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Diffusion anomalies in ac driven Brownian ratchets
Jakub Spiechowicz1and Jerzy Luczka1, 2
1Institute of Physics, University of Silesia, 40-007 Katowice, Poland
2Silesian Center for Education and Interdisciplinary Research, University of Silesia, 41-500 Chorz´ow, Poland∗
We study diffusion in ratchet systems. As a particular experimental realization we consider an
asymmetric SQUID subjected to an external ac current and a constant magnetic flux. We analyze
mean-square displacement of the Josephson phase and find that within selected parameter regimes
it evolves in three distinct stages: initially as superdiffusion, next as subdiffusion and finally as
normal diffusion in the asymptotic long-time limit. We show how crossover times that separates
these stages can be controlled by temperature and an external magnetic flux. The first two stages
can last many orders longer than characteristic time scales of the system thus being comfortably
detectable experimentally. The origin of abnormal behavior is noticeable related to the ratchet form
of the potential revealing an entirely new mechanism of emergence of anomalous diffusion. Moreover,
a normal diffusion coefficient exhibits non-monotonic dependence on temperature leading to an
intriguing phenomenon of thermal noise suppressed diffusion. The proposed setup for experimental
verification of our findings provides a new and promising testing ground for investigating anomalies
in diffusion phenomena.
PACS numbers: 05.10.Gg, 05.40.-a, 05.40.Ca, 05.60.-k, 85.25.Cp, 85.25.Dq
I. INTRODUCTION
The theory of Brownian motion has played a guiding
role in the development of statistical physics. It provides
a link between the microscopic dynamics and the observ-
able macroscopic phenomena such as diffusion. The lat-
ter has been in the research spotlight already for over
100 years [1]. One century after pioneering Einstein’s
work it remains both a fundamental open issue and a
continuous source of developments for many areas of sci-
ence. Recent prominent examples include stochastic res-
onance [2], ratchet effects [3], enhancement of diffusion
[4] or efficiency [5], to name but a few. Due to universal
character of diffusion and its ubiquitous presence both
in classical and quantum systems as well as in molecular
biology it is still a subject of intensive studies. In last
years its anomalous character has been one of the main
research topic in various fields [6–9]. In particular, dif-
fusion of Brownian particles in deterministic periodic or
random potentials, or a combination of both has been
investigated [10–13]. Anomalous behavior may not sur-
vive until the asymptotic long time regime nonetheless
lately its transient nature has been predicted theoreti-
cally and observed experimentally [14–17]. In this work
we study a model which belongs to an archetypal class of
Brownian ratchets [3]. Despite its simplicity it is able to
exhibit an extremely rich dynamics and variety of anoma-
lous transport features like the absolute negative mobility
in a linear response regime [18–20], the negative mobility
in a nonlinear response regime and the negative differ-
ential mobility [21]. However, here we focus on diffusion
anomalies occurring in this system.
Our model consists of four relevant components and
can be formulated in terms of (i) a classical inertial Brow-
∗jerzy.luczka@us.edu.pl
nian particle of mass M, (ii) moving in a deterministic
ratchet potential U(x) = U(x+L) of period L, (iii)
driven by an unbiased time-periodic force acos (ωt) of
amplitude aand angular frequency ω, and (iv) affected
by thermal noise of temperature T. There are exam-
ples of experimentally accessible physical systems that
can be described by this type of a model. An impor-
tant representative that comes to mind is transport of
ions through nanopores [22], cold atoms in optical lattices
[23–25], type II superconducting devices based on motion
of Abrikosov vortices [26, 27], Josephson vortices [28–30]
and a superconducting phase in weak links and SQUIDs
[31–33], to give only a few. To maintain a close link
with recent experimental research in this field and chal-
lenge experimentalists to put our theoretical predictions
into a reality check from now on we stick to a particular
realization of a rocking ratchet mechanism, namely the
asymmetric SQUID [34, 35]. This Josephson-Brownian
ratchet offers some advantages over other setups: (i) pre-
FIG. 1. The asymmetric SQUID composed of three Joseph-
son junctions and driven by the external current I(t). The
external constant magnetic flux is Φ and the instantaneous
voltage across the SQUID is V(t).
