Radiophysics and Quantum Electronics, Vol. 44, No. 4, 2001

RELAXATION TIME OF MATERIAL DENSITY IN A MEDIUM WITH ARBITRARYSPACE-VARYING DIFFUSION COEFFICIENT AND POTENTIAL PROFILE

A. N. Malakhov∗ and E. L. Pankratov UDC 539.219.3:669

We determine the temporal characteristics of material relaxation in a medium with arbitraryspace-varying diffusion coefficient and potential profile and analyze some examples.

1. FORMULTION OF THE PROBLEM

Let a one-dimensional medium be bounded (0 ≤ x ≤ L) and have reflecting boundaries. The diffusioncoefficient D(x) of the medium is an arbitrary function of the coordinate and takes finite nonzero valuesinside the medium. The distribution of the potential ϕ(x) inside the medium is a dimensionless function ofthe coordinate. The initial density f(x) of the material (admixture) of unit mass is a specified function ofthe coordinate. The admixture-density distribution tends to its equilibrium value C(x,∞) in the course oftime. We should find the relaxation time of the equilibrium density at the specified point x = ` inside themedium.

The material density C(x, t) satisfies the continuity equation

∂C(x, t)∂t

= div{D(x) [C(x, t) gradϕ(x) + gradC(x, t)]

}= −div G(x, t), (1a)

whereG(x, t) = −D(x) [C(x, t) gradϕ(x) + gradC(x, t)] (1b)

is the material flux.The diffusion equation should be complemented by the boundary conditions G(0, t) = G(L, t) = 0.

The initial density of the admixture is specified as C(x, 0) = f(x).

2. METHOD OF SOLUTION

Let us determine the relaxation time Θ by analogy with the exit time introduced in [1], i.e., asthe interval between t = 0 and the time of the jump-like variation of the function approximating thedensity evolution C(x, t) of the admixture with the minimum error. A function of the type φ(x, t,Θ) =a0(x) + a1(x) [1(t)− 1(t− Θ(x))] is used as the approximating function, in which 1(t) is the unit function.We omit the arguments of the amplitudes a0 and a1 of the approximating function and relaxation timeΘ, assuming that they are dependent on the coordinate. The optimal values of the parameters of such anapproximating function satisfy the condition that the functional

U =

tN∫0

[C(x, t)− φ(x, t,Θ)]2 dt (2)

∗Deceased.

N. I. Lobachevsky State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Ucheb-nykh Zavedenii, Radiofizika, Vol. 44, No. 4, pp. 367–373, April, 2001. Original article submitted November 17,2000.

0033-8443/01/4404-0339$25.00 c© 2001 Kluwer Academic/Plenum Publishers 339

is minimal. Here tN is the time of the process observation. It is well known that the necessary condition ofextremum with respect to the parameters a0, a1, and Θ has the form

∂U/∂a0 = 0, ∂U/∂a1 = 0, ∂U/∂Θ = 0. (3)

The first condition yields∫ tN0 [C(x, t)− a0 − a1 (1(t)− 1(t−Θ))] dt = 0. We transform this condition

astN∫0

C(x, t) dt = a0tN + a1Θ. (4a)

The condition of minimum of the functional U with respect to Θ is written as

C(x,Θ) = a0 + a1/2 (4b)

and with respect to a1, as

Θ∫0

C(x, t) dt = (a0 + a1)Θ. (4c)

.Although the proposed estimate is nonlinear, it does not result in pronounced difficulties when pro-

cessing the results of physical or mathematical experiments. An increase in the observation time tN allowsus to improve the estimates. In doing so we observe slight variations in the amplitudes a0 and a1 and theshift of the time Θ of the jump-like variation of the approximating function.

