Exact solutions for diffusion-reaction processes in one dimension: II. Spatial distributions

1989 ◽  
Vol 22 (22) ◽  
pp. 5051-5051
Author(s):  
J L Spouge
Keyword(s):  
Exact Solutions ◽  
Diffusion Reaction ◽  
One Dimension ◽  
Spatial Distributions ◽  
Reaction Processes

RETARDATION EFFECTS IN A FINITE DIFFUSION-REACTION SYSTEM

2008 ◽  
Vol 22 (12) ◽  
pp. 1947-1959 ◽  
Author(s):  
KNUD ZABROCKI ◽  
STEFFEN TRIMPER ◽  
MICHAEL SCHULZ
Keyword(s):  
Reaction System ◽  
Reaction Diffusion ◽  
Diffusion Reaction ◽  
One Dimension ◽  
Anisotropic Behavior ◽  
Production Term ◽  
Spatial Point ◽  
Delay Effects ◽  
Spatiotemporal Delay ◽  
Reaction Processes

The reaction-diffusion process is generalized by including spatiotemporal delay effects. As a first example, we study the influence of a constant production term which is switched off after a finite time. In a second case, all diffusion-reaction processes within a distance R(t) = κtα around a certain spatial point are assumed to contribute to the instantaneous dynamics of the system. There occurs a competition between reaction-diffusion and the accumulation process which leads to a non-trivial stationary state. The evolving concentration profiles are calculated analytically for both a ballistic behavior with α = 1 and a diffusion-like transport with α = 1/2. Because the spatiotemporal delay breaks the reflection symmetry, the profiles reveal an anisotropic behavior. The exact solution in one dimension is supported by numerical simulations.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (04) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (4) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

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