arXiv:1506.00105v1 [cond-mat.stat-mech] 30 May 2015
2
cise experimental control of applied driving forces here
in the form of external currents, (ii) detection of directed
motion manifested in a non-zero long-time dc voltage,
(iii) access to studies over a wide frequency range of adi-
abatic and non-adiabatic external perturbations and fi-
nally (iv) both underdamped and overdamped dynamics
can be investigated by proper junction fabrication and
variation of system parameters. We want to emphasize
that our findings are universal in the sense that they ap-
ply to a broad selection of physical setups and could be
observed in variety of experimental realizations of a rock-
ing ratchet mechanism. In this context, we recommend
readers refer to the papers [36, 37], where normal and
anomalous diffusion in ac driven systems of cold atoms
in dissipative optical lattices has been studied both ex-
perimentally and theoretically. Moreover, in Ref. [37],
the survey of previous experiments on anomalous diffu-
sion in such systems is presented.
Our work is organized as follows. In Sec. II we intro-
duce the model and all quantities of interest. Sec. III
contains a detailed description of our results, in particu-
lar control of anomalous diffusion by temperature and
an external magnetic field as well as explanation of the
mechanism that stands behind observed diffusion anoma-
lies. In Sec. IV we discuss temperature dependence of
the diffusion coefficient. Finally, Sec. V is devoted to
summary and conclusions.
II. MODEL
As an exemplary real system we consider a SQUID
presented in Fig. 1. It is a loop with three resistively
and capacitively shunted Josephson junctions: two iden-
tical are placed in one arm whereas the third is located
in the other arm. Additionally, the SQUID is threaded
by an external constant magnetic flux Φ and driven by a
time periodic current I(t). There is a one-to-one corre-
spondence between this setup and a classical Brownian
particle. The particle position xtranslates to the phase
ϕ=ϕ1+ϕ2, where ϕ1and ϕ2are the Josephson phases
of two junctions located in the same arm. The particle
velocity v= ˙xtranslates to the voltage V∝˙ϕacross the
SQUID, the external force to the current I(t), the particle
mass Mto the capacitance M∝Cand the friction coef-
ficient γto the normal conductance γ∝G= 1/R. Here
we present only the dimensionless form of the Langevin
equation governing the phase dynamics. For details we
refer the reader to our recent paper [35]. It reads
˜
C¨x(t) + ˙x(t) = −U0(x(t)) + acos(ωt) + p2Q ξ(t),(1)
where the dot and prime denotes a differentiation with
respect to the dimensionless time tand the rescaled phase
x= (ϕ+π)/2, respectively. Other dimensionless quanti-
ties appearing in this formula are: the capacitance ˜
Cof
the device, the amplitude aand the frequency ωof the ex-
ternal ac current I(t). Johnson-Nyquist thermal noise is
−2π−π π 2π
U(x)
x
˜
Φe= 0 ˜
Φe=π
2˜
Φe=−π
2
FIG. 2. The potential (2) for j= 1/2 in the symmetric case
˜
Φe= 0 (solid red line) in comparison with the ratchet po-
tential for two values of the external magnetic flux ˜
Φe=π/2
(dashed green line) and ˜
Φe=−π/2 (dotted blue line).
modelled by symmetric and unbiased δ-correlated Gaus-
sian white noise ξ(t) of average hξ(t)i= 0 and the correla-
tion function hξ(t)ξ(s)i=δ(t−s). Its intensity Q∝kBT
is proportional to thermal energy, where Tand kBis the
system temperature and the Boltzmann constant, respec-
tively. The spatially periodic potential U(x) of period 2π
is in the following form [35]
U(x) = −sin(x)−j
2sin(2x+˜
Φe−π/2).(2)
The parameter j=J2/J1is a ratio of critical currents
of two junctions in opposite arms and ˜
Φeis the dimen-
sionless external constant magnetic flux. If j6= 0 the
potential is generally asymmetric and its reflection sym-
metry is broken, see Fig. 2. However, even when j6= 0
there are certain values of the external magnetic flux ˜
Φe
for which it is still symmetric.
