Reaction-telegraph equation

Reaction-Cattaneo Systems and Reaction-Telegraph Equations... 

We call the hyperbolic system (2.17) and (2.18) a reaction-Cattaneo system. Eu and Al-Ghoul have derived such systems from generalized hydrodynamic theory [9, 7, 8, 6]. Reaction-Cattaneo systems can also be obtained from extended irreversible thermodynamics [223], see for example [282]. If we differentiate (2.17) with respect to t and (2.18) with respect to x and eliminate mixed second derivatives, we obtain the so-called reaction-telegraph equation. 

Remark 2.1 The reaction-telegraph equation can also be derived as the kinetic equation for a branching random evolution, see [101]. 

Remark 2.2 Nomenclature in this field is unfortunately not uniform, and some authors use the term hyperbolic reaction-diffusion equations for reaction-telegraph equations. 

Note that the reaction-telegraph equation (2.19) differs from the ad hoc HRDE (2.15) by the additional term —xF p) dp/dt) on the left-hand side. It can be shown that solutions of (2.19) converge to solutions of the reaction-diffusion equation (2.3) as T 0 [494]. Traveling wave front solutions for the reaction-telegraph equation have been investigated by several authors [201, 176, 282, 291, 285, 136, 288, 137, 114, 116, 115, 117]. 

Even if p x, 0) > 0, the solution p x, t) of (2.19) will in general assume also negative values [178], which is unacceptable for a tme density, (ii) In order to ensure the dissipative character of the reaction-telegraph equation (2.19), the damping coefficient I - tF ip) must be positive, i.e.. 

This relation between the relaxation time t of the flux and the time scale l/F (p) of the reaction appears to be a purely mathematical requirement. The following mesoscopic approach will shed light on the foundational problems of the reaction-Cattaneo system (2.17) and (2.18) and the reaction-telegraph equation (2.19) hinted at by points (i) and (ii). 

For n-variable systems, the reaction-Cattaneo systems and reaction-telegraph equations read, i = 1,... 

Substitution of (3.146) into (3.150) and inversion of the Fourier and Laplace transforms leads to the RD equation (2.3) with D = j l (t>- If we substitute (3.147) into (3.150) and invert the Fourier and Laplace dansforms, we obtain the reaction-telegraph equation ... 

If one considers the exponential memory kernel in (5.41) and takes the temporal derivative, the reaction-telegraph equation (2.19) is recovered. 

In this section we consider the problem of front propagation for the three-dimensional reaction-telegraph equation involving KPP kinetics. The goal is to derive the equation governing the evolution of the front in the long-time large-scale limit as in the previous sections and to show that this equation is identical in form to the relativistic Hamilton-Jacobi equation. 

Consider the reaction-telegraph equation (2.19) in three dimensions. 

Consider the reaction-telegraph equation (5.48) in one dimension. Show that the Hamilton-Jacobi equation is given by... 

Remark 10.2 These equations do not constitute a reaction-Cattaneo system because of the contribution of the intrinsic death rates f (p , Py), / = 1, 2, to the decay rate of the flows. Further, no reaction-telegraph equations can be derived for the total densities, unless the death rates /i (p , Py) are constants. 

Hadeler, K.R Reaction telegraph equations and random walk systems. In van Strien, S.J., Verduyn Lunel, S.M. (eds.) Stochastic and Spatied Structures of Dynamical Systems, pp. 133-161. North-HoUand, Amsterdam (1996)... 

From a mathematical viewpoint, the origin of the infinitely fast spreading of local disturbances in the diffusion equation can be traced to its parabolic character. This can be addressed in an ad hoc manner by adding a small term rdffp to the diffusion equation or the reaction-diffusion equation to make it hyperbolic. From the diffusion equation (2.1) we obtain the telegraph equation, a damped wave equation. 

If reactant velocity does not influence the rate of reaction when an encounter pair is formed (see Sect. 2.4), the effect of velocity may be removed from an analysis of the solute motion. Davies [447] showed that, when the velocity distribution is of no interest, the position and time distribution of a solute is described by the telegrapher s equation. It is a diffusion-like process, but one where the particle has a limiting velocity so that a wave of solute probability spreads out with a... 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

---