Indefinite Potential Solutions of Stochastic Particle Motions

in a Finite Domain with Absorbing Barrier Conditions

Fethi Bin Muhammed Belgacem, Ahmed Abdullatif Karaballi

Kuwait University, P.O. Box 5969, Safat 130

Kuwait

In this paper, we show that despite their distinction, both the Stratonovich and Ito calculi lead to the same reactive Fokker-Planck Equation:

(1)

describing the stochastic dynamics of a particle moving under the influence of an indefinite potential , a drift and a constant diffusion . We treat the periodic-parabolic eigenvalue problem (1) for finite domains having absorbing barriers. We show that under conditions required by the maximum principle, the positive principal eigenvalue (and the negative principal eigenvalue ) is connected to the probability eigendensity function by a Raleigh-Ritz like formulation. In the process, we establish the manner of effect of the drift and any inducing potential on the size of the principal eigenvalue. We show that the degree of convexity of the potential plays a major role in this regard.

In the absence of randomness, the directed motion of a particle on the real line due to an existing differentiable potential [1] is given by:

(1.1)

The drift function is then the velocity with which the particle travels and in the event that B does not vary with time, the motion of the particle is steady, and satisfies the equation

(1.2)

Assuming an additional random component for the motion of the particle without violating its continuity, equation (1.1) can be expanded to the Langevin equation:

(1.3)

Here, the function is rapidly fluctuating with time, having null statistical mean and Dirac variance function. The quantity is commonly known to be closely related to the Wiener process [2]. The function indicates the scale of randomness in the particle motion and is connected with the phenomenon of diffusion in the prescribed medium [3]. In fact the diffusion coefficient of the medium D(x,t) is known to be given by the relation:

(1.4)

Clearly is a non-negative function, and in this regard the Langevin equation may technically be called the stochastic drift-diffusion equation, and be written:

(1.5)

The problem at hand is to estimate the influence of the drift on the behavior of the particle motion described by the Langevin equation in a finite interval having absorbing ends. Assuming that the particle is subjected to a reactive potential that changes sign in the interval, we may ask how is the long term behavior of the particle affected if the drift is induced by . The standing conjecture [4,5] is that for the reflecting barrier case a drift along should settle the particle mostly in areas where the potential is optimally positive. This is treated in [6]. In this paper we only consider the absorbing barrier case.

References

1. Karaballi, A. A. On the ejection of solutions in a central force field. Celestial Mechanics, 41, 1988, 323-332.

2. Gardiner, C. W. Handbook of stochastic methods for physics, chemestry and natural sciences, 3^rd Ed., Springer-Verlag, New York, 1988.

3. Eckstein, E. C., Belgacem, F. Model of platelet transport in flowing blood with drift and diffusion terms. Biophysical Journal, Vol.60, July 1991, 53-69.

4. Belgacem, F. Boundary Value Problem with indefinite Weight and Applications. Internat. J. “Problems of Nonlinear Analysis in Engineering Systems”, Vol.10, No.2, 1999, 51-58.

5. Belgacem, F., Cosner C. The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments. Canadian Applied Math. Quaterly Journal Vol.3, Number 4, fall 1995, 379-397.

Belgacem, F., Karaballi, A.A. The Motion of a Particle in a Finite Interval Subject to an Indefinite Potential and the Stochastic Cosner Conjecture. Submitted.