Thermophoresis as persistent random walk - Saint Anselm College

Physics Letters A 373 (2009) 2122–2124
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Physics Letters A
www.elsevier.com/locate/pla
Thermophoresis as persistent random walk
A.V. Plyukhin
Department of Mathematics, Saint Anselm College, Manchester, NH 03102, USA
article info abstract
Article history:
Received 16 January 2009
Received in revised form 20 March 2009
Accepted 11 April 2009
Availableonline17April2009
Communicated by C.R. Doering
In a simple model of a continuous random walk a particle moves in one dimension with the velocity
fluctuating between +v and −v. If v is associated with the thermal velocity of a Brownian particle
and allowed to be position dependent, the model accounts readily for the particle’s drift along the
temperature gradient and recovers basic results of the conventional thermophoresis theory.
© 2009 Elsevier B.V. All rights reserved.
PACS:
05.40.-a
66.10.C-
82.70.Dd
Thermophoresis [1–4] is the systematic drift of a Brownian particle
caused by a temperature gradient ∇T . The steady-state thermophoretic
velocity acquired by the particle is given by
V T =−D T ∇T , (1)
where D T is generally referred to as thermal diffusion coefficient.
A similar phenomenon exists on the molecular level in gas mixtures
and is known as the Soret effect. In most cases D T is positive,
i.e. the particle moves toward a colder region. However, thermophilic
behavior, D T < 0, has also been observed experimentally
[5–7]. Among other well-known field-driven transport effects, thermophoresis
is perhaps the most subtle: it takes place in a stationary
state when the pressure is uniform, and therefore the average
force on a particle which is fixed in space is strictly zero. Interesting
in itself, thermophoresis has many applications, in particular in
the context of molecular Brownian motors driven by thermal fluctuations
[8–10].
Despite many efforts, a realistic theoretical model of thermophoresis,
valid over a wide region of densities of the host fluid,
seems to be lacking, and even the physical origin of the effect is
still under debate [1]. Many authors argue that the effect cannot
be described by a simple modification of the over-damped diffusion
equation in the configurational space only (Smoluchowski
equation), but requires a kinetic description. Well-known example
of such approach is that by van Kampen [11], which takes as a
starting point the Kramers equation with position-dependent temperature
∂ f
∂t =−v ∂ f
∂x − F ∂ f
m ∂ v + γ ∂ (
vf + kT(x) )
∂ f
. (2)
∂ v m ∂ v
E-mail address: aplyukhin@anselm.edu.
Here f (x, v, t) is the joint probability density of position and velocity,
F is an external force, and γ is the friction coefficient which
may depend on x and is assumed to be large. Using the expansion
in powers of γ −1 , one can obtain to the lowest order the
corresponding overdamped Smoluchowski equation for the spatial
density f (x, t) in the form
∂ f
∂t + ∂ (
∂x J = 0, J = μF − D ∂ )
∂x − D ∂ T
T f (x, t), (3)
∂x
where the mobility μ = 1/mγ , the diffusion coefficient D = kT μ,
and the thermal diffusion coefficient is
D T =
k
mγ = D = kμ. (4)
T
Therefore, the thermophoretic force F T on the particle, defined
through the relation μF T =−D T ∇T ,equals
F T =−k∇T . (5)
The result (4) often underestimates the value of D T by up to one
order of magnitude [1], and cannot, of course, account for thermophilic
behavior corresponding to a negative D T .Anotherrestriction
of the approach is that the validity of the Kramers equation
for the case of nonuniform temperature is postulated rather than
proved. More “microscopic” approaches lead to the expression for
D T with additional terms involving rather complicated correlation
functions [12–14]. However, conceptually the result (4) is important
as a demonstration that the expression for the current J
in the Smoluchowski equation (3) cannot be obtained by simple
insertion of the position dependence into the conventional driftdiffusion
expression J = (μF − D∂/∂x) f for an isothermal system.