The most important quantity characterizing diffusion
of the phase x(t) is its mean square displacement (MSD)
h∆x2(t)idefined as
h∆x2(t)i=[x(t)− hx(t)i]2,(3)
where h·i indicates an average over initial conditions and
thermal noise realizations. Although the dynamics may
not be normal diffusion at all times nonetheless a time-
dependent diffusion coefficient D(t) can be defined as
D(t) = h∆x2(t)i
2t.(4)
Information on the diffusive process is also contained in
the slope of the MSD which can be obtained from the
power-law fitting [38]
h∆x2(t)i ∼ tα.(5)
The exponent αcharacterizes a type of diffusion. Normal
diffusion is for α= 1. There are two distinct regimes of
anomalous diffusion: subdiffusion for 0 < α < 1 and
3
102
104
106
108
1010
1012
102104106
h∆x2(t)i
t
Q= 0.0001
Q= 0.0002
Q= 0.0003
Q= 0.0004
Q= 0.001
Q= 0.002
Q= 0.005
100
102
104
106
102104106
D(t)
t
Q= 0.0001
Q= 0.0002
Q= 0.0003
Q= 0.0004
Q= 0.001
Q= 0.002
Q= 0.005
103
104
105
τ1
Q
FIG. 3. Impact of temperature Q∝Ton the diffusion process. (a) The mean square displacement h∆x2(t)iof the Joseph-
son phase. (b) The diffusion coefficient D(t). (c) The crossover time τ1separating superdiffusion and subdiffusion stages.
Parameters are: ˜
C= 6, a= 1.899, ω= 0.403, ˜
Φe=π/2 and j= 0.5.
superdiffusion for α > 1. In the former case the MSD
increases over time slower and in the latter case faster
than the rate of normal diffusion. Another special case
is ballistic diffusion when α= 2. When the value of α
is guaranteed to be unity, the time-independent diffusion
coefficient Dcan be determined as
D= lim
t→∞ D(t).(6)
Otherwise the above definition is not constructive be-
cause Dis either zero (subdiffusion) or diverges to infinity
(superdiffusion). In the case of SQUID, the phase diffu-
sion can be investigated experimentally via measurement
of the power spectrum of voltage fluctuations [38–40].
III. RESULTS
The Fokker-Planck equation corresponding to the
Langevin equation (1) cannot be solved by use of any
known analytical methods [41]. Moreover, even in the
deterministic limit of vanishing thermal noise intensity
Q= 0 this system exhibits very complex dynamics in-
cluding chaotic regimes [42, 43]. Therefore, In order to
investigate the diffusion process we have carried out com-
prehensive numerical simulations of the driven Langevin
dynamics determined by (1). All numerical calculations
were done by use of a CUDA environment implemented
on a modern desktop GPU. This proceeding allowed for
a speed-up of a factor of the order of 103as compared
to a common present day CPU method. For up-to-date
review of this scheme we refer the reader to [44].
The system described by (1) has a 6-dimensional
parameter space {˜
C, a, ω , j, ˜
Φe, Q}which is too large
to analyze numerically in a systematic fashion even
with the help of our innovative computational meth-
ods. We therefore have decided to focus our atten-
tion on the effect of temperature Q∝Tand the im-
pact of the external constant magnetic flux ˜
Φein the
remarkable regime of the thermal noise enhanced rec-
tification efficiency of the SQUID studied in detail in
Ref. [45]. Unless stated otherwise this case corresponds
to the following set of parameters {˜
C, a, ω , j, ˜
Φe, Q}=
{6,1.899,0.403,0.5, π/2,0.0004}.
