The telegraphers equation

If the velocity dependence of the rate of a reaction could be assumed to be constant and equal to k for velocities in excess of u0 and zero below 0, then reaction could be regarded as bleeding-off those reactant (Brownian) particles which have an energy in excess lmti02. This perturbs the velocity distribution of reactants and hence of solvent molecules [446]. Under such circumstances, the Fokker—Planck equation should be used to describe the chemical reaction. If this simple form of representing reaction is incorrect, there is little that can be done currently. The Fokker—Planck equation contains too much information about Brownian motion. In particular, the velocity dependence of the Brownian particles distribution is relatively unimportant. Davies [447] reduced the probability 

Davies showed that the Fokker—Planck equation could be reduced to an equation of the form 

Monchick [525] has used the telegrapher s equation to describe chemical kinetics. Rice [484] solved the field-free form of the telegrapher s equation for the Smoluchowski boundary conditions, supplemented by n f=Q = 0, to find the rate coefficient as 

D = 10 Sm2s l, T = 300 K, m = 2 X 10 2S kg, hence the velocity relaxation time m/f — 0.05 ps. Little difference in these rate coefficients persist beyond 0.05 ps. 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

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Furthermore, there is approach by Camacho using a viewepoint of Irreversible Thermodynamics and leading to the Telegraph equation. For more details we refer to Camacho (1993a, b, c) and to the doctoral thesis Berentsen (2003). We plan to address this subject in the near future and extend our result in this direction. 

Monchick [36, 273] has used the diffusion equation and radiation boundary conditions [eqns. (122) and (127)] to discuss photodissociative recombination probabilities. His results are similar to those of Collins and Kimball [4] and Noyes [269]. However, Monchick extended the analysis to probe the effect of a time delay in the dissociation of the encounter pair. It was hoped that such an effect would mimic the caging of an encounter pair. Since the cage oscillations have periods < 1 ps, and the diffusion equation is hardly adequate over such times (see Chap. 11, Sect. 2), this is a doubtful improvement. Nor does using the telegraphers equation (Chap. 11, Sect. 3.3) help significantly as it is only valid for times longer than a few picoseconds. 

As a consequence, the telegraph equations (8.17) become the equations of diffusion... 

As we have already stated, Eq. (572) is called the telegraph equation.1 Applying Laplace transformation to Eq. (571), we obtain... 

Editor s Note A peculiarity of the telegraph equation, when used to approximately include inertial effects, is that it yields the correct result for the variance while yielding a poorer approximation for the distribution function than the Smoluchowski equation [see Ref. 248 and Risken (loc.cit.).]... 

The problem is reduced to solving a linear partial differential equation of hyperbolic type (the telegraph equation [482]) in the region t > 0, 0 < Y < oo. Specifically, we have the equation... 

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. 

Differentiating (2.29) with respect to t and (2.30) with respect to x and eliminating the mixed second derivatives, we obtain the telegraph equation... 

Note that the parabolic (3.6) predicts moisture propagation at an infinite speed. To overcome this unrealistic result, it has been suggested (Taitel 1972) that the above expression be modified by the telegraph equation, namely. 

During the derivation of the Fokker—Planck equation and telegrapher s equation, the derivatives with respect to distance were dropped at the second order. However, there is a case for including the third- and fourth-order spatial derivatives, though the third-order derivatives W2 will average to zero. This procedure leads to the Burnett equation... 

The original theory of Brownian motion by Einstein was based on the diffusion equation and was valid for long times. Later, a more general formulism including short times also, has been developed. Instead of the diffusion equation, the telegrapher s equation enters. Again, an indeterminacy relation results, which, for short times, gives determinacy as a limit. Physically, this simply means that a Brownian particle s... 

Here only n satisfies the telegrapher s equation, while j obeys a somewhat different equation because div grad grad div.]... 

Incidentally, the telegrapher s equation (17) with x = ihjlmc2 is satisfied by the Klein-Gordon (also Dirac) wavefunction for a free particle, if the factor exp [—(ijh)mc2f is split off from it. Thus, the time lag according to relativity corresponds to an imaginary relaxation time x. 

Equation (71) reduces to the telegrapher s-type equation found in the Brownian limit a = 1 [115]. In the usual high-friction or long-time limit, one recovers the fractional Fokker-Planck equation (19). The generalized friction and diffusion coefficients in Eq. (19) are defined by [75]... 

In a conducting medium (c / 0) with no source of charge (p = 0), the Maxwell equations yield the first telegraph equation (so named from a transmission-line theory for long-range telegraphy developed by Heaviside) [14] ... 

This is the celebrated telegrapher s equation, whose phenomenological pedigree dates back to Maxwell. His (Maxwell s) argument was to include relaxation into the wave equation and did not require the invocation of microscopic dynamics. However, his use of dissipation was compatible with the action of infinitely many degrees of freedom in the medium supporting the wave motion. 

Thus, for times shorter than T, the evolution of the Liouville density consists of two peaks traveling in opposite directions at the same speed, W. Note that this is the same early-time solution one would obtain for the solution to the telegrapher s equation [51]. 

For colored noise sources the derivation of evolution equations for the probability densities is more difficult. In a Markovian embedding, i.e. if the Ornstein-Uhlenbeck process is defined via white noise (cf. chapter 1.3.2) and v t) is part of the phase space one again gets a Fokker-Planck equation for the density P x,y, Similarly, one finds in case of the telegraph... 

In 1957 Marcel Golay published a paper entitled Vapor Phase Chromatography and the Telegrapher s Equation [Anal. Chem., 29 (1957) 928]. His equation predicted increased number of plates in a narrow open-tubular column with the stationary phase supported on the inner wall. Band broadening due to multiple paths (eddy diffusion) would be eliminated. And in narrow columns, the rate of mass transfer is increased since molecules have small distances to diffuse. Higher flow... 

Thus, for points in the medium, where the electric and magnetic properties do not vary spatially we have obtained equations involving only the electric or magnetic fields. The two equations are of identically the same form, being the second order in partial derivatives. They are sometimes known as telegraph equations for a conductive medium. 

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. 

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. 

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