The simplicity of the result (4) suggests that it might have
a transparent physical interpretation. Yet the transition from the
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A.V. Plyukhin / Physics Letters A 373 (2009) 2122–2124 2123
Kramers equation (2) in phase space to Eq. (3) for the spatial distribution
function f (x, t) = ∫ dv f (x, v, t) (the elimination of the
fast variable v) isnotatrivialstep[15,16], which makes the origin
of the result (4) somewhat obscure. The same perhaps may
be said of the Luttinger’s method of fictitious external fields [17],
which also leads to the result (4) [18,19]. The aim of this Letter is
to formulate a minimal qualitative model of thermophoresis, which
leads to the expression (4) elementary and directly. Besides pedagogical
merits, such model might be useful for numerical modeling
of stochastic processes at nonuniform temperature.
The model is a slightly generalized version of the well-known
stochastic process of the continuous persistent random walk
[20–22]. In its simplest setting, the process describes a particle
moving in one dimension with fixed speed v suffering occasionally
a complete reversal of direction. Let f + (x, t) and f − (x, t) be
the probability density for the particle moving to the right and to
the left, respectively. Reversals of velocity are Poisson distributed,
i.e. occurring with a constant rate 1/2τ , so that the probability for
reversal in a time interval dt is dt/2τ . For an infinitesimal time
step one can write
f + (x, t + dt) = f + (x − vdt, t) − f + (x − vdt, t) dt/2τ
+ f − (x + vdt, t) dt/2τ , (6)
and a similar equation for f − (x, t). The corresponding differential
equations read
∂ f +
∂t
=−v ∂ f +
∂x
− f + − f −
,
2τ
∂ f −
∂t
= v ∂ f −
∂x
+ f + − f −
, (7)
2τ
and lead to the telegrapher’s equation for the total density f =
f + + f − ,
∂ 2 f
∂t + 1 2 τ
∂ f
∂t = v2 ∂2 f
∂x 2 . (8)
The same equation holds also for the difference Δ = f + − f − , and
therefore for the components f + and f − separately.
Suppose the particle at t = 0 is at the origin with equal probability
to be in each of the two velocity states,
f ± (x, 0) = 1 2 δ(x), ∂ f ± (x, 0)
=∓ v ∂
δ(x). (9)
∂t 2 ∂x
Here the second initial condition follows from the first one and
Eq. (7). Respectively, the initial conditions for the total density f =
f + + f − are
∂ f (x, 0)
f (x, 0) = δ(x), = 0. (10)
∂t
The corresponding solution of the telegrapher’s equation is well
known [26]. In the long-time limit t ≫ τ , vt ≫ x it coincides exactly
with solution of the Smoluchowski equation
)
f (x, t) ≈ (4π Dt) −1/2 exp
(− x2
(11)
4Dt
with the diffusion coefficient D = τ v 2 . However, unlike the overdamped
Smoluchowski equation (3), Eq.(7) incorporates effects of
inertia of the particle. This advantage, which was recognized and
used beneficially in many previous works (see [22] and references
therein), allows to account for thermophoresis in a particularly
simple way.
To link the model to the problem of thermophoresis, it is natural
to identify v with a typical thermal speed of a Brownian
particle. The choice is ambiguous. For instance, one can set v equal
to the root mean square velocity v rms =〈v 2 〉 1/2 = √ 3kT/m. However,
one can show that in this case the final result for the thermophoretic
force would differ from Eq. (5) by the factor 3/2 (see
Eq. (16) below). As will be shown, a perfect agreement with the
standard results D T = D/T and F T =−k∇T can be achieved if v
is identified not with v rms but with the most probable speed of a
Brownian particle v mp (for which the Maxwell speed distribution
has a maximum):
v = v mp = √ 2kT/m. (12)
The second parameter of the model 1/τ should be identified with
the friction coefficient γ which appears in the Kramers equation
(2) and in the corresponding Langevin equation ˙v =−γ v + ξ(t).
For nonuniform temperature, the velocity in Eq. (7) is position
dependent, v(x) = √ 2kT(x)/m. In this case instead of the telegrapher’s
equation (8) one obtains [23]
∂ 2 f
∂t + 1 (
∂ f
2 τ ∂t = ∂2 f
v2 ∂x + 2 v dv ) ∂ f
dx ∂x . (13)
Suppose the temperature gradient is constant, so that T (x) = T +
x ∇T and
v(x) = √ (
2kT(x)/m = v
1 + ∇T
T x ) 1/2
, (14)
where v is the thermal velocity corresponding to the temperature
T , v = √ 2kT/m. ThenEq.(13) reads as
∂ 2 f
∂t + 1 2 τ
(
∂ f
∂t = v2 1 + ∇T
T
) ∂ 2 x f ∇T
+ v2
∂x2 2T
∂ f
∂x .