A. Control of anomalous diffusion regimes by
temperature
In panel (a) and (b) of Fig. 3 we show time evolu-
tion of the MSD h∆x2(t)iand the diffusion coefficient
D(t), respectively, for selected values of the noise inten-
sity Q∝T. Especially in the latter the reader may eas-
ily distinguish between the type of diffusion: superdif-
fusion occurs in the interval where D(t) increases, the
case of decreasing D(t) corresponds to subdiffusion and
for non-varying D(t) normal diffusion takes place. In the
4
101
102
103
104
105
102104106
D(t)
t
˜
Φe= 0.32π
˜
Φe= 0.36π
˜
Φe= 0.38π
˜
Φe= 0.51π
˜
Φe= 0.73π
˜
Φe= 0.77π
˜
Φe= 0.806π
˜
Φe= 0.815π
101
102
103
104
102104106
D(t)
t
˜
Φe= 0.492π
˜
Φe= 0.495π
˜
Φe= 0.499π
˜
Φe= 0.502π
˜
Φe= 0.506π
˜
Φe= 0.51π
103
104
105
106
0.49π0.5π0.51π
τ1
˜
Φe
FIG. 4. Control of diffusion by the external magnetic flux ˜
Φe. (a) The diffusion coefficient D(t) for selected values of the flux
˜
Φe. (b) Sensitivity of the diffusion coefficient D(t) in the vicinity of the magnetic flux ˜
Φe=π/2. (c) The crossover time τ1as
a function of ˜
Φein the vicinity of ˜
Φe=π/2. Note the colossal increase of order 103in the crossover time τ1when the magnetic
flux changes in the small interval (0.49π, 0.51π). Other parameters are the same as in Fig. 3 except Q= 0.0004.
low temperature limit the lifetime of superdiffusion is ex-
tremely long, see the case of Q= 0.0001 in panel (b). It
might lead one incorrectly to conclude that this anoma-
lous diffusion regime occurs in the stationary regime.
This statement is true only in the deterministic case when
formally Q= 0. However, in such a case the considered
SQUID model is not correct. The deflection from the
early superdiffusive behavior can expressively be noted
as temperature increases. The evolution can be divided
into three time-domains: the early period of superdiffu-
sion τ1, the intermediate interval τ2where subdiffusion
is developed and the asymptotic long time regime where
normal diffusion occurs. The crossover times τ1and τ2
separating these domains can be controlled by tempera-
ture. For example, when Q= 0.0004, τ1≈1.52 ·104
and τ2≈0.9985 ·107. If temperature is increased to
Q= 0.002, the crossover times are reduced to τ1≈103
and τ2≈0.99 ·105. In panel (c) of the same figure we
present the dependence of the crossover time τ1separat-
ing super- and subdiffusion on temperature Q∝T. It is
remarkable that superdiffusion lifetime τ1changes nearly
three order of magnitude when the thermal noise inten-
sity Qvaries in the interval (0.0002,0.002). For higher
temperatures the phase motion is initially superdiffusive
and next normal diffusion takes place, as e.g. in the case
Q= 0.005 in panel (b) of Fig. 3.
B. Control of anomalous diffusion regimes by
external magnetic field
From the experimental point of view it is more conve-
nient to manipulate transport properties of the SQUID
by the external magnetic flux ˜
Φe. In Fig. 4, its impact
on the diffusion process is illustrated. In some regimes,
two crossover times τ1and τ2are identified and their
magnitudes can be changed by variation of the exter-
nal magnetic flux ˜
Φe. For example, when ˜
Φe= 0.73π,
the superdiffusion lifetime is τ1≈1.56 ·107while for
˜
Φe= 0.492πit is τ1≈1.8·103, i.e. four orders shorter.
However, in contrast to the case when temperature Qis
varied the dependence of the crossover times τ1and τ2on
˜
Φeis non-monotonic. It can be concluded from panel (a)
of Fig. 4. Moreover, for some intervals of the magnetic
flux these times are extremely sensitive to small changes
of ˜
Φe. This situation is exemplified in panels (b) and (c):
small changes of order 10−2in ˜
Φeare accompanied by
the giant increase of order 103in the crossover time τ1.
For our particular parameter regime the crossover time
τ2is often so long that its numerical estimation may be
controversial due to surely limited stability of the uti-
lized algorithms leading to uncontrolled propagation of
round-off and truncation errors [44]. Therefore we do
not present its dependence on temperature or the exter-
nal magnetic flux.
