The dynamics of diffusion mediated reactions processes has been extensively studied  for many years. It was the objective of these studies the proposal of theoretical models for describing chemical reactions. The schema so developed has also shown useful for the description of diverse phenomena in physics, biology or echology     . The reaction process is modeled as a trapping reaction or and it is said diffusion controlled when the reactives have to diffuse to meet in space and then react.
One hundred years ago, Smoluchowski  proposed a simplified model assuming a very dilute species (the minority species denoted here as T, the trap) so that it can be considered as a single isolated spherical particle T sorrounded by a swarm of diffusing pointlike particles A. In Smoluchowski’s model particles A diffusion coeficient is the sum of the individual coeficients of A’s and T’s respectively. In the original model the trapping is inmediate (perfect trapping) upon the collision of an A particle with the sphere T. The reaction rate is computed from the flux of A’s into the sphere.
Since this pioneering work different extensions have been proposed to include diffusion in disordered media    or to give a better description for the short time behaviour introducing a finite reaction time  (imperfect trapping). In imperfect trapping the particles may even separate after en encounter without reacting  .
The discrete formulation of the problem is also of interest in many physical problems   and trapping models have also been formulated based on Random Walks on lattices. Among these models we mention imperfect trapping   with a finite reaction rate and dynamic or gated trapping      when the reaction is also modulated by an another independent reaction that switches the trap between an active and an inactive state.
Recently it has also been considered an evanescent Random Walk   when the walkers have a finite lifetime.
Another extension of Smoluchowski’s model is to consider mobile traps. Koza and Taitelbaum  have proposed a model of a mobile imperfect trap in the continuous space with the same diffusion coeficient for both species. Sánchez et al.  and Sánchez  have considered a mobile trap that diffuses but with no explicit diffusion model. In these works, the problem is considered in the trap reference system and in the laboratory system respectively.
In this article, we propose a generalized Continuous Time Random Walk for imperfect trapping with both species diffusing with different coeficients. We assume separable Random Walks and a time dependent reaction rate. In Section 2, we present a noninteracting two particles Random Walk to introduce the main magnitudes of the model. In Section 3, we include the reaction process in the model through an statiscally independent process. It is calculated in this section the local and global probability density for the time of reaction or the absorption probability density. In Section 4, an initial uniform distribution of the majority species is considered and we calculate the local and global reaction rate. It is verified that the global reaction rate coincides with the reaction rate for an immobile trap. The time dependent concentration of particles A is also calculated. In Section 5, we present the results from the trap reference system. In this frame we recover some results of the immobile trap model. Finally in Section 6, we illustrate the results of our model with a one dimensional Random Walk. Discussion and conclusios follow.
2. Two Particles Random Walk
Let us consider a CTRW of two non-interacting particles on an homogeneous infinite lattice. The particles will be identified as A and T respectively. The position of each particle is given by an integer vector: for particle A and for particle T. Each particle makes a generalized, separable, statistically independent RW, with transitions rates for a transition of particle A and for a transition of particle T res- pectively. The sojourn probability of the walkers at a given configuration is
with . In this way the diffusion coeficients for a lattice parameter a are respectively and .
In turn the probability for a transition from configuration to con- figuration between t and after a sojourn time t is given by with the probability density
Here we are neglecting simultaneous transitions of both particles for being of second order in . This is the main magnitude in our model since all the results will be expressed in terms of this density.
Let be the conditional probability of reaching con- figuration between t and given that the walkers started at the configuration . The probability density satisfies the recursive rela- tion
since the walkers may reach the configuration by a transition from a previous configuration or is the initial configuration, these being mutually exclusive events.
We are representing here by the symbol the convolution product
is the Green’s function for the diffusion problem of two particles.
An analytic solution for the Green’s function is obtained by taking Fourier transform in the spatial coordinates and Laplace transform in the time variable
Hereafter the superindex L represents the Laplace transform in the temporal variable
and the symbol denotes the Fourier transform in spatial variables
In this way the Green’s function Fourier-Laplace ( ) transform in (5) is given in terms of the transform of the transition probability density (2) only.
