Two-dimensional optical thermal ratchets based on Fibonacci spirals
Abstract.
An ensemble of symmetric potential energy wells arranged at the vertices of a Fibonacci spiral can serve as the basis for an irreducibly two-dimensional thermal ratchet. Periodically rotating the potential energy landscape through a three-step cycle drives trapped Brownian particles along spiral trajectories through the pattern. Which spiral is selected depends on the angular displacement at each step, with transitions between selected spirals arising at rational proportions of the golden angle. Fibonacci spiral ratchets therefore display an exceptionally rich range of transport properties, including inhomogeneous states in which different parts of the pattern induce motion in different directions. Both the radial and angular components of these trajectories can undergo flux reversal as a function of the scale of the pattern or the rate of rotation.
§ I. Introduction
Systems that rely on time-dependent forces to rectify thermal noise are called “ thermal ratchets” (1); (2); (3); (4); (5); (6). Introduced with James Clerk Maxwell's thought experiments in the 1860's, thermal ratchets recently have enjoyed a resurgence of interest because of their relevance to biological molecular motors, and have been realized experimentally both for molecular (7); (8), micrometer-scaled (9); (10); (11); (12); (13); (15); (16); (17); (18); (19) and for quantum objects (22). Virtually all of these studies, however, have focused on one-dimensional systems (9); (10); (11); (12); (13); (15); (16); (17); (18); (19), or on systems that can be projected onto one dimension (7); (21). Comparatively little attention has been paid to thermal ratchets in two or higher dimensions.
The present study introduces an irreducibly two-dimensional thermal ratchet based on rotational symmetries of the Fibonacci spiral (23). Section II presents the rich phenomenology of Fibonacci spiral ratchets comprised of discrete symmetric potential energy wells. This system features a deterministic regime in which the ratchet-induced flux spirals through the pattern of traps, and a stochastic regime in which diffusion-assisted transport admits flux reversals in both the radial and azimuthal degrees of freedom. These predictions are tested in Sec. III through experiments on colloidal spheres diffusing in water through rotating arrays of holographic optical traps arranged on the vertexes of Fibonacci spirals.
§ II. Fibonacci spiral ratchet
§ II.1. Fibonacci spiral
The Fibonacci spiral (23) is a set of points in the plane
whose -th member falls at polar coordinates
![]() |
![]() |
(1) | ||
![]() |
![]() |
(2) |
for .
It differs from a more general
Archimedean spiral both by the square-root increase in radius
and also by the choice of the Golden angle
![]() |
(3) |
for the inter-node angular separation, where
![]() |
(4) |
is the irrational number known as the Golden mean.
A typical example is plotted in Fig. 1.
The overall scale factor does not enter into the
definition of a Fibonacci spiral but plays a crucial
role in its operation as a thermal ratchet.
This particular set of points is the densest packing of identical circles within a circular region (24). As a result, it appears with great regularity in natural systems, most famously in the distribution of seeds within the seed-heads of sunflowers (25); (26).


The dense packing of nodes in a Fibonacci spiral gives rise to
a host of intriguing symmetries not found in other space-filling
two-dimensional patterns.
The principal Fibonacci spiral is defined by Eqs. (1)
and (4) for integers separated by unity.
Higher-order spirals appear within this pattern for sequences
of
that increase by steps equal to one of the
Fibonacci numbers,
, which are defined recursively
by
(27); (25).
No other integers define regular spirals within the underlying pattern.
The Fibonacci numbers also are noteworthy because their ratio
approaches the Golden mean in the limit of large index,
.
Within a Fibonacci spiral,
the -th Fibonacci number,
, defines a set of
intertwined
spirals, which have come to be known in the botanical context
as parastichies of order
(28). By definition, the
-th parastichy appears for radii
.
Which parastichies are available at a given radius
substantially influences the behavior of thermal ratchets based
on Fibonacci spirals.