When constructing analytical dependences, we should pay attention to asymptotically optimal es-timates. They assume an unlimited duration of the observed process (tN → ∞). In these estimates, weimpose an additional condition on the jump amplitude which is taken equal to the difference between theasymptotic and initial values of the approximated function, i.e., a0 = C(x,∞) and a1 = C(x, 0)− C(x,∞).Only the time of the jump-like variation of the function should be determined. In such a method, the desiredquantity is calculated by solving the system of linear equations and is a linear estimate of the parameter ofthe process evolution.

To obtain an analytical solution of the system of equations (4), we consider them for tN → ∞. Forthis asymptotics we should allow for the fact that the stationary density C(x,∞) is the limit of a0 fortN →∞. Equation (4) has the following asymptotic form for tN →∞:

Θ(`) =

∫ ∞

0[C(`,∞)− C(`, t)] dt

C(`,∞)− C(`, 0). (5)

Therefore, we have obtained the well-known approximation of the density evolution by the rectangleof equal area, which is an asymptotically optimal estimate, as is obvious from the above discussion. Anotherwell-known asymptotically optimal estimate results from Eq. (4b) and is written in the form

C(x,Θ) = [C(x, 0) + C(x,∞)]/2.

However, this estimate of Θ yields an analytical solution much less frequently than that in the form of arectangle of equal area. We should note that asymptotically optimal estimates hold only for monotonicevolutions (see Figs. 1 and 2).

To find the relaxation time, we use the approximation of the density evolution by a rectangle of equalarea and seek the relaxation time using the method presented in [2, 3]. To find the relaxation time, weintroduce the Laplace transforms of the density and flux

340

Fig. 1 Fig. 2

Y (x, s) =

∞∫0

C(x, t)e−st dt, (6)

G(x, s) =

∞∫0

G(x, t)e−st dt. (7)

We write continuity equation (1a) and Eq. (5) in terms of the Laplace transforms and with allowancefor the initial condition in the form

div[D(x) gradY (x, s)]− sY (x, s) = −f(x), (8)

Θ(`) = lims→0

C(`,∞)− sY (`, s)s [C(`,∞)− C(`, 0)]

. (9)

To find the relaxation time (9), it is sufficient to know the asymptotic values of the density Y (x, s)and flux G(x, s) for s→ 0 rather than for all values of the Laplace parameter s.

Let us expand the functions sY (x, s) and sG(x, s) in the power series of s:

sY (x, s) = Z0(x) + sZ1(x) + s2Z2(x) + . . . , sG(x, s) = H0(x) + sH1(x) + s2H2(x) + . . . . (10)

Substituting Eq. (10) into Eqs. (8) and (9), we obtain the following expression for the relaxationtime:

Θ(`) = lims→0

C(`,∞)− Z0(`)− sZ1(`)− s2Z2(`)− . . .

s [C(`,∞)− C(`, 0)]. (11)

The limiting theorems of the Laplace-transform theory [4] allow us to obtain the following relations:

lims→0

[sY (x, s)] = C(x,∞) = Z0(x), lims→0

[sG(x, s)] = G(x,∞) = H0(x). (12)

Therefore, the steady-state values of the material flux and density are equal to H0(x) = 0 and Z0(x),

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respectively. With allowance for relations (12), Eq. (11) has the form

Θ(`) =Z1(`)

C(`, 0)− C(`,∞). (13)

To find the relaxation time, we should find the functions Z0(x) and Z1(x). Substituting Eq. (12) intoEq. (9), we obtain general equations for the functions Zk(x) and Hk(x)

div{D(x) [Z0(x) gradϕ(x) + grad Z0(x)]

}= 0,

div{D(x) [Z1(x) gradϕ(x) + grad Z1(x)]

}= Z0(x)− f(x),

div{D(x) [Zk(x) gradϕ(x) + gradZk(x)]

}= Zk−1(x), k ≥ 2. (14)

In this case, the functions Zk(x) and Hk(x) are related as follows:

dZk(x)dx

+ Zk(x) gradϕ(x) = −Hk(x)D(x)

. (15)

