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Nonlinear heat equation and thermodiffusion
Vol. 46 (2000)
REPORTS
NONLINEAR
HEAT
ON
MATHEMATICAL
EQUATION
AND
No. 112
PHYSICS
THERMODIFFUSION
R. WOJNAR Institute
of Fundamental Technological Research, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warszawa, Poland (e-mail: [email protected]) (Received October 6, 1999)
A Streater model for diffusion of Brownian particle in a liquid under the influence of an exterior field is considered for the case when thermodiffusion effects are taken intro account. It is shown that for such a model the first law of thermodynamics holds true but the second law is modified to include the probability density function and temperature.
1.
Introduction It was pointed
out in [l, 21 that
falling in the gravity field becomes diffusion and thermal
processes,
the friction
an irreversible
notwithstanding
work performed
the weaker cross-effects
thermodiffusion which were neglected in [l, 21. Streater [3] performed an analysis of DufourSoret model of a gas of molecules {macroscopic)
coefficient
of the microscopic expressed
with repulsive occurs,
hopping
by the diffusion
In the present
namely
rate,
2.
density
namely
effects coefficient
conductivity
of Dufour- Soret
for the special
statistical
which has interpretation and cross-coefficients
and a maximum
thermodiffusion
arc
density.
effect on the process described
of the thermodiffusion
those of diffusion,
particles
In this model only one phenomenological
the diffusion
paper, a thermodiffusion
We do not enter into micromechanics logical coefficients,
cores.
and the heat
coefficient,
by Brownian
heat source, and it plays a role in linking
and exploit
in (I] is studied. only phenomeno-
and heat conductivity.
The basic equations Let Q be a bounded,
sufficiently
regular
domain
with boundary
80.
Let the function
f = f(x, t) represent the probability density of finding a tagged Brownian particle at x at time t. Let j and q denote the flux of Brownian particle diffusion and heat flow, Let the diffusion coefficient be D, the specific respectively, considered in that domain. heat be C and the heat conductivity mutually interrelated, cf. Eq. position x, time t, temperature
be K, while M and N be thermodiffusion
coefficients,
(4.5). We admit that D, K, M and N are functions T and probability density f.
P951
of
296
R. WOJNAR
Consider the motion of the Brownian particle at its terminal velocity in a fluid at temperature T, under the influence of a conservative field of force F = -VV, where V denotes a potential. The equations for diffusion and heat fluxes are as follows
(2.1) The corresponding continuity equations read
(2.2) where r denotes a heat source. Substituting (2.1) 1 into (2.2)i and (2.1)~ into (2.2)~ we get
af
dt=
‘g =
0ijC.f.j + kvj)
1,i ,
+ MijT,j
+r.
Nij(f,j+Gl/j)+K,jTj
(2.3)
I ,i
Instead of the single diffusion equation with drift, known as the Einstein-Smoluchowski equation, we have the set of two equations (2.3). In (2.2) and (2.3) the term r represents the rate at which the Brownian particle does work converted subsequently into heat, r=j.F
r = -j . VV.
or
(2.4)
We make the following boundary assumptions: j=O
3.
and
on
q=o
dR.
(2.5)
The first law of thermodynamics By Eqs. (2.2)1,2 and (2.4) we obtain the following continuity relation
g+j,4;=o, where e denotes the energy density and jE e-Vf+CT
(3.1)
the current of total energy, namely
and
j:
= Vji + qi .
(3.2)
According to (2.5), jE=O
on
dR
(3.3)
NONLINEAR
HEAT
EQUATION
and, as in the case without thermodiffusion E+/dx By integration
=
s,
297
AND THERMODIFFUSION
the energy of the system is
[v(x)f(x,
(3.4)
t)l dx.
t) + C(x)T(x,
, Jjs:dx
of (3.1) we find 8E -_= at
R
or by virtue of the divergence theorem and vanishing boundary condition (3.3) we get, dE -_= f3t Thus the first law of thermodynamics
(3.5)
O.
holds true.