For a small gradient (∇T /T )x ≪ 1, the equation is simplified to
the form
∂ 2 f
∂t + 1 ∂ f
2 τ ∂t = ∂2 f
v2 ∂x − F T ∂ f
2 m ∂x , (15)
where the thermophoretic force F T coincides (thanks to the setting
v = √ 2kT/m) with the expression (5) of the standard theory:
F T =− 1 ∇T
mv2 =−k∇T . (16)
2 T
It can be shown that in the long-time limit the solution of Eq. (15)
with the boundary conditions (10) coincides with the solution of
the overdamped equation (3) and describes the drift along the
temperature gradient, superimposed on the diffusion
f (x, t) ≈
1
√ 4π Dt
exp
(− (x − V T t) 2 )
4Dt
(17)
with the drift velocity V T = F T τ /m. Using(16) and recalling D =
τ v 2 ,onegetsV T =−D T ∇T with D T = D/T , thus recovering the
result (4) of the conventional theory.
The asymptotic solution (17) can be obtained as follows. After
applying the transformation f (x, t) = φ(x, t) exp(−t/2τ +
F T x/2mv 2 ),Eq.(15) takes the form
∂ 2 φ
∂t 2
= ∂2 φ
v2 ∂x + 1 φ, 2 τ∗
2 (18)
with 1/τ 2 ∗ = 1/4τ 2 − F T /4m 2 v 2 , while the boundary conditions
corresponding to (10) read
φ(x, 0) = δ(x),
∂φ(x, 0)
∂t
= 1
2τ δ(x).
Note that the model makes sense only under the assumption τ∗ 2 >
0, which guarantees the positiveness of f + and f − [25]. Eq.(18) is
the modified telegrapher’s equation whose solution is well known
[26]. Transforming back from φ to f ,theresultcanbewrittenin
the following form
(
f (x, t) = exp
− t
2τ +
F T x
2mv 2 )
(φ 1 + φ 2 + φ 3 ), (19)

2124 A.V. Plyukhin / Physics Letters A 373 (2009) 2122–2124
where the functions φ i (x, t) are
φ 1 (x, t) = 1 [ ]
δ(x − vt) + δ(x + vt) ,
2
φ 2 (x, t) = 1 ( 1 √ )
4vτ I 0 v
vτ 2 t 2 − x 2 θ ( vt −|x| ) ,
∗
φ 3 (x, t) =
t
2τ ∗
√
v 2 t 2 − x 2 I 1
( 1
vτ ∗
√
v 2 t 2 − x 2 )
θ ( vt −|x| ) , (20)
and θ(x) is the unit step function. Using the asymptotic form of
the modified Bessel functions for large argument I α (x) ≈ 1 √ 2π x
e x ,
it is an easy matter to prove that in the limit of strong damping
F T τ /mv ≪ 1 and of long time t ≫ τ , vt ≫ x, the exact solution
(19) is reduced to the simple form (17).
One interesting problem related to thermophoresis is that of
Brownian motors driven by position-dependent temperature. In
particular, the Büttiker–Landauer [27,28] motor is essentially a
Brownian particle diffusing in the periodic potential field subject
to a spatially inhomogeneous temperature. In this context it is of
interest to generalize the model to the case of the presence of
an external force F . As discussed in [25], in this case Eq. (7) for
f ± (x, t) should be extended as follows
∂ f ±
∂t
∂ 2 f
∂t + 1 2 τ
=∓v ∂ f ±
∂x
∓ f + − f −
2τ
∂ f
∂t = ∂2 f
v2 ∂x − F T ∂ f
2 m ∂x − ∂ ∂x
± F
2mv ( f + + f − ). (21)
Respectively, Eq. (15) for the total density f = f + + f − is generalized
to the form
( ) F
m f . (22)
This equation differs from the overdamped equation (3) of the
standard theory by the presence of the term f tt . This term, related
to inertial effects, is unimportant in the long time limit, but
may be responsible for wave-like behavior at short times [22].
