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| The '''Mason–Weaver equation''' (named after [[Max Mason]] and [[Warren Weaver]]) describes the [[sedimentation]] and [[diffusion]] of solutes under a uniform [[force]], usually a [[gravitation]]al field.<ref name="mason_1924" >{{cite journal | last = Mason | first = M | coauthors = Weaver W | year = 1924 | title = The Settling of Small Particles in a Fluid | journal = [[Physical Review]] | volume = 23 | pages = 412–426 | doi = 10.1103/PhysRev.23.412 | bibcode=1924PhRv...23..412M}}</ref> Assuming that the [[gravitation]]al field is aligned in the ''z'' direction (Fig. 1), the Mason–Weaver equation may be written
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| :<math>
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| \frac{\partial c}{\partial t} =
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| D \frac{\partial^{2}c}{\partial z^{2}} +
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| sg \frac{\partial c}{\partial z}
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| </math>
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| where ''t'' is the time, ''c'' is the [[solution|solute]] [[concentration]] (moles per unit length in the ''z''-direction), and the parameters ''D'', ''s'', and ''g'' represent the [[solution|solute]] [[diffusion constant]], [[sedimentation coefficient]] and the (presumed constant) [[acceleration]] of [[gravitation|gravity]], respectively.
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| The Mason–Weaver equation is complemented by the [[boundary conditions]]
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| :<math>
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| D \frac{\partial c}{\partial z} + s g c = 0
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| </math>
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| at the top and bottom of the cell, denoted as <math>z_{a}</math> and <math>z_{b}</math>, respectively (Fig. 1). These [[boundary conditions]] correspond to the physical requirement that no [[solution|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
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| the [[Cartesian coordinate system|Cartesian axes]] (Fig. 1), so that the net [[flux]] through the side walls is likewise
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| zero. Hence, the total amount of [[solution|solute]] in the cell
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| :<math>
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| N_{tot} = \int_{z_{b}}^{z_{a}} dz \ c(z, t)
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| </math>
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| is conserved, i.e., <math>dN_{tot}/dt = 0</math>.
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| [[Image:Mason Weaver cell.png|frame|left|Figure 1: Diagram of Mason–Weaver cell and Forces on Solute]]
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| ==Derivation of the Mason–Weaver equation==
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| A typical particle of [[mass]] ''m'' moving with vertical [[velocity]] ''v'' is acted upon by three [[force]]s (Fig. 1): the
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| [[drag (physics)|drag force]] <math>f v</math>, the force of [[gravitation|gravity]] <math>m g</math> and the [[buoyancy|buoyant force]] <math>\rho V g</math>, where ''g'' is the [[acceleration]] of [[gravitation|gravity]], ''V'' is the [[solution|solute]] particle volume and <math>\rho</math> is the [[solvent]] [[density]]. At [[mechanical equilibrium|equilibrium]] (typically reached in roughly 10 ns for [[molecule|molecular]] [[solution|solutes]]), the
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| particle attains a [[terminal velocity]] <math>v_{term}</math> where the three [[force]]s are balanced. Since ''V'' equals the particle [[mass]] ''m'' times its [[partial specific volume]] <math>\bar{\nu}</math>, the [[mechanical equilibrium|equilibrium]] condition may be written as
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| :<math>
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| f v_{term} = m (1 - \bar{\nu} \rho) g \ \stackrel{\mathrm{def}}{=}\ m_{b} g
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| </math>
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| where <math>m_{b}</math> is the [[buoyant mass]].
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| We define the Mason–Weaver [[sedimentation coefficient]] <math>s \ \stackrel{\mathrm{def}}{=}\ m_{b} / f = v_{term}/g</math>. Since the [[drag coefficient]] ''f'' is related to the [[diffusion constant]] ''D'' by the [[Einstein relation (kinetic theory)|Einstein relation]]
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| :<math>
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| D = \frac{k_{B} T}{f}
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| </math>,
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| the ratio of ''s'' and ''D'' equals
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| :<math>
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| \frac{s}{D} = \frac{m_{b}}{k_{B} T}
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| </math>
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| where <math>k_{B}</math> is the [[Boltzmann constant]] and ''T'' is the [[temperature]] in [[kelvin]]s.
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| The [[flux]] ''J'' at any point is given by
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| :<math>
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| J = -D \frac{\partial c}{\partial z} - v_{term} c
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| = -D \frac{\partial c}{\partial z} - s g c.
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| </math>
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| The first term describes the [[flux]] due to [[diffusion]] down a [[concentration]] gradient, whereas the second term
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| describes the [[convective flux]] due to the average velocity <math>v_{term}</math> of the particles. A positive net [[flux]] out of a small volume produces a negative change in the local [[concentration]] within that volume
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| :<math>
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| \frac{\partial c}{\partial t} = -\frac{\partial J}{\partial z}.
