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(equation 3.25). Making these substitutions gives

If we say that the concentration of particles to the right of area A is taken at location x + 8 [C(R) = C(x + 8)] and the concentration of particles to the left is actually taken at area A [C(L) = C(x)], the fraction in equation 3.33 takes the form of a first derivative:

(A straightforward refresher on methods and interpretations for derivatives can be found in Newby (1980).) As ^ the distance between x and 8 shrinks toward zero (lim), the flux at any point x becomes 8^0

x dx

In total, the flux is determined by the product of the rate at which particles spread out from a given point and the rate of change in concentration along the axis of diffusion. The sign of the flux tells the direction of net movement of particles. The flux is positive, meaning the number of particles in the area to the right of A will increase after one time step, if more particles move in from the left than move out going right. As we would expect, there is net movement of particles from areas of higher concentration into areas of lower concentration. For example, a higher concentration of particles to the left of point x in Fig. 3.12 will result in a positive flux, meaning that after one time step of diffusion there will be an increase in the number of particles at point x. When the concentration of particles is the same everywhere along the axis of diffusion, the flux must be zero since the numbers that move into and out of any area are equal.

The derivation for particle flux hid one detail that we now need to reveal in order to continue. The flux depends on both the mean movement and the net movement of particles. Imagine for example that the ink particles in Fig. 3.12 were positively charged and one of the boundaries was negatively charged. The ink particles would diffuse but at the same time the whole cloud of particles would be moving on average toward the negatively charged boundary. In such a case the flux at any point x would also need to account for changes to the mean position of particles

M(x), so that J = M(x)di - D. This mean change x dx was neglected earlier since the mean position is zero if particles are moving left or right at random and are not influenced by some force changing the mean location of all particles. Substituting this full version of the flux (remember that C(x,i) is now ^(x,t)) gives

Diffusion coefficient (D) Half the rate at which the variance in particle position (or allele frequency in a single population) changes as time advances. Flux (Jx) The net number of particles (or populations) moving through a defined area (or allele frequency) per time interval.

Let's now take all the concepts of particle diffusion and apply them to genetic drift in an ensemble population. We want to predict the change in the chance that a population has a given allele frequency, just as we did with the Markov chain model. In this case allele frequency is a continuous variable and the chance that a population has an allele frequency between x and x + 8 at a given time t is called the probability density, symbolized by ^(x,t). (^ is pronounced "phi".) Probability density for populations is just like concentration for particles, so ^(x,t) is the analog of C(x,t). The probability density ^(x,t) at any point along the axis of allele frequency will depend on the net difference between populations which drift into allele frequency x and those which drift out of allele frequency x. This is the flux in allele frequency at point x and time t. To know the probability at all points along the axis of diffusion, we need the rate of change in the probability density with change in allele frequency. This is the rate of change in the flux of populations with change in allele frequency:

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