Rozenbaum, V. M.

Shapochkina, I. V.

English

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c© 2010 Proceedings of the F&ANS-2010 Conference-School
Nonlinear and nonadiabatic phenomena in
Brownian motors
I. V. Shapochkina
Belarusian State University, Minsk, Belarus
V. M. Rozenbaum
BInstitute of Surface Chemistry, NAS of Ukraine, Kiev, Ukraine
The term “Brownian motor” or “thermal ratchet” appeared in the literature due to attempts
of theoretical physics to describe some phenomena in biological systems [1,2]. One can determine
a Brownian motor as a nanoparticle which can move directionally under the nonequilibrium
fluctuations of its characteristics. Such a fluctuations can be caused by chemical reactions
occurring on a particle or due to various external processes [3]. There are three important
requirements for the directed-motion generation: nonequilibrium fluctuations, energy pumping
to the system induced by them, and the asymmetry of the potential profile. It is of importance
that Brownian motor operation is a typical nonlinear process.
A fluctuation-induced transport of Brownian particles is widely discussed in the literature
and is described by models (generically called ratchets) of various levels of complexity. A so called
flashing ratchet operates through the fluctuations of the spatially asymmetric periodic potential
profile [4]. In our work we consider deterministic temporal fluctuations of the potential energy.
An essential step in understanding the mechanism of influence of space–time variations of
the potential energy on the characteristics of directed motion was the adiabatic motor (reversible
ratchet) introduced by Parrondo [5]. This motor operates without any energy consumptions,
provided that it performs no effective work. A striking property of this motor is that the system
is at thermodynamic equilibrium with a zero instantaneous particle flux at every time, while
the flux averaged over the period τ of the adiabatically slow variation of the potential energy
is nonzero. Necessary condition of the operation of this motor is that the slow variation of the
potential energy should be specified by more than one function of time.
At the same time, the adiabaticity of the process, i.e., the absence of heat transfer between
the system and the environment, may be realized also for an instantaneous change in the potential
profile (so called adiabatically fast process). In this case, for the existence of the nonzero flux, it
is sufficient that the potential energy V (x, σ(t)) periodic in the coordinate space is specified by
a single function of the time σ(t) having at least one jump over its period τ in the limit τ →∞.
The significant difference of this motor from the adiabatic one is the presence of heat exchange
102 Proceedings of the F&ANS-2010 Conference-School, (2010) 102 - 107
Nonlinear and nonadiabatic phenomena in Brownian motors 103
occurring after the jump of the potential energy.
A typical example of a Brownian motor with a stepwise change in the potential energy is a
dichotomic process characterized for a deterministic case by the function σ(t) having two jumps
over its period τ and two regions with constant values between them. However, the assumption
of the instantaneous change is an idealization never present in nature. In reality, there is always
a transient process with the duration τ0 characterizing the rate of the variation of the potential
energy (i.e., there is always nonadiabaticity of the process). The smallness of the parameter τ0
is established by its comparison with the characteristic time parameter of the system; such a
parameter for the Brownian motor with a smooth potential profile is a characteristic diffusion
time of a particle τD = L
2/D over the potential period L (D is the diffusion coefficient).
In work presented here, we demonstrate how to calculate corrections to the flux and velocity
if the potential profile does not change instantaneously. Such a nonadiabatic corrections are
determined by the characteristic features of the potential profile and are important both from
applied and from fundamental points of view.
For this purpose, we use the formalism of Brownian dynamics described by the Smolu-
chowski equation for the distribution function ρ (x, t). The average velocity of a Brownian par-
ticle with potential energy periodic in space and time can be determined through the fraction of
particles Φ(x) crossing the point x during the time τ . For the steady periodic processes (with
ρ (x, τ) = ρ (x, 0)) and σ(τ) = σ(0)), the function Φ(x) is independent of x, and the quantity
Φ(0), i.e., the fraction of particles crossing the point x = 0, certainly determines the average
flux 〈J〉τ ≡ τ−1Φ(0) and average velocity L 〈J〉τ .
We have obtained Φ(0) as a sum of adiabatic contribution ∆Φabad coinciding with the Par-
rondo’s result [5] and nonadiabatic correction ∆Φabnonad for the system with sharp change of the
potential energy plotted in Fig.1 (line M1M2M4) [6]:
∆Φab = ∆Φabad +∆Φ
ab
nonad ,
∆Φabad =
L∫
0
dxρ+(x, b)
x∫
0
dy [ρ−(y, b)− ρ−(y, a)] ,
∆Φabnonad = −12β3D2
L∫
0
dxρ+(x, b)ρ−(x, a) [V ′(x, a) + V ′(x, b)]×
×
τ0∫
0
dt [V ′(x, b)− V ′(x, σ(t))]
t∫
0
dt′ [V ′(x, σ(t′))− V ′(x, a)] ,
ρ±(x, σ) = exp [±βV (x, σ)]
/
L∫
0
dx exp [±βV (x, σ)]
(1)
(β = (kBT )
−1, kB is the Boltzmann constant and T is the absolute temperature). Here we
assumed that the system at the time t = 0, when σ(0) = a, is in the equilibrium state with the
distribution function ρ− (x, a); after the sharp change from V (x, a) to V (x, b) in the short range
Proceedings of the F&ANS-2010 Conference-School, 2010
104 I. V. Shapochkina, V. M. Rozenbaum
 