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| </math>
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| Substituting the equation for the [[flux]] ''J'' produces the Mason–Weaver equation
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| :<math>
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| \frac{\partial c}{\partial t} =
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| D \frac{\partial^{2}c}{\partial z^{2}} +
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| sg \frac{\partial c}{\partial z}.
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| </math>
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| ==The dimensionless Mason–Weaver equation==
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| The parameters ''D'', ''s'' and ''g'' determine a length scale <math>z_{0}</math>
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| :<math>
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| z_{0} \ \stackrel{\mathrm{def}}{=}\ \frac{D}{sg}
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| </math>
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| and a time scale <math>t_{0}</math>
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| :<math>
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| t_{0} \ \stackrel{\mathrm{def}}{=}\ \frac{D}{s^{2}g^{2}}
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| </math>
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| Defining the [[dimensionless]] variables <math>\zeta \ \stackrel{\mathrm{def}}{=}\ z/z_{0}</math> and <math>\tau \ \stackrel{\mathrm{def}}{=}\ t/t_{0}</math>, the Mason–Weaver equation becomes
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| :<math>
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| \frac{\partial c}{\partial \tau} =
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| \frac{\partial^{2} c}{\partial \zeta^{2}} +
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| \frac{\partial c}{\partial \zeta}
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| </math>
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| subject to the [[boundary conditions]]
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| :<math>
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| \frac{\partial c}{\partial \zeta} + c = 0
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| </math>
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| at the top and bottom of the cell, <math>\zeta_{a}</math> and
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| <math>\zeta_{b}</math>, respectively.
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| ==Solution of the Mason–Weaver equation==
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| This partial differential equation may be solved by [[separation of variables]]. Defining <math>c(\zeta,\tau) \ \stackrel{\mathrm{def}}{=}\ e^{-\zeta/2} T(\tau) P(\zeta)</math>, we obtain two ordinary differential equations coupled by a constant <math>\beta</math>
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| :<math>
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| \frac{dT}{d \tau} + \beta T = 0
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| </math>
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| :<math>
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| \frac{d^{2} P}{d \zeta^{2}} +
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| \left[ \beta - \frac{1}{4} \right] P = 0
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| </math>
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| where acceptable values of <math>\beta</math> are defined by the [[boundary conditions]]
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| :<math>
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| \frac{dP}{d\zeta} + \frac{1}{2} P = 0
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| </math>
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| at the upper and lower boundaries, <math>\zeta_{a}</math> and <math>\zeta_{b}</math>, respectively. Since the ''T'' equation
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| has the solution <math>T(\tau) = T_{0} e^{-\beta \tau}</math>, where <math>T_{0}</math> is a constant, the Mason–Weaver equation is reduced to solving for the function <math>P(\zeta)</math>.
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| The [[ordinary differential equation]] for ''P'' and its [[boundary conditions]] satisfy the criteria
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| for a [[Sturm–Liouville theory|Sturm–Liouville problem]], from which several conclusions follow. '''First''', there is a discrete set of [[orthonormal]] [[eigenfunction]]s
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| <math>P_{k}(\zeta)</math> that satisfy the [[ordinary differential equation]] and [[boundary conditions]]. '''Second''', the corresponding [[eigenvalue]]s <math>\beta_{k}</math> are real, bounded below by a lowest | |
| [[eigenvalue]] <math>\beta_{0}</math> and grow asymptotically like <math>k^{2}</math> 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 [[eigenfunction]]s form a complete set; any solution for <math>c(\zeta, \tau)</math> can be expressed as a weighted sum of the [[eigenfunction]]s
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| :<math>
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| c(\zeta, \tau) =
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| \sum_{k=0}^{\infty} c_{k} P_{k}(\zeta) e^{-\beta_{k}\tau}
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| </math>
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| where <math>c_{k}</math> are constant coefficients determined from the initial distribution <math>c(\zeta, \tau=0)</math>
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| :<math>
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| c_{k} =
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| \int_{\zeta_{a}}^{\zeta_{b}} d\zeta \
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| c(\zeta, \tau=0) e^{\zeta/2} P_{k}(\zeta)
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| </math>
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| At equilibrium, <math>\beta=0</math> (by definition) and the equilibrium concentration distribution is
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| :<math>
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| e^{-\zeta/2} P_{0}(\zeta) = B e^{-\zeta} = B e^{-m_{b}gz/k_{B}T}
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| </math>
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| which agrees with the [[Boltzmann distribution]]. The <math>P_{0}(\zeta)</math> function satisfies the [[ordinary differential equation]] and [[boundary conditions]] at all values of <math>\zeta</math> (as may be verified by substitution), and the constant ''B'' may be determined from the total amount of [[solution|solute]]
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| :<math>
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| B = N_{tot} \left( \frac{sg}{D} \right)
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| \left( \frac{1}{e^{-\zeta_{b}} - e^{-\zeta_{a}}} \right)
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| </math>
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| To find the non-equilibrium values of the [[eigenvalue]]s <math>\beta_{k}</math>, we proceed as follows. The P equation has the form of a simple [[harmonic oscillator]] with solutions <math>P(\zeta) = e^{i\omega_{k}\zeta}</math> where
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| :<math>
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| \omega_{k} = \pm \sqrt{\beta_{k} - \frac{1}{4}}
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| </math>
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| Depending on the value of <math>\beta_{k}</math>, <math>\omega_{k}</math> is either purely real (<math>\beta_{k}\geq\frac{1}{4}</math>) or purely imaginary (<math>\beta_{k} < \frac{1}{4}</math>). Only one purely imaginary solution can satisfy the [[boundary conditions]], namely, the equilibrium solution. Hence, the non-equilibrium [[eigenfunctions]] can be written as
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| :<math>
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| P(\zeta) = A \cos{\omega_{k} \zeta} + B \sin{\omega_{k} \zeta}
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| </math>
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| where ''A'' and ''B'' are constants and <math>\omega</math> is real and strictly positive.