5
10−610−510−410−310−210−1
α
Q
˜
Φe= 0
˜
Φe=π/2
−π−π
20π
2π
α
˜
Φe
Q= 10−6Q= 4 ·10−4
FIG. 5. The power exponent αin dependence on temperature
Q∝Tand the external magnetic flux ˜
Φe. Parameters are the
same as in Fig. 3.
C. Comment on the power exponent
Let us now ask a complementary question, how the
power exponent αdefined by the relation (5) depends on
temperature and the external magnetic flux. This expo-
nent was fitted from time evolution of the MSD h∆x2(t)i
over ≈105dimensionless time units at a number of Qand
˜
Φevalues. The results are presented in Fig. 5. Panel
(a) depicts the dependence of αon temperature Qfor
two different potential profiles U(x), namely symmetric
(˜
Φe= 0) and ratchet (˜
Φe=π/2). In the low tempera-
ture limit the crucial role of the potential asymmetry for
the emergence of anomalous diffusion is observed. When
potential is reflection symmetric then α≈0 and there is
no diffusion at all. Contrary, for the ratchet case, α= 2
and diffusion is ballistic indicating the wave-like phase
motion and revealing an entirely new mechanism respon-
sible for anomalous diffusion. This one should be clearly
distinguished from the well known that appears in disor-
dered systems [10–13]. The latter form of the potential
manifests also in the fact that there is a finite window of
temperature for which the motion is super and subdiffu-
sive. One can alter the regime of diffusion in the SQUID
by proper adjustment of thermal noise intensity Q. In
the high temperature limit diffusion is normal regardless
of the potential symmetry. This observation agrees with
our naive intuition since then the impact of both the con-
servative U0(x) and the time-dependent acos (ωt) forces
is negligible in comparison to thermal noise. In panel (b)
the same quantity is presented as a function of the ex-
ternal magnetic flux ˜
Φefor two selected temperatures Q.
The unique feature is the ability to control the type of
phase diffusion covering normal and anomalous regimes:
from subdiffusion, to superdiffusion and finally the bal-
listic motion just by experimentally doable variation of
˜
Φe. Moreover, in agreement with our previous statement
we note that the phase diffusion regime is very sensitive
to changes of this parameter and observe the rapid vari-
ability of αin the vicinity of ˜
Φe= 0 and ˜
Φe=π/2.
D. Mechanism for transient anomalous diffusion
To gain insight into the origin of transient anomalous
diffusion, let us consider the deterministic limit of vanish-
ing thermal noise intensity Q→0 and study the struc-
ture of basins of attraction for the asymptotic long time
phase velocity v= ˙xaveraged over the period of the
external ac driving, to be specific
hvi= lim
t→∞
ω
2πZt+2π/ω
t
ds ˙x(s).(7)
The result is shown in the upper panel of Fig. 6. There
exist only three attractors: the running state with either
positive or negative phase velocity hvi=±0.4 (marked
by red and blue color, respectively) or the locked state
hvi= 0 when the phase motion is limited to a finite
number of potential wells (green color). The examples
of corresponding trajectories are depicted in the bottom
panel of the same figure together with the ensemble av-
eraged temporal evolution of the phase across the device.
This unexpected simplicity of attractors is crucial for the
occurrence of the ballistic transport. In particular, qual-
itatively, due to the existence of two counter-propagating
solutions with equal constant velocity hvi= 0.4 the con-
tribution of the average trajectory hx(t)ito the mean
square displacement h∆x2(t)iis vanishingly small in com-
parison to its second moment hx2(t)i, see the bottom
panel of Fig. 6. As a consequence the mean square dis-
placement is proportional to time in the second power
h∆x2(t)i ∼ t2. The application of thermal noise generally
smooths out the complex structure of boundaries demar-
cating the coexisting attractors. In a qualitative picture,
if temperature start to increase the phase is kicked out of
its deterministic trajectory at random time and its mean
value corresponds to the crossover time τ1. When tem-
perature grows τ1becomes to decrease what is exposed in
Fig. 3. Moreover, stochastic escape events among attrac-
tors give rise to other forms of the anomalous diffusion
[13].