We find the solution of the first equation of system (14). The first integral of this equation has theform

D(x)[dϕ(x)

dxZ0(x) +

dZ0(x)dx

]= c, (16)

where c is the integration constant, equal to zero in accordance with the boundary conditions. In the finalform, the first integral of the first equation of system (14) is written as

dϕ(x)dx

Z0(x) +dZ0(x)

dx= 0. (17)

The solution of Eq. (17) is written in the form of the function

Z0(x) = c1e−ϕ(x). (18)

Using the boundary conditions, we find the integration constant c1 and uniquely determine the stationarydensity allowing for the normalization:

Z0(x) = e−ϕ(x)/

ρ(L) , (19)

where ρ(L) =∫ L0 e−ϕ(v) dv.

We find the solution of the second equation of system (14). The first integral of this equation isdetermined from the following relation:

D(x)[dϕ

dxZ1 +

dZ1

dx

]= P (x)−

x∫0

f(v) dv + c2 = −H1(x), (20)

where P (x) = ρ(x)/ρ(L). The integration constant c2 is equal to zero in accordance with the boundaryconditions. The second integral of the second equation of system (14) has the form

Z1(x) = e−ϕ(x)

x∫0

eϕ(y)P (y)D(y)

dy − e−ϕ(x)

x∫0

eϕ(y)

D(y)

y∫0

f(v) dv dy + c3e−ϕ(x). (21)

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The integration constant c3 is obtained from the boundary conditions and Eq. (15)

H2(0)−H2(L) =

L∫0

e−ϕ(x)

x∫0

eϕ(y)P (y)D(y)

dy dx−L∫

0

e−ϕ(x)

x∫0

eϕ(y)

D(y)

y∫0

f(v) dv dy dx + c3ρ(L) = 0,

whence we obtain

c3 =

L∫0

e−ϕ(x)

x∫0

eϕ(y)

ρ(L)D(y)

y∫0

f(v) dv dy dx−L∫

0

e−ϕ(x)

x∫0

eϕ(y)P (y)ρ(L)D(y)

dy dx,

or, changing the integration order,

c3 =

L∫0

eϕ(y)

D(y)[1− P (y)]

y∫0

f(v) dv dy −L∫

0

eϕ(y)

D(y)P (y) [P (y)− 1] dy.

Now, the function Z1(x) is uniquely determined and written as

Z1(x) = e−ϕ(x)

x∫

0

eϕ(y)

D(y)

P (y)−y∫

0

f(v) dv

dy +

L∫0

eϕ(y)

D(y)[1− P (y)]

y∫0

f(v) dv dy

−− e−ϕ(x)

L∫

0

eϕ(y)

D(y)P (y) [P (y)− 1] dy

.

After obtaining the functions Z0(x) and Z1(x) we can write the relaxation time in the form (13)

Θ(`) =

`∫

0

eϕ(y)

D(y)

P (y)−y∫

0

f(v) dv

dy +

L∫0

eϕ(y)

D(y)

y∫0

f(v) dv [1− P (y)] dy

−−

L∫0

eϕ(y)

D(y)P (y) [P (y)− 1] dy

1ρ(L)f(`)eϕ(`) − 1

. (22)

Example of the initial distribution. Let initially the admixture be at a single point inside themedium: f(x) = δ(x− x0). In this case, from Eq. (22) we obtain

Θδ(` ≥ x0) =

L∫0

eϕ(y)

D(y)

P (y)

L∫y

e−ϕ(v) dv

dy −L∫

`

eϕ(y)

D(y)

L∫y

e−ϕ(v) dv dy −x0∫0

eϕ(y)

D(y)ρ(y) dy, (23)

Θδ(` ≤ x0) =

L∫0

eϕ(y)

D(y)

P (y)

L∫y

e−ϕ(v)dv

dy −L∫

x0

eϕ(y)

D(y)

