Fronts in Direction-Dependent Reaction Walks

If we replace Brownian motion by its simplest generalization, the persistent random walk, we obtain direction-independent reaction walks as the simplest generalization of reaction-diffusion equations. Both describe chemical reactions in the reaction-limited or activation-controlled regime. However, the activation barrier is only implicitly taken into account it is incorporated into the kinetic coefficients 

In gas-phase reactions, molecules must collide with sufficient kinetic energy to overcome the activation barrier. Since velocity is not defined for Brownian motion, this requirement cannot be taken into account explicitly in a reaction-diffusion 

For the branching-coalescence kinetics, this amounts to replacing the kinetic scheme (1.56) and (1.57) by 

Here /c, = A, = FiZ, and P is the steric factor and Zj the collision frequency factor [350]. We will consider only situations where the steric factor ensures that the kinetics are reaction-limited. We assume that the pool species A is in equilibrium, i.e., /0+a = /0, a = In dimensionless variables we obtain the following 

The uniform steady states of the Fisher DDRW (5.80) and their stability properties are identical to those of the Fisher DIRW (5.61). We again look for propagating front solutions of (5.80a) and (5.80b), i.e., solutions of the form (5.62) and (5.63), and we obtain 

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