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| By introducing the oscillator [[amplitude]] <math>\rho</math> and [[phase (waves)|phase]] <math>\phi</math> as new variables,
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| :<math>
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| u \ \stackrel{\mathrm{def}}{=}\ \rho \sin(\phi) \ \stackrel{\mathrm{def}}{=}\ P
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| </math>
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| :<math>
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| v \ \stackrel{\mathrm{def}}{=}\ \rho \cos(\phi) \ \stackrel{\mathrm{def}}{=}\ - \frac{1}{\omega}
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| \left( \frac{dP}{d\zeta} \right)
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| </math>
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| :<math>
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| \rho \ \stackrel{\mathrm{def}}{=}\ u^{2} + v^{2}
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| </math>
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| :<math>
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| \tan(\phi) \ \stackrel{\mathrm{def}}{=}\ v / u
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| </math>
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| the second-order equation for ''P'' is factored into two simple first-order equations
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| :<math>
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| \frac{d\rho}{d\zeta} = 0
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| </math>
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| :<math>
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| \frac{d\phi}{d\zeta} = \omega
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| </math>
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| Remarkably, the transformed [[boundary conditions]] are independent of <math>\rho</math> and the endpoints <math>\zeta_{a}</math> and <math>\zeta_{b}</math>
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| :<math>
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| \tan(\phi_{a}) =
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| \tan(\phi_{b}) = \frac{1}{2\omega_{k}}
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| </math>
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| Therefore, we obtain an equation
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| :<math> | |
| \phi_{a} - \phi_{b} + k\pi = k\pi =
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| \int_{\zeta_{b}}^{\zeta_{a}} d\zeta \ \frac{d\phi}{d\zeta} =
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| \omega_{k} (\zeta_{a} - \zeta_{b})
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| </math>
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| giving an exact solution for the frequencies <math>\omega_{k}</math>
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| :<math>
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| \omega_{k} = \frac{k\pi}{\zeta_{a} - \zeta_{b}}
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| </math>
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| The eigenfrequencies <math>\omega_{k}</math> are positive as required, since <math>\zeta_{a} > \zeta_{b}</math>, and comprise the set of [[harmonic]]s of the [[fundamental frequency]] <math>\omega_{1} \ \stackrel{\mathrm{def}}{=}\ \pi/(\zeta_{a} - \zeta_{b})</math>. Finally, the [[eigenvalue]]s <math>\beta_{k}</math> can be derived from <math>\omega_{k}</math>
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| :<math>
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| \beta_{k} = \omega_{k}^{2} + \frac{1}{4}
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| </math>
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| Taken together, the non-equilibrium components of the solution correspond to a [[Fourier series]] decomposition of the initial concentration distribution <math>c(\zeta, \tau=0)</math>
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| multiplied by the [[weight function|weighting function]] <math>e^{\zeta/2}</math>. Each Fourier component decays independently as <math>e^{-\beta_{k}\tau}</math>, where <math>\beta_{k}</math> is given above in terms of the [[Fourier series]] frequencies <math>\omega_{k}</math>.
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| ==See also==
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| * [[Lamm equation]]
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| * The Archibald approach , and a simpler presentation of the basic physics of the Mason-Weaver equation than the original. <ref>{{cite web |url=http://prola.aps.org/abstract/PR/v53/i9/p746_1 |title=Phys. Rev. 53, 746 (1938): The Process of Diffusion in a Centrifugal Field of Force |format= |work= |accessdate=}}</ref>
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| ==References==
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| {{reflist|1}}
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| {{DEFAULTSORT:Mason-Weaver equation}}
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| [[Category:Laboratory techniques]]
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| [[Category:Partial differential equations]]
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