nonlinea insananeous em : M [] Memoy em: convoluion ye K memoy kenel L linea funcional of DPG Fühjahsagung Regensbug

7 Secial ealizaion Time local em M [] consiss of anso em and eacion em Tanso mechanism: diffusion D, Reacion em: olynom in descibing gain and loss wih aoiae aes R[] Linea oeao L ime deivaive o ge feedback couling of aes D R[ ] d d K,,, DPG Fühjahsagung Regensbug

8 Quesion: choice of he memoy kenel K Unil now: Reacion-limied case no saial flucuaions, homogeneous 1 Evoluion model wih a cumulaive feedback couling PRE 65, 5616 Memoy diven Ginzbug-Landau model PRE 66, 6114 u 3 u d d DPG Fühjahsagung Regensbug

9 3 Delay-conolled eacions subm. o Phys. Le. A u K d - ime local em: sonaneous ceaion of aicles and single-secies ai annihilaion - wo choices of memoy kenel: exenal feedback eesened by exlici funcions a Discee ime delay K b Fading memoy K ex c Damed eiodic K ex cos DPG Fühjahsagung Regensbug

10 b Exonenial kenel: equivalen oblem second ode o.d.e. dimensionless equaion: a damed oscillao in anhamonic oenial ~ u d d ~ u d d 1 1 d d 1 U oenial sae-deenden daming, 1,.7,1.,1.3 Sabiliy domain of s 1.5, 1,.7,1.,1.3.5, DPG 1, Fühjahsagung.7,1.,1. 3 Regensbug

11 DPG Fühjahsagung Regensbug - Inenal feedback eesened by sae- deenden memoy kenel K=K[]= - saionay soluion - Tivial soluion unsable d 1 1 ; s dimensionless fom u

12 Sabiliy domain of he non-ivial fixed oin DPG Fühjahsagung Regensbug

13 DPG Fühjahsagung Regensbug Diffusion wih memoy Inclusion of a delayed eacion em in a diffusion equaion Saially local, emoal consan kenel Linea inhomogeneous equaion K d d D,,,,, K ], [,, D

14 Inhomogeneiy due o iniial disibuion, Wihou inhomogeneiy diffusive moving sonaneously decaying aicles wih ae Exac o aoximae soluion of he equaion fo diffeen saing disibuions and saial dimensions d=1,,3 a Dela-Funkion b Gauß-Veeilung d ex c Exonenielle Veeilung ex Non-ivial saionay soluion caused by he inhomogeneiy DPG Fühjahsagung Regensbug

15 Dela-disibued iniial funcion d, D 1, d 1, 1.5, D 1, d 1, 1 1, D 1, d 1, 1 3, D 1, d 1, 1 Invese lengh scale Saionay solulion,.5,1.,. D DPG Fühjahsagung Regensbug

16 Oen Quesion and Summay Evoluion equaions wih memoy em of convoluion ye Concee sysems exhibiing memoy Aoiae choices of memoy kenel Ohe ossibiliies modelling memoy Bounded domains bounday condiions Secial iniial disibuions Memoy induced aen DPG Fühjahsagung Regensbug