6
01
2π π 3
2π2π
x
v
x
t
hx(t)i
FIG. 6. Basins of attraction for the asymptotic long time
phase velocity hviaveraged over the period of the external
ac driving acos ωt are presented in the upper panel. The
bottom one depicts a number of sample realizations of the
phase motion together with its ensemble averaged trajectory.
Parameters are the same as in Fig. 3 except the thermal noise
intensity which is fixed to zero Q= 0.
IV. TEMPERATURE SUPPRESSED
DIFFUSION
Last but not least, let us analyze the diffusion coef-
ficient Din the normal diffusion regime. This scenario
surely takes place in the limiting case of relatively high
temperature Q > 0.002 since in such a case for times
longer than ≈105the diffusion coefficient Ddoes not
change with time (see Fig. 3b and Fig. 5a). Then it
has a well established physical interpretation and can be
conveniently computed by use of formula (6). In Fig. 7
we present its dependence on temperature, Q∝T. The
striking feature is its non-linear and non-monotonic be-
havior. For low temperature, Dinitially increases as Q
grows, passes through its local maximum and next starts
to decrease reaching its local minimum at some charac-
teristic temperature Qc. For temperatures higher that
Qcthe diffusion coefficient is a monotonically increas-
ing function of Q. This temperature suppressed diffusion
phenomenon is in clear contrast with the Einstein rela-
tion D∝Tas well as with the other known formulas as
e.g for a Brownian particle moving in a periodic poten-
tials [46, 47] and in a tilted periodic potentials (under
additional presence of an external constant force) [4, 48].
V. SUMMARY
In this work we have investigated diffusion processes in
the archetypal model of an inertial Brownian ratchet. As
a particular realization we picked the asymmetric SQUID
device driven by the time periodic current and pierced
by the external constant magnetic flux. We have found
selected parameter regimes for which the MSD of the
Josephson phase evolves in three distinct stages: ini-
tially as superdiffusion, next as subdiffusion and finally
as normal diffusion in the asymptotic long-time limit.
We have shown that crossover times separating these
three stages can be controlled by temperature and the
external magnetic flux. The latter parameter is espe-
cially useful for this purpose as these times are particu-
larly sensitive to its changes. Despite the fact that these
abnormal processes have only transient nature they last
many order longer than characteristic time scales of the
system and thus they are comfortably detectable exper-
imentally. Moreover, we have studied the origin of the
discussed anomalous diffusion behavior and revealed the
entirely new mechanism of its emergence which is based
on breaking of reflection symmetry of the potential. This
effect is particularly evident for low temperature regimes
and should be clearly contrasted with the one that op-
erates in disordered systems. In particular, in the de-
terministic limit of vanishing thermal noise the origin of
the ballistic phase diffusion lies in the existence of two
counter-propagating attractors. Finally, we have studied
also the opposite limiting scenario of high thermal noise
intensity and presented new manifest of the fascinating
interplay between nonlinearity and thermal fluctuations,
namely the phenomenon of noise suppressed diffusion.
Our findings can be corroborated experimentally with
2
4
6
8
10
10−210−1100101
D
Q
FIG. 7. The diffusion coefficient Dversus the thermal noise
intensity Qin the high temperature limit. Other parameters
are the same as in Fig. 3.
7
a wealth of physical systems outlined in the introduc-
tion. One of the most promising setup for this purpose
is the asymmetric SQUID device which has already been
constructed [32, 33]. It is due to its good experimental
control as well as easy detection of the discussed diffu-
sion behavior in the power spectrum of the voltage fluc-
tuations. Such an experiment will probably not be as
spectacular as those based on the single particle tracking
technique [49–51], however, probably it will be much eas-
ier to perform and therefore we think that our work may
suggest a completely new testing ground for investigating
anomalies in diffusion phenomena.
ACKNOWLEDGEMENT
This work was supported by the MNiSW pro-
gram Diamond Grant (J. S.) and NCN grant DEC-
2013/09/B/ST3/01659 (J. L.)
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