The mean velocity of the Brownian particle is, by Eq. Remark 1: divergence theorem, d
u2=, J
xif(x,
t)dx =
R
x$??&x = _ dt
JR
So, similarly as in [l], v = J, jdx,
J
xjkkdx
R
=
I[R
J
X,jknkdA
where the boundary condition (2.5)i Dij (f,j + $)
+ MtjT,j
If D,j, A&j, T and VV are constants, Remark 2: we get, as in [l], Einstein’s formula
+
and the
jidx.
an
1
Eq.(2.1)1, vi = -
_
(2.2)i
1. dx
is used. or, by (3.6)
and f = 0 on 130, from Eq. (3.6)
vi = $D~~F~,
(3.7)
where the relation s, f(x) dx = 1 was used. 4.
The second law of thermodynamics We have the following expression for the entropy of our model
s=
-f(x,
I[R
where To denotes the temperature dS
dt= or
S[ -g
t) + C(x)
of the initial state. lnf(x,
t) - 2
Q
t?S -=
at
t) lnf(x,
I[ R
hi lnf(X,t) +ji,i f
lny]
dx,
(4.1)
Hence
+ C(x)$g
1
dx
$ (-Si,, + r) I dx.
(4.2)
298
R. WOJNAR
We again use the divergence theorem and find
and after using the boundary conditions (2.5) we get iAS
dt=* I[
’f f,i -
-ji
T,i qi + k T dx. I
&
(4.3)
Substituting T from (2.4)~ into (4.3) we get LhS -=
-ji I[
or
’
f,i - $
From (4.4) we see that the thermodynamic Remark 3: heat fluxes j and q are
Remark 4: as follows:
dx
g”S;-jii:[f,i+k vi]- qif 1
and
1
T,i qi - $ j,yi
&VT,
Ti)dx
(4.4)
forces for the diffusion and
respectively.
Thus the proper form of equations for diffusion and heat fluxes (2.1) is
ji
=
-Dij
f $ (f,j + $vj)
- Mij T2 &T,j
)
4i
=
-Nij
f $ (f,j + GQ)
- Kij T2 &T,j
)
and the symmetry of kinetic coefficients demands MijT2=Nijf.
(4.5)
The condition (4.5) depends not only on coefficients characterizing the system M and N but also on the unknown fields f and T. Next, from Eq. (2.1)r we find f,i + fyi where D = D-’
= -Dij
(jj
+ MjmT,,)
)
(4.6)
and after Eq. (2.1)~ we can write Qi = NijBikj,,
- (Kim - NijDikMkm) T,m .
(4.7)
NONLINEAR
HEAT
EQUATION
AND THERMODIFFUSION
299
Using (4.6) and (4.7) we find ji(j’,z+h,i)
= qi Ti
Substituting dS dt=
/{o
Therefore,
=
-Oijjij,
-
N,jDik
jk
DijMjmjiT,,,
Tsi -
(Kim
-
NijBjkMkm)
T,,T,
these relations into (4.4) we obtain lu,.j f
jijj
+
(
fMii
- $Nij
>
jk T,i + (Kz,
- NijDjkMk,)
T,iT,
dx. (4.8)
if (4.5) is satisfied and the matrix Kim - NijDjkMkrn
(4.9)
Or
[
DM NK
1
are positive definite, and the second law of thermodynamics a!!? ,,,O
(4.10)
holds true in the system. 5.
Driven
system in a slab
Again, treading in the footsteps of [l], we look for a stationary solution far from equilibrium. Consider one-dimensional isotropic model for which a Brownian particle is confined to 0 < x 5 1. We impose the condition that the particle current is zero, j=o
(5.1)
and that time derivatives vanish. From (2.4)~ and (5.1), we get r=O
Also we assume that there is a nonzero flow of heat driven by maintaining the interval [0, l] at different temperatures, To and TI. Eqs. (2.1)r and (2.1)~ for our case (l-dim, isotropic) read
(5.2)
the ends of
(5.3)
R. WOJNAR
300
or after the use of (4.5) in (5.3)r we have
T,,=
0.
