In the chapter, in order to obtain the velocity of a...

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.

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.

Key Concepts

Master Equation

The master equation is a fundamental tool in describing the time evolution of the probability distribution for systems with discrete states. It provides a set of coupled differential equations that govern how probability flows between states over time and is central to the analysis of stochastic processes, such as the two-state motor system in this problem.

Dispersion Relation

A dispersion relation is a connection between the frequency and the wave-number of a perturbation in a system, deriving from substituting a trial or ansatz solution into the governing equations. In the context of the two-state motor, it relates the parameters of the trial solution (such as K and ?) and ensures that the solution adheres to the underlying dynamics captured by the master equation.

Drift-Diffusion Equation

The drift-diffusion equation describes the dynamics of particles that experience both systematic drift and random diffusion. It is a partial differential equation that models how probability densities evolve in space and time, and its associated dispersion relation illustrates the contributions of both drift (through linear terms in wave-number) and diffusion (through quadratic terms).

Trial Solution Approach

The trial solution approach involves assuming a solution form—typically an exponential function of space and time—to simplify solving the governing equations. By inserting this ansatz into the differential equations, one obtains constraints that lead to a dispersion relation. This method is particularly useful for linear equations and helps to avoid directly solving the master equation.

Taylor Expansion

Taylor expansion is a mathematical method used to approximate functions by expanding them in powers of a small parameter. In the context of the dispersion relation for the two-state motor, Taylor expanding ?(K) for small values of the wave-number (K << 1) allows one to extract the drift velocity and diffusion coefficient by matching coefficients order by order, hence connecting the microscopic dynamics to macroscopic transport properties.

Two-State Motor Model

The two-state motor model illustrates how a system with two distinct conformational or operational states can produce net motion. By analyzing the transitions between these states using a master equation and applying methods like the trial solution substitution, one can derive meaningful physical quantities such as the motor's drift speed and its effective diffusion coefficient, linking microscopic kinetics to observable behavior.

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