Front (physics)

In physics, a front{{cite book|last1=Pismen|first1=L. M.|title=Patterns and interfaces in dissipative dynamics|date=2006|publisher=Springer|location=Berlin|isbn=978-3-540-30430-2}}{{cite book|last1=Horsthemke|first1=Vicenç Mendéz, Sergei Fedotov, Werner|title=Reaction-transport systems : mesoscopic foundations, fronts, and spatial instabilities|date=2010|publisher=Springer|location=Heidelberg|isbn=978-3642114427}} is an interface between two different possible states (either stable or unstable) in a physical system. For example, a weather front is the interface between two different density masses of air, in combustion where the flame is the interface between burned and unburned material or in population dynamics where the front is the interface between populated and unpopulated places. Fronts can be static or mobile depending on the conditions of the system, and the causes of the motion can be the variation of a free energy, where the most energetically favorable state invades the less favorable one, according to Pomeau{{cite journal |last1=Pomeau |first1=Y. |date=1986 |title=Front motion, metastability and subcritical bifurcations in hydrodynamics |journal= Physica D: Nonlinear Phenomena|volume=23 |issue=1–3 |pages=3–11 |doi=10.1016/0167-2789(86)90104-1 |bibcode=1986PhyD...23....3P }} or shape induced motion due to non-variation dynamics in the system, according to Alvarez-Socorro, Clerc, González-Cortés and Wilson.{{cite journal |last1=Alvarez-Socorro |first1=A. J. |last2=Clerc|first2=M.G.|last3= González-Cortés|first3=G|last4=Wilson|first4=M.|date=2017 |title= Nonvariational mechanism of front propagation: Theory and experiments |journal= Physical Review E|volume=95 |issue=1 |pages=010202 |doi=10.1103/PhysRevE.95.010202 |pmid=28208393 |bibcode=2017PhRvE..95a0202A |hdl=10533/232239 |hdl-access=free }}

From a mathematical point of view, fronts are solutions of spatially extended systems connecting two steady states, and from dynamical systems point of view, a front corresponds to a heteroclinic orbit of the system in the co-mobile frame (or proper frame).

File:Moving magnetic domains by Zureks.gif

Fronts connecting stable - unstable homogeneous states

The most simple example of front solution connecting a homogeneous stable state with a homogeneous unstable state can be shown in the one-dimensional Fisher–Kolmogorov equation:

:: $N_{t} = D N_{xx} + r N (N_0-N)$

that describes a simple model for the density $N(x,t)$ of population. This equation has two steady states, $N = 0$ , and $N = N_0$ . This solution corresponds to extinction and saturation of population. Observe that this model is spatially-extended, because it includes a diffusion term given by the second derivative. The state $N \equiv N_0$ is stable as a simple linear analysis can show and the state $N = 0$ is unstable. There exist a family of front solutions connecting $N=N_0$ with $N=0$ , and such solution are propagative. Particularly, there exist one solution of the form $N(t,x)=N(x-vt)$ , with $v$ is a velocity that only depends on $D$ and $r$ {{cite journal |last1=Uchiyama |first1=Kohei |title=The behavior of solutions of the equation of Kolmogorov–Petrovsky–Piskunov |journal=Proceedings of the Japan Academy, Series A, Mathematical Sciences |date=1977 |volume=53 |issue=7 |pages=225–228 |url=https://www.researchgate.net/publication/38389672|doi=10.3792/pjaa.53.225 |doi-access=free }}

File:Front Solution.png

File:Propagating front to left.png

References

Category:Concepts in physics