The Random Walk and Diffusion Theory
The random walk, a classical example of Markov -chains, is used as an entry-point for a more involved discussion of diffusion theory. After a complete analysis of the random walk, the Wiener process and its relation to Brownian motion is presented. In fac
- PDF / 784,041 Bytes
- 25 Pages / 439.36 x 666.15 pts Page_size
- 41 Downloads / 197 Views
The Random Walk and Diffusion Theory
17.1 Introduction Diffusion is one of the most widely spread processes in science. Its occurrence ranges from random motion of dust particles on fluid surfaces, historically known as Brownian motion, to the motion of particles in numerous physical systems [1, 2], the spreading of malaria by migration of mosquitoes [3], or even to the description of fluctuations in stock markets [4]. For instance, let us regard N neutral, identical, classical particles which solely interact through collisions, for instance an H2 -gas in a box, where N D NA 6:022 1023 . We are interested in the dynamics of one particle under the influence of all others and under no influence by an external force; we expect that diffusion will be the dominating process. From the microscopic point of view such a situation can be described with the help of N coupled NEWTON’s equations of motion. (See Chap. 7.) Anyhow, such a task will not be feasible due to the size of the system – the magnitude of N. However, a statistical description can be obtained from BOLTZMANN’s equation [5] ˇ @ d ˇ f .r; ; t/ D f .r; ; t/ˇ ; coll. dt @t
(17.1)
where f .r; ; t/ is the phase space distribution function. Hence, f .r; ; t/drd is the number of particles of momentum within the phase-space volume drd which is centered around position r at time t. We have, in particular: @ @ @ f .r; ; t/ C f .r; ; t/ C F f .r; ; t/ D CΠf .r; ; t/ : @t m @r @
(17.2)
Here CŒ f .r; ; t/ is the collision integral and F describes an external force. In cases where collisions result solely from two-body interactions between particles that © Springer International Publishing Switzerland 2016 B.A. Stickler, E. Schachinger, Basic Concepts in Computational Physics, DOI 10.1007/978-3-319-27265-8_17
271
272
17 The Random Walk and Diffusion Theory
are assumed to be uncorrelated prior to the collision,1 the collision integral can be described by Z
Z CΠf .r; ; t/ D
d1
Z d3 g.1 ; 2 ; 3 ; / Œf .r; 1 ; t/f .r; 2 ; t/
d2
f .r; ; t/f .r; 3 ; t/ ;
(17.3)
where g.1 ; 2 ; 3 ; / accounts for the probability that a collision between two particles of initial moments 1 and 2 and final momenta 3 and occurs. This function depends on the particular type of particles under investigation and has, in general, to be determined from a microscopic theory.2 We now define the particle density .r; t/ as a function of space r and time t via Z .r; t/ D
d f .r; ; t/ :
(17.4)
A complicated mathematical analysis of Eq. (17.1) results in a diffusion equation of the well-known form @2 @ .r; t/ D D 2 .r; t/ ; @t @r
(17.5)
if collisions dominate the dynamics (diffusion limit). Here D D const is the diffusion coefficient of dimension length2 time1 . Note that Z dr.r; t/ D N ;
(17.6)
is the number of particles within our system.3 Thus, in our example we can interpret diffusion as the average evolution of the integrated phase space distribution function governed by collisions between particles. Such an interpretation w
Data Loading...