L∫y

e−ϕ(v) dv dy −`∫

0

eϕ(y)

D(y)ρ(y) dy. (24)

The maximum relaxation time is attained for the most distant source and observation points x0 = 0and ` = L

Θmax =

L∫0

eϕ(y)P (y)D(y)

L∫y

e−ϕ(v) dv dy. (25)

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The relaxation time decreases as the source and observation points approach each other.Examples of potentials. If the potential is assumed to be constant for the above source, the first

terms on the right-hand sides of Eqs. (1a) and (1b) are dropped, and Eqs. (23)–(25) take the form

Θδ(` ≥ x0) =

L∫0

(L− y) y

LD(y)dy −

L∫`

L− y

D(y)dy −

x0∫0

y

D(y)dy, (26)

Θδ(` ≤ x0) =

L∫0

(L− y) y

LD(y)dy −

L∫x0

L− y

D(y)dy −

`∫0

y

D(y)dy. (27)

Θmax =

L∫0

(L− y) y

LD(y)dy. (28)

As expected, Eqs. (26), (27), and (28) coincide with Eqs. (22), (23), and (25), respectively, in [2].In the case of the simplest model of a constant diffusion coefficient D(x) = D0 = const, Eqs. (26)

and (27) yield

Θδ(` ≥ x0) = Θ0

(`

L− 1

3− `2

2L2− x2

0

2L2

), Θδ(` ≤ x0) = Θ0

(x0

L− 1

3− x2

0

2L2− `2

2L2

),

where Θ0 = Θmax = L2/(6D0) is the characteristic scale of the relaxation time for the case where thediffusion coefficient and potential are constant.

As the second example, we consider the linear distribution of the potential ϕ(x) = ax + b. In thiscase, calculation of the relaxation time yields

Θδ(` ≥ x0) =

L∫0

(eax − 1)(e−ax − e−aL

)aD(x)(1− e−aL)

dx−L∫

`

1− ea(x−L)

aD(x)dx−

x0∫0

eax − 1aD(x)

dx, (29)

Θδ(` ≤ x0) =

L∫0

(eax − 1)(e−ax − e−aL

)aD(x)(1− e−aL)

dx−L∫

x0

1− ea(x−L)

aD(x)dx−

`∫0

eax − 1aD(x)

dx, (30)

Θmax =

L∫0

(eax − 1) (e−ax − e−aL)aD(x)(1− e−aL)

dx. (31)

In the simplest case of a constant diffusion coefficient D(x) = D0, Eqs. (29)– (31) yield

Θδ(` ≥ x0) =aL (1 + e−aL) + 2 (1− e−aL)

a2D0 (1− e−aL)+

1− e−a(`−L)

a2D0− ` + x0 − L

aD0+

1− e−ax0

a2D0, (32)

Θδ(` ≤ x0) =aL (1 + e−aL) + 2 (1− e−aL)

a2D0 (1− e−aL)+

1− e−a(x0−L)

a2D0− ` + x0 − L

aD0+

1− e−a`

a2D0, (33)

Θmax =aL (1 + e−aL) + 2 (1− e−aL)

a2D0 (1− e−aL). (34)

If the initial distribution f(x) is arbitrary but differs from the delta function, the relaxation time,due to the linearity of the diffusion equation, can be obtained by the superposition method rather than by

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substituting the new model of the initial distribution into Eq. (22):

Θ =

L∫0

Θδ(x0)f(x0) dx0.

Here Θδ(x0) is the relaxation time obtained for the initial density distribution in the form of a delta function.

This work was supported by the Russian Foundation for Basic Research (project Nos. 99–02–17544and 00–15–96620) and the Ministry of Higher Education (project No. 992874).

REFERENCES

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2. A.N. Malakhov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 40, No. 7, 889 (1997).

3. A.N. Malakhov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 42, No. 6, 581 (1999).

4. G.A.Korn and T.M.Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill NewYork (1968).

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