Jointly the conditional probability of a given configuraion at time t of the particles is
after reaching the confiugration at time the walkers must stay in these positions at least for a time .
The marginal probability for the A particle position
is in the transform representation
the transition probability density and sojourn probability for the A walker. This result is consistent with the assumption of no interaction between the particles: particle A diffuses with diffusion coeficient on an infinite lattice of parameter a.
A similar result is obtained for the position marginal probability of particle T position
If we have an uniform distribution of A particles, in each lattice site, the particle concentration at site at time t is given by
and, replacing (5) in (9) and then in (14), we get in the transform representation
with d the dimension of space.
So it is verified that the uniform distribution is the equilibrium distribution for the noninteracting particles system.
3. The Reaction Process
In this section we include in the model the possibility of the reaction when both particles are at the same position in the lattice. In this case the particles may separate by one of them changing position or they may react with a time dependent reaction rate . Assuming diffusion and reaction as statiscally independent processes, the sojourn probability of the walkers at the same position is 
Concurrently the probability density for the time of separation of the particles is
and the probability density for the time of reaction is
When particles A and T are not at the same position the state of the system can only change by one of the particles jumping to another position with probability density as given by (2).
We denote by the Green’s function for the diffusion of the particles when the reaction is allowed. It must satisfy the recursive relation
This equation is similar to (3) but here
We obtain an analytic solution of Equation (21) in the transform representation by an extension of the local inhomogeneity method   and making use of the identity 
with d the dimension of the space.
The probability density for the time of both particles arriving at the same position is in the transform repre- sentation
with the definition
while the Green’s function is in the transform representation
More details of this calculation is given in Appendix A.
For the reaction to take place the walkers have to be at the same position. So the probability density for the time of reaction at a particular position or local absorption probability density is given by the convolution product
since for the reaction to take place both particles have to meet at the same lattice site and react before they move away.
In the representation we get from (18) and (24)
With this result we may acknowledge Equation (26) as the transform of
i.e. the Green’s function for the reaction process is the Green’s function of the two particles RW (3) minus the contribution of those realizations with a reaction at a previous time. Here
is the probability density for the time of reaching the configuration after making at least one transition. For and this density is the equivalent of the probability density of the time of return to the origin for one particle random walk.
It is interesting to note that in the case of inmediate reaction upon encounter
The probability density of the meeting time (in this case the first meeting due to inmediate reaction) independent of the lattice position is
Taking Laplace transform, evaluating Equation (24) at and using (31) we get
may be identified with Siegert’s formula for the First Passage Time for the lattice position of a walker that started a RW at with transition probability density
by recognizing the identity
the Green’s function of one particle Random Walk with transition probability density (34) on an infinite lattice  . Thus it is verified Smoluchowski’s assumption that for the global reaction process we may consider one of the particles fixed on the lattice while the other diffuses with coeficient the sum of the individual diffusion coeficients.
The conditional probability of finding the walkers in configuration at time t assuming that they started at configuration is given by the convolution product
i.e. as in the case of (9), after reaching the configuration at time the walkers must stay in this configuration at least for a time .
4. Reaction Rate
We consider in this section a distribution of particles A with initial uniform concentration on the lattice. This is the equilibrium concentration in the absence of reaction for the diffusion model under consideration. We assume that the A particles do not interact among them. Then the local reaction rate at at time t when T starts at ; i.e. the number of particles that react at with T between t and is with the probability density
In the transform representation we get from (28)
is the transform of Green’s function of the isolated particle T random walk.
The global reaction rate, defined as is in the Laplace transform representation
Note that the global reaction rate does not depend on the initial trap position as should be in a homogeneous lattice.
This reaction rate is coinciding with that of a set of walkers in the presence of a fixed trap when the walkers diffuses with the equivalent transition density (34).
In the presence of reaction the marginal probability of an A particle position is
with in (35). From this marginal probability we calculate the concentration of A’s at a particular site
transforming this equation, using (15) and (36) and noting that
In the space-time representation this result corresponds to
the local concentration is the uniform initial concentration deducted the contri- bution of those A particles that have reacted at a previous time .