 
 
 
 
 
 
 
 
 
 
 
FIG. 1.
τ0 < t < τ , which is described by the function σ(t), the potential energy stays equal to V (x, b)
in the range τ0 < t < τ . For τ >> τD, the system has the time to relax to a new state with the
new potential profile corresponding to the equilibrium distribution function ρ− (x, b).
These equations allowed us to write average flux (in the low-frequency limit τ−1 → 0)
for a deterministic dichotomic process with two transient states (line M1M2M3M5M6 in Fig.1),
〈J〉dikhτ = τ−1
(
∆Φab +∆Φba
)
, and to analyze the cycle process, which includes regions (a, b)
with the sharp and (b, a) with the smooth variations of the potential energy (line M1M2M6 in
Fig. 1). It is shown that for antisymmetric potential profiles, the average flux in the last case
is equal to half the contribution from the dichotomic process in which two transient states are
identical. Thus, the Brownian motors with the potential energy periodic in the coordinate space
specifying by a single function of the time σ(t) operate only at the cost of step. The smooth
region of the potential does not matter.
We have also obtained the explicit expressions for the adiabatic contribution and nonadi-
abatic correction for the saw-tooth antisymmetric potential profile (see Fig.2a) [6] which lead
to the following conclusions: (i) the nonadiabatic correction reduces the absolute value of the
flux; (ii) for the extremely asymmetric potential profile (l → 0 or l → L) the adiabatic flux is
Proceedings of the F&ANS-2010 Conference-School, 2010
Nonlinear and nonadiabatic phenomena in Brownian motors 105
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FIG. 2.
maximal and has a finite value, while the nonadiabatic correction diverges. The latter means a
new characteristic time of the system τ˜D ≡ l(L − l)/D which is less than τD is occurred and it
determines the time interval τ0 << τ˜D in which the nonadiabatic correction ∆Φ
ab
nonad is valid.
For τ˜D < τ0 << τ , we are beyond the framework of the adiabatic approximation. In this case,
the average flux is proportional to τ−2 [7].
The singularity of the potential profile can induce both small and large renormalized time
parameters. For example, if the spatial period of the potential profile includes several narrow
high barriers (Vbar >> kBT , Vbar is the barrier height), and the profile variations are slow (e.i.,
τ >> τD), then the time is sufficient for the local thermodynamic equilibrium to be established
in every well separated by the barriers and the directed motion becomes of a hopping type.
Under these conditions, the system can be described in the kinetic approach, which deals with
the time dependencies of the probabilities of the occupation of potential wells and with the
transfer rates between the wells. In the framework of kinetic approach a new long characteristic
time of the system such as τbar ∼ τD exp βVbar appears in addition to short times such as τ˜D.
Proceedings of the F&ANS-2010 Conference-School, 2010
106 I. V. Shapochkina, V. M. Rozenbaum
The new time parameter τbar sets the time necessary for a Brownian particle to overcome the
barrier.
We use very popular in literature model [8] with the potential profile with two wells and
two barriers (as it shown in Fig.2b). The potential well minima and the barrier maxima are
described by the functions of time u(t) and v(t), which are measured in units of kBT . If functions
u (t) and v (t) are changed following the rectangular path with vertices ±u0, ±v0 (see Fig.2c),
then the flux is equal to 〈J〉adτ = τ−1 tanh v0 tanhu0 and requires no energy. This is an adiabatic
motor. It is interesting that the same flux may be created also by a dichotomic process, when the
functions u(t) and v(t) change synchronously and instantaneously from certain values −u0, −v0
