In the chapter, in order to obtain the velocity of a two-state motor, we made use of a trick to circumvent solving the master equation directly. Here we take up this task and in the process also derive an expression for the diffusion constant.
(a) Consider a trial solution of the system of equations for $p_{0}(n, t)$ and $p_{1}(n, t),$ given by Equations 16.33 and 16.34
Find a relation between $K$ and $\omega$ that guarantees the existence of a solution of this form. This is the so-called dispersion relation.
(b) By substituting the trial solution $\mathrm{e}^{\mathrm{i}[(K / a)-\omega t]}$ into the differential equation for diffusion with drift (Equation 13.54 ,
p. 530 , show that the dispersion relation in this case is
$$\omega=v \frac{K}{a}-i D \frac{K^{2}}{a^{2}}$$
where $v$ is the drift velocity and $D$ is the diffusion constant.
(c) Demonstrate that in the limit $K \ll 1$, the dispersion relation for the two-state motor is the same as that for diffusion with drift. To do this, Taylor-expand $\omega(K)$ in $K$ and solve the equation for $\omega$ obtained in (a) order by order in $K$ which amounts to computing the coefficients in the Taylor expansion. Compare your result with the dispersion relation for dispersion with drift and read off the diffusion coefficient for the motor and its speed. Check that the formula for the speed matches the one obtained in the chapter.