Absorbing boundary, probability distributions

Using the boundary condition BCl, where both boundaries atm = Oandm = N are absorbing, the probability distribution of the first passage time is given by Equation 6.116,... 

The Transition Probability. Suppose we have a Brownian particle located at an initial instant of time at the point xo, which corresponds to initial delta-shaped probability distribution. It is necessary to find the probability Qc,d(t,xo) = Q(t,xo) of transition of the Brownian particle from the point c 0 Q(t,xo) = W(x, t) dx + Jrf+ X W(x, t) dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary. Here we suppose that c and d are arbitrary chosen points of an arbitrary potential profile (x), and boundary conditions at these points may be arbitrary W(c, t) > 0, W(d, t) > 0. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

To summarize, in order to calculate the probability density of the FPTs to state from state 5., one needs to solve the FME or FPE for the auxiliary process with the absorbing boundary condition at obtain the probability current J t) into the absorbing state which then provides the FPT distribution through the relation F T,Sf S ) = J Sf,t +r S, Q. 

For the combination (BC3) of an absorbing boundary at x = L and a mixed boundary condition at x = 0, the same procedure as above can be carried out to And the various quantities associated with the FPT. For the sake of completeness with respect to different boundary conditions, we provide the formulas for the probability distribution function and the unconditional mean FPT. Other quantities can also be calculated in an analogous manner. 

One Absorbing Barrier and One Reflecting Barrier Let the boundary at m = 0 be reflecting and the boundary at m=be absorbing, namely the boundary condition BC2 in Table 6.1. The probability distribution function for the first passage time is given by Equation 6.124 as... 

The initial condition assumes that the particles are initially homogeneously distributed in space with a number concentration Nq. The first boundary condition requires that the number concentration of particles infinitely far from the particle absorbing sphere not be influenced by it. Finally, the boundary condition at r = 2 Rp expresses the assumption that the fixed particle is a perfect absorber, that is, that particles adhere at every collision. Although little is known quantitatively about the sticking probability of two colliding aerosol particles, their low kinetic energy makes bounce-off unlikely. We shall therefore assume here a unity sticking probability. 

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