to u0, v0, respectively, and from u0, v0 back to −u0, −v0, respectively. The flux in this case
requires an energy income 4kBTτ
−1u0 tanhu0 per unit time [8].
In the adiabatic limit (τ >> τbar), the nonadiabatic correction to the average flux corre-
sponding to the linear (for simplicity) variation of the functions u(t) and v(t) during a short
time interval τ0 is given by the expression [6]
〈J〉nonadτ = −
u0 sinh 2v0 − v0 sinh 2u0
(v20 − u20) cosh v0 cosh u0
wτ0
τ
(2)
(w = w0 exp(−V¯ ) ∼ τ−1bar; w0 is the pre-exponential factor, which depends on the diffusion
coefficient and the curvature of the potential profile at the maxima and minima.)
The significant difference of the nonadiabatic correction in the kinetic approach given by this
Eq. appropriate for τD << τ0 << τbar from the corrections given by Eq. (1) and appropriate
for τ0 << τ˜D < τD is that the former and latter corrections are linear and quadratic in τ0,
respectively. This difference is due to the fact that at the time scale of kinetic description τ0 is
enough to establish the equilibrium distribution in the smooth potential regions. Thus, we have
new time scale and new hierarchy of the characteristic times of the system and new region in
which the calculated nonadiabatic corrections are valid: τD << τ0 << τbar. τbar will stand as
τD.
In conclusion, we note that adiabatically slow and adiabatically fast variations of the poten-
tial profile initiate the transport of Brownian particles in quite a similar way, although there are
certain significant differences. The same flux can be induced in both processes. The main dif-
ference of adiabatically slow process from adiabatically fast one is that at least a two-parameter
time dependence of the potential energy is required to initiate the directed motion in the former,
while a one-parameter dependence is sufficient for the latter processes. It is also important that
the realization of the adiabatically fast process is quite simple. The dichotomic process is an
example. At the same time, this process is always an idealization of the real situation, when
the transition between two states occurs during a nonzero time τ0. That is why it is important
Proceedings of the F&ANS-2010 Conference-School, 2010
Nonlinear and nonadiabatic phenomena in Brownian motors 107
to take into account the variation of the average velocity of a Brownian particle induced by the
nonadiabaticity of the process. It is always exists in reality and depends on transient process du-
rations and characteristic system times. The nonadiabatic corrections also dominate in limiting
the high motion rectification coefficients for the high-efficiency motor operation mode [6]. Thus,
the study of the nonadiabatic nonlinear effects gives a possibility to determine the hierarchy of
the characteristic times of the system generated by the peculiarities of the potential profile.
References
[1] P. Reimann, Phys. Rep. 361, 57 (2002).
[2] P. Ha¨nggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
[3] R. D. Astumian, Science 276, 917 (1997).
[4] R. D. Astumian and M. Bier, Phys. Rev. Lett. 72, 1766 (1994).
[5] J. M. R. Parrondo, Phys. Rev. E 57, 7297 (1998).
[6] V.M.Rozenbaum, I.V.Shapochkina. Pis’ma Zh. Eksp. Teor. Fiz. 92, 124 (2010) [JETP Letters.
92, 120 (2010)].
[7] I.V. Shapochkina, V. M. Rozenbaum, Vestn. Bel. Gos. Univ., Ser. 1, No. 2, 43 (2009).
[8] V. M. Rozenbaum, Yu. A. Makhnovskii, D.-Y. Yang, et al., J. Phys. Chem. B 114, 1959 (2010).
Proceedings of the F&ANS-2010 Conference-School, 2010


Nonlinear and nonadiabatic phenomena in Brownian motors

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Nonlinear and nonadiabatic phenomena in Brownian motors

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