5. Description from the Trap Reference System
We present the previous results as described from the trap position. Let us start calculating the conditional probability of the relative position to the trap T of an A particle given by the coordinate . From (35) by the variable trans- formation method  
and in the representation
From this probablity we calculate the concentration of A’s, that in the representation is
Going back to the space time representation we may understand this result
in the following way: the concentration of A’s at a given distance from the trap is given by the initial concentration subtracting the contribution of those realizations with a reaction at a previous time.
Note that the concentration as seen from the trap position is coinciding with the concentration for a fixed trap position as could be expected.
From this result we may conclude that the description of the reaction process from the trap position is comparable to the model of the trap at a fixed position on the lattice.
The main difference among the results at the reference systems appears in the A’s concentration as given in (43) in the laboratory frame when compared with (47) in the trap frame.
6. One Dimensional Random Walk
We present here the results of our model when we consider an one dimensional Random Walk as schematically shown in Figure 1. The probability density for the change of configuration is
so that particle A makes a biased Random Walk. With this probability density the integral in (25) is
where we have defined
Figure 1. One dimensional Random Walk scheme. Transition rate for the trap T is and for particle A is . Particle T makes a symmetric Random Walk while particle A makes a biased Random Walk: is the probability of jumping to the left and is the probability of jumping to the right.
For the reaction process we assume an exponential dynamics with a constant reaction rate
For this one dimensional model the local reaction rate in the repre- sentation is
and the A’s concentration in the laboratory frame is
In turn the A’s concentration in the trap frame is
In Figure 2 we plot the normalized concentration in both reference frames: the laboratory frame and the trap frame as a function of the distance to the initial trap position, , and as a function of the relative position to the trap, , respectively. We include several curves with an adimensional time as a paramenter.
Figure 2. Majority species concentration in presence of a mobile imperfect trap vs. relative position to the trap, trap reference frame, (filled symbols in (a), thick lines in (b)) or vs. relative position to the initial position of the trap, laboratory reference frame, (hollowed symbols in (a), thin lines in (b)). It can be appreciated a deeper and narrower depletion zone around the trap in the trap reference frame.
As can be seen there is a greater scavenging at the trap position when the concentration is calculated in the reference frame of the trap. A more extended scavenging region is observed around the initial trap position in the laboratory frame. This apparent disagreement can be understood if we take into account that in the laboratory frame we are considering an average realization of the trap Random Walk while in the trap frame we are just on the trap.
7. Discussion and Conclusions
We have presented a theoretical study of diffusion mediated reaction processes to extend previous treatments of imperfect trapping. Here both species, the minority and majority species, diffuse with arbitrary constants. We have calculated global and local absorption probability densities, reaction rates and concentration of the majority species in the laboratory and trap reference frames. All the results are expressed in terms of the relevant magnitudes of the model, the transition probability density between configurations of the system and the reaction dynamics. We have got analytical results in the Fourier Laplace trans- form representation.
Fixed trap absorption probability density and reaction rate are reobtained when we consider the global behaviour in the laboratory frame or in the trap frame of reference. The main difference between mobile and fixed trap models emerges when we consider the majority species concentration as can be appreciated in the one dimensional case of Section 6. In the trap frame, there is a deeper and spatially more reduced scavenging zone when compared with the concentration in the laboratory reference frame where the scavenging is not so deep but more extended spatially. This difference can be understood when we take into account that in the laboratory frame we are considering an average behaviour of the trap diffusion process, while in the trap frame we are con- sidering a particular realization.
The authors thank SeCyT-UTN for partial support of this project.
Appendix A: Local Inhomogeneity Method
Starting with the recursive relation (21) in the Laplace representation we work out the sum . Noting that in the Laplace representation
and making use of the recursive relation (3) we get
It should be noted that in the right hand side it appears as defined in (24). Then we evaluate (56) at and take Fourier transform obtaining (24). For this calculation use have been made of identity (23) and definition (25).
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