(5.4)
Hence
After integration (5.5) Using (5.4) we write (5.3)~ in the form
,_!?f
T
D
7-‘2
9
>
1 ,2
=
0.
(5.6)
’
(5.7)
After integration we get K_!!?f
T
DT2
>
lx
=a
where a = const and f is given by (5.5). Thus (5.7) is complicated nonlinear integrodifferential equation for T which can be solved by approximative methods. We can write an alternative form of (5.7) T(x)
=
To
+
a
I
2
0
dl
&!I!2 D
(5.8) T2
where for N + 0 we have expression from [l]. 6.
System in a slab with given temperature
We still consider one-dimensional isotropic model for which a Brownian particle at t = 0 is at x = ~0 and we impose the condition that the particle current is zero, j = 0, at the bottom of the slab x = 0. We assume that the temperature distribution is known: T = T(z). Evolution of the distribution function is described by Eq. (2.3)r which for (l-dim, isotropic) case reads g= or, after use of (4.5), we have
b(f.+;
I%)
+MTZ],Z
(6.1)
NONLINEAR HEAT EQUATION AND THERMODIFFUSION
301
Or
(6.2) where
is a given function. If N and D are constants, we have
(6.3)
li=V+$lnT,
which means that thermodiffusion leads to a modification of the external potential V. Solution of equations of type (6.2) was discussed by Smoluchowski [4], cf. also Cherkasov [5] and Ricciardi [6]. REFERENCES [I] R. F. Streater: Rep. Math. Phys. 40, 557--564 (1997). [2] R. F. &eater: J. Stat. Phys. 88, 447-469 (1997). [3] R. F. Streater: The Soret and Dufour effects in statistical
dynamics, Preprint KCL- MTH98-32, Department of Mathematics, King’s College, London. [4] M. Smoluchowski: Physik. Zeitschrft 17, 557-599 (1916). (51 I. D. Cherkasov: Theo? Probab. Its Applic. 2, 373-377 (1957). [6] L. M. Ricciardi: J. Math. Analysis and Applications
54,
185-199 (1976).
REPORTS
NONLINEAR
HEAT
ON
MATHEMATICAL
EQUATION
AND
No. 112
PHYSICS
THERMODIFFUSION
R. WOJNAR Institute
of Fundamental Technological Research, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warszawa, Poland (e-mail: [email protected]) (Received October 6, 1999)
A Streater model for diffusion of Brownian particle in a liquid under the influence of an exterior field is considered for the case when thermodiffusion effects are taken intro account. It is shown that for such a model the first law of thermodynamics holds true but the second law is modified to include the probability density function and temperature.
1.
Introduction It was pointed
out in [l, 21 that
falling in the gravity field becomes diffusion and thermal
processes,
the friction
an irreversible
notwithstanding
work performed
the weaker cross-effects
thermodiffusion which were neglected in [l, 21. Streater [3] performed an analysis of DufourSoret model of a gas of molecules {macroscopic)
coefficient
of the microscopic expressed
with repulsive occurs,
hopping
by the diffusion
In the present
namely
rate,
2.
density
namely
effects coefficient
conductivity
of Dufour- Soret
for the special
statistical
which has interpretation and cross-coefficients
and a maximum
thermodiffusion
arc
density.
effect on the process described
of the thermodiffusion
those of diffusion,
particles
In this model only one phenomenological
the diffusion
paper, a thermodiffusion
We do not enter into micromechanics logical coefficients,
cores.
and the heat
coefficient,
by Brownian
heat source, and it plays a role in linking
and exploit
in (I] is studied. only phenomeno-
and heat conductivity.
The basic equations Let Q be a bounded,
sufficiently
regular
domain
with boundary
80.
Let the function
f = f(x, t) represent the probability density of finding a tagged Brownian particle at x at time t. Let j and q denote the flux of Brownian particle diffusion and heat flow, Let the diffusion coefficient be D, the specific respectively, considered in that domain. heat be C and the heat conductivity mutually interrelated, cf. Eq. position x, time t, temperature
be K, while M and N be thermodiffusion
coefficients,
(4.5). We admit that D, K, M and N are functions T and probability density f.
