The Mason-Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field.^{[1]} Assuming that the gravitational field is aligned in the z direction (Fig. 1), the Mason-Weaver equation may be written

where t is the time, c is the soluteconcentration (moles per unit length in the z-direction), and the parameters D, s, and g represent the solute diffusion constant, sedimentation coefficient and the (presumed constant) acceleration of gravity, respectively.

The Mason-Weaver equation is complemented by the boundary conditions

at the top and bottom of the cell, denoted as z_{a} and z_{b}, respectively (Fig. 1). These boundary conditions correspond to the physical requirement that no solute pass through the top and bottom of the cell, i.e., that the flux there be zero. The cell is assumed to be rectangular and aligned with
the Cartesian axes (Fig. 1), so that the net flux through the side walls is likewise
zero. Hence, the total amount of solute in the cell

A typical particle of mass m moving with vertical velocity v is acted upon by three forces (Fig. 1): the
drag forcefv, the force of gravity mg and the buoyant force ρVg, where g is the acceleration of gravity, V is the solute particle volume and ρ is the solventdensity. At equilibrium (typically reached in roughly 10 ns for molecularsolutes), the
particle attains a terminal velocity v_{term} where the three forces are balanced. Since V equals the particle mass m times its partial specific volume , the equilibrium condition may be written as

where m_{b} is the buoyant mass.

We define the Mason-Weaver sedimentation coefficient . Since the drag coefficient f is related to the diffusion constant D by the Einstein relation

The first term describes the flux due to diffusion down a concentration gradient, whereas the second term
describes the convective flux due to the average velocity v_{term} of the particles. A positive net flux out of a small volume produces a negative change in the local concentration within that volume

Substituting the equation for the flux J produces the Mason-Weaver equation

The dimensionless Mason-Weaver equation

The parameters D, s and g determine a length scale z_{0}

and a time scale t_{0}

Defining the dimensionless variables and , the Mason-Weaver equation becomes

subject to the boundary conditions

at the top and bottom of the cell, ζ_{a} and
ζ_{b}, respectively.

Solution of the Mason-Weaver equation

This equation may be solved by separation of variables. Defining , we obtain the two equations coupled by a constant β

where acceptable values of β are defined by the boundary conditions

at the upper and lower boundaries, ζ_{a} and ζ_{b}, respectively. Since the T equation
has the solution T(τ) = T_{0}e^{ − βτ}, where T_{0} is a constant, the Mason-Weaver equation is reduced to solving for the function P(ζ).

The ordinary differential equation for P and its boundary conditions satisfy the criteria
for a Sturm-Liouville problem, from which several conclusions follow. First, there is a discrete set of orthonormal eigenfunctions
P_{k}(ζ) that satisfy the ordinary differential equation and boundary conditions. Second, the corresponding eigenvalues β_{k} are real, bounded below by a lowest
eigenvalue β_{0} and grow asymptotically like k^{2} where the nonnegative integer k is the rank of the eigenvalue. (In our case, the lowest eigenvalue is zero, corresponding to the equilibrium solution.) Third, the eigenfunctions form a complete set; any solution for c(ζ,τ) can be expressed as a weighted sum of the eigenfunctions

where c_{k} are constant coefficients determined from the initial distribution c(ζ,τ = 0)

At equilibrium, β = 0 (by definition) and the equilibrium concentration distribution is

which agrees with the Boltzmann distribution. The P_{0}(ζ) function satisfies the ordinary differential equation and boundary conditions at all values of ζ (as may be verified by substitution), and the constant B may be determined from the total amount of solute

To find the non-equilibrium values of the eigenvalues β_{k}, we proceed as follows. The P equation has the form of a simple harmonic oscillator with solutions where

Depending on the value of β_{k}, ω_{k} is either purely real () or purely imaginary (). Only one purely imaginary solution can satisfy the boundary conditions, namely, the equilibrium solution. Hence, the non-equilibrium eigenfunctions can be written as

P(ζ) = Acosω_{k}ζ + Bsinω_{k}ζ

where A and B are constants and ω is real and strictly positive.

By introducing the oscillator amplitude ρ and phase φ as new variables,

the second-order equation for P is factored into two simple first-order equations

Remarkably, the transformed boundary conditions are independent of ρ and the endpoints ζ_{a} and ζ_{b}

Therefore, we obtain an equation

giving an exact solution for the frequencies ω_{k}

The eigenfrequencies ω_{k} are positive as required, since ζ_{a} > ζ_{b}, and comprise the set of harmonics of the fundamental frequency . Finally, the eigenvalues β_{k} can be derived from ω_{k}

Taken together, the non-equilibrium components of the solution correspond to a Fourier series decomposition of the initial concentration distribution c(ζ,τ = 0)
multiplied by the weighting function e^{ζ / 2}. Each Fourier component decays independently as , where β_{k} is given above in terms of the Fourier series frequencies ω_{k}.