Summarizing, in this Letter we discussed a stochastic process
which underlies thermophoresis in a way similar to that as the discrete
random walk underlies isothermal diffusion. The model leads
to the telegrapher’s equation whose asymptotic solution coincides
with the solution of the overdamped Smoluchowski equation (3).
This is consistent with an observation that the telegrapher’s equation
is reduced to the Smoluchowski equation under conditions
v →∞, τ → 0, v 2 τ = const. Since the model takes into account
inertial effects, one might hope that it can resolve difficulties of the
overdamped theory of thermally driven Brownian motors [8–10].
However, similar to the Kramers equation (2), the model is based
on the assumption that the thermalization of the particle to a local
temperature is faster than any other process involved. Since the
characteristic time scales for inertial and thermalization effects are
typically the same, the application of the model beyond the overdamped
regime, strictly speaking, is not justified, and should be
undertaken with caution. Despite of this limitation, the mapping
of thermophoresis onto a random walk problem may offer some
advantage for both analytical and numerical modeling, in particular
for problems with complicated or velocity-dependent [24,25]
boundary conditions.
Acknowledgements
I thank Richard Bowles, Stephen Shea, and the referees for stimulating
comments.
References
[1] R. Piazza, A. Parola, J. Phys.: Cond. Matter 20 (2008) 153102.
[2] M. Braibanti, D. Vigolo, R. Piazza, Phys. Rev. Lett 100 (2008) 108303.
[3] S. Duhr, D. Braun, Phys. Rev. Lett. 96 (2006) 168301.
[4] E. Bringuier, A. Bourdon, Phys. Rev. E 67 (2003) 011404;
E. Bringuier, A. Bourdon, Physica A 385 (2007) 9.
[5] M. Giglio, A. Vendramini, Phys. Rev. Lett. 38 (1977) 26.
[6] B.-J. de Gans, R. Kita, B. Müller, S. Wiegand, J. Chem. Phys. 118 (2003) 8073.
[7] S. Iacopini, R. Piazza, Europhys. Lett. 63 (2003) 247.
[8] I. Derényi, R.D. Astumian, Phys. Rev. E 59 (1999) R6219.
[9] T. Hondou, K. Sekimoto, Phys. Rev. E 62 (2000) 6021.
[10] R. Benjamin, R. Kawai, Phys. Rev. E 77 (2008) 051132.
[11] N.G. van Kampen, IBM J. Res. Develop. 32 (1988) 107;
N.G. van Kampen, J. Phys. Chem. Solids 49 (1988) 673.
[12] G. Nicolis, J. Chem. Phys. 43 (1965) 1110.
[13] D.N. Zubarev, A.G. Bashkirov, Physica 39 (1968) 334.
[14] J.-E. Shea, I. Oppenheim, J. Phys. Chem. 100 (1996) 19035.
[15] M.E. Widder, U.M. Titulaer, Physica A 154 (1989) 452.
[16] J.M. Sancho, M. San Miguel, D. Dürr, J. Stat. Phys. 28 (1982) 291.
[17] J.M. Luttinger, Phys. Rev. 135 (1964) A1505.
[18] A.L. Efros, Sov. Phys. JETP 23 (1966) 536.
[19] A.G. Bashkirov, Theor. Math. Phys. 1 (1969) 213.
[20] S. Goldstein, Quart. J. Mech. Appl. Math. 4 (1951) 129.
[21] M. Kac, Rocky Mountain J. Math. 4 (1974) 497.
[22] G.H. Weiss, Physica A 311 (2002) 381.
[23] J. Masoliver, G.H. Weiss, Phys. Rev. E 49 (1994) 3852.
[24] J. Masoliver, J.M. Porra, G.H. Weiss, Phys. Rev. E 45 (1992) 2222.
[25] A.V. Plyukhin, K.S. Kim, Phys. Rev. E 61 (2000) 3207.
[26] E. Zauderer, Partial Differential Equations of Applied Mathematics, Wiley, New
York, 1983.
[27] M. Büttiker, Z. Phys. B 68 (1987) 161.
[28] R. Landauer, J. Stat. Phys. 53 (1988) 233.