P951
of
296
R. WOJNAR
Consider the motion of the Brownian particle at its terminal velocity in a fluid at temperature T, under the influence of a conservative field of force F = -VV, where V denotes a potential. The equations for diffusion and heat fluxes are as follows
(2.1) The corresponding continuity equations read
(2.2) where r denotes a heat source. Substituting (2.1) 1 into (2.2)i and (2.1)~ into (2.2)~ we get
af
dt=
‘g =
0ijC.f.j + kvj)
1,i ,
+ MijT,j
+r.
Nij(f,j+Gl/j)+K,jTj
(2.3)
I ,i
Instead of the single diffusion equation with drift, known as the Einstein-Smoluchowski equation, we have the set of two equations (2.3). In (2.2) and (2.3) the term r represents the rate at which the Brownian particle does work converted subsequently into heat, r=j.F
r = -j . VV.
or
(2.4)
We make the following boundary assumptions: j=O
3.
and
on
q=o
dR.
(2.5)
The first law of thermodynamics By Eqs. (2.2)1,2 and (2.4) we obtain the following continuity relation
g+j,4;=o, where e denotes the energy density and jE e-Vf+CT
(3.1)
the current of total energy, namely
and
j:
= Vji + qi .
(3.2)
According to (2.5), jE=O
on
dR
(3.3)
NONLINEAR
HEAT
EQUATION
and, as in the case without thermodiffusion E+/dx By integration
=
s,
297
AND THERMODIFFUSION
the energy of the system is
[v(x)f(x,
(3.4)
t)l dx.
t) + C(x)T(x,
, Jjs:dx
of (3.1) we find 8E -_= at
R
or by virtue of the divergence theorem and vanishing boundary condition (3.3) we get, dE -_= f3t Thus the first law of thermodynamics
(3.5)
O.
holds true.
The mean velocity of the Brownian particle is, by Eq. Remark 1: divergence theorem, d
u2=, J
xif(x,
t)dx =
R
x$??&x = _ dt
JR
So, similarly as in [l], v = J, jdx,
J
xjkkdx
R
=
I[R
J
X,jknkdA
where the boundary condition (2.5)i Dij (f,j + $)
+ MtjT,j
If D,j, A&j, T and VV are constants, Remark 2: we get, as in [l], Einstein’s formula
+
and the
jidx.
an
1
Eq.(2.1)1, vi = -
_
(2.2)i
1. dx
is used. or, by (3.6)
and f = 0 on 130, from Eq. (3.6)
vi = $D~~F~,
(3.7)
where the relation s, f(x) dx = 1 was used. 4.
The second law of thermodynamics We have the following expression for the entropy of our model
s=
-f(x,
I[R
where To denotes the temperature dS
dt= or
S[ -g
t) + C(x)
of the initial state. lnf(x,
t) - 2
Q
t?S -=
at
t) lnf(x,
I[ R
hi lnf(X,t) +ji,i f
lny]
dx,
(4.1)
Hence
+ C(x)$g
1
dx
$ (-Si,, + r) I dx.
(4.2)
298
R. WOJNAR
We again use the divergence theorem and find
and after using the boundary conditions (2.5) we get iAS
dt=* I[
’f f,i -
-ji
T,i qi + k T dx. I
&
(4.3)
Substituting T from (2.4)~ into (4.3) we get LhS -=
-ji I[
or
’
f,i - $
From (4.4) we see that the thermodynamic Remark 3: heat fluxes j and q are
Remark 4: as follows:
dx
g”S;-jii:[f,i+k vi]- qif 1
and
1
T,i qi - $ j,yi
&VT,
Ti)dx
(4.4)
forces for the diffusion and
respectively.
Thus the proper form of equations for diffusion and heat fluxes (2.1) is
ji
=
-Dij
f $ (f,j + $vj)
- Mij T2 &T,j
)
4i
=
-Nij
f $ (f,j + GQ)
- Kij T2 &T,j
)
and the symmetry of kinetic coefficients demands MijT2=Nijf.
(4.5)
The condition (4.5) depends not only on coefficients characterizing the system M and N but also on the unknown fields f and T. Next, from Eq. (2.1)r we find f,i + fyi where D = D-’
= -Dij
(jj
+ MjmT,,)
)
(4.6)
and after Eq. (2.1)~ we can write Qi = NijBikj,,
- (Kim - NijDikMkm) T,m .
(4.7)
NONLINEAR
HEAT
EQUATION
AND THERMODIFFUSION
299
Using (4.6) and (4.7) we find ji(j’,z+h,i)
= qi Ti
Substituting dS dt=
/{o
Therefore,
=
-Oijjij,
-
N,jDik
jk
DijMjmjiT,,,
Tsi -
(Kim
-
NijBjkMkm)
T,,T,
these relations into (4.4) we obtain lu,.j f
jijj
+
(
fMii
- $Nij
>
jk T,i + (Kz,
- NijDjkMk,)
T,iT,
dx. (4.8)
if (4.5) is satisfied and the matrix Kim - NijDjkMkrn
(4.9)
Or
[
DM NK
1
are positive definite, and the second law of thermodynamics a!!? ,,,O
(4.10)
holds true in the system. 5.
Driven
system in a slab
Again, treading in the footsteps of [l], we look for a stationary solution far from equilibrium. Consider one-dimensional isotropic model for which a Brownian particle is confined to 0 < x 5 1. We impose the condition that the particle current is zero, j=o
(5.1)
and that time derivatives vanish. From (2.4)~ and (5.1), we get r=O
Also we assume that there is a nonzero flow of heat driven by maintaining the interval [0, l] at different temperatures, To and TI. Eqs. (2.1)r and (2.1)~ for our case (l-dim, isotropic) read
(5.2)
the ends of
(5.3)
R. WOJNAR
300
or after the use of (4.5) in (5.3)r we have
T,,=
0.
(5.4)
Hence
After integration (5.5) Using (5.4) we write (5.3)~ in the form
,_!?f
T
D
7-‘2
9
>
1 ,2
=
0.
(5.6)
’
(5.7)
After integration we get K_!!?f
T
DT2
>
lx
=a
where a = const and f is given by (5.5). Thus (5.7) is complicated nonlinear integrodifferential equation for T which can be solved by approximative methods. We can write an alternative form of (5.7) T(x)
=
To
+
a
I
2
0
dl
&!I!2 D
(5.8) T2
where for N + 0 we have expression from [l]. 6.
System in a slab with given temperature
We still consider one-dimensional isotropic model for which a Brownian particle at t = 0 is at x = ~0 and we impose the condition that the particle current is zero, j = 0, at the bottom of the slab x = 0. We assume that the temperature distribution is known: T = T(z). Evolution of the distribution function is described by Eq. (2.3)r which for (l-dim, isotropic) case reads g= or, after use of (4.5), we have
b(f.+;
I%)
+MTZ],Z
(6.1)
NONLINEAR HEAT EQUATION AND THERMODIFFUSION
301
Or
(6.2) where
is a given function. If N and D are constants, we have
(6.3)
li=V+$lnT,
which means that thermodiffusion leads to a modification of the external potential V. Solution of equations of type (6.2) was discussed by Smoluchowski [4], cf. also Cherkasov [5] and Ricciardi [6]. REFERENCES [I] R. F. Streater: Rep. Math. Phys. 40, 557--564 (1997). [2] R. F. &eater: J. Stat. Phys. 88, 447-469 (1997). [3] R. F. Streater: The Soret and Dufour effects in statistical
dynamics, Preprint KCL- MTH98-32, Department of Mathematics, King’s College, London. [4] M. Smoluchowski: Physik. Zeitschrft 17, 557-599 (1916). (51 I. D. Cherkasov: Theo? Probab. Its Applic. 2, 373-377 (1957). [6] L. M. Ricciardi: J. Math. Analysis and Applications
54,
185-199 (1976).