§ II.2. Three-state ratchet
We define a Fibonacci spiral ratchet by placing a spatially symmetric
potential energy well of width at each vertex in the Fibonacci spiral.
The resulting potential energy landscape is transformed
into a ratchet potential through its time evolution.
To break spatio-temporal symmetry we adopt the three-state protocol
introduced originally for studies of one-dimensional ratchets
(29); (18).
Rather than translating the pattern, as in earlier studies
(29); (18); (19), we rotate it about its center:
![]() |
(5) |
with time depedence
![]() |
(6) |
This cyclic pattern has period and holds the
landscape stationary for the duration
of each state.
For simplicity, we also choose identical angular
steps
between the states.
The arrangement of traps and the three-state time evolution
define the Fibonacci spiral ratchet whose
performance dependends on the three control parameters,
,
, and
.
§ II.3. Deterministic regime
To study transport properties arising purely from the ratchet's geometry,
we first select to be small enough that no force-free region
exists within the pattern. Any object released
from a vertex in one state will be drawn into the nearest potential
well (vertex) in the next state.
Cycling through a sequence of discrete angular steps therefore causes
the object to move from one site on the lattice to another,
a process that may be represented as a mapping of the spiral onto itself.
This mapping depends on the sequence of nearest-neighbor steps
in the intermediate states,
which is determined by the angle of rotation,
.
In this deterministic regime we assume that diffusion may be ignored.
Defining trajectories through the rotating Fibonacci spiral
therefore involves finding the index of the rotated
site that is closest to the vertex
in the previous state.
We use the notation
![]() |
![]() |
|||
![]() |
(7) |
to represent the
-th site of the lattice in the
-th state (
).
With this definition,
the inter-vertex distance between the
-th site
in the initial state
and the
-th site after one step is
![]() |
![]() |
|||
![]() |
(8) |
We assume that a particle initially at
site will advance to the site
whose index
minimizes
.
Repeating the process for the second and third rotations
yields the trajectory
.
After one complete cycle, therefore, a particle initially at
site
undergoes a change of index
![]() |
(9) |
The single-cycle mapping can be composed into
trajectories that particles will follow through the array
as the pattern cycles repeatedly through its sequence of states.
Empirically, we observe that
only
takes on values
related to the Fibonacci numbers,
and consequently that particles are conveyed along parastichies
by the three-state ratchet cycle.
Four examples are plotted in
Fig. 2(a) through (d).
Arrows in these plots indicate the direction of motion,
with red traces indicating outward
motion (
) and blue indicating inward motion
(
).
Trajectories characterized by large jumps in index are possible
only for sufficiently large radii.
Consequently, different parasticies may be selected at different
radii within the spiral.
Figure 2(e) shows which trajectory is selected
as a function of rotation angle and radius within the spiral.
Depending on the angular step,
,
trajectories spiral inward or outward, clockwise or counterclockwise.
In some cases, such as Fig. 2(d), there is no
motion at all,
.
Each domain in Fig. 2(e) is labeled according to the
parastichy
along which particles travel, with positive numbers
indicating trajectories that spiral outward.
The direction of motion also is indicated by the domains' shading.
The number of accessible domains increases in Fig. 2(e)
as the number of nodes in the spiral increases and
additional Fibonacci numbers become accessible.
Consequently, the selected parastichy tends to change with radius
for all
.
This behavior is evidient in Fig. 2(b), whose
inner region spirals outward, and whose outer region spirals inward.
The only exception arises for particular rotation angles
![]() |
(10) |
which correspond to motionless states, .
The deterministic Fibonacci spiral ratchet thus exhibits far richer transport properties than any one-dimensional ratchet. This provides the foundation for the still more varied properties that arise when Fibonacci spiral is used as the basis for a thermal ratchet.
§ II.4. Stochastic regime
When is large enough that traps in consecutive states do not
overlap, a Brownian particle released from a trap at
the end of one state finds itself in a force-free region at the
beginning of the next state.
The particle can only advance through the pattern by diffusing.
In one-dimensional ratchets,
such a diffusive contribution to the transport process creates
possibilities for temperature-dependent flux reversals
(30); (5); (18).
Observing that particles travel along parastichies in the
Fibonacci spiral ratchet's deterministic limit suggests that
similar reversals should emerge in its stochastic limit.
Unlike one-dimensional models, however, the stochastic limit
of the Fibonacci spiral ratchet also can permit transport along
directions not permitted in the deterministic limit.
To explore these possibilities, we model the trap at each vertex as a Gaussian potential well
![]() |
(11) |
where is the effective width of a trap, and
is its depth.
The time-evolution of the probability density
for finding
a particle at position
is described by the
Fokker-Planck equation (31)
![]() |
(12) |
where ,
is the
particles' diffusion coefficient, and
is their mobility.
Equation (12) can be solved numerically using
the finite-difference method (32) for any starting distribution
.
From this, the instantaneous local drift velocity induced by the
ratchet potential can be computed as
![]() |
(13) |
To characterize the ratchet's performance, we start with the traps occupied,
![]() |
(14) |
and compute the mean full-cycle velocity field
![]() |
(15) |
This velocity field can be integrated to obtain trajectories that may be compared with the deterministic mapping. The same numerical model thus can be used to explore behavior in both the deterministic and stochastic regimes.






The results in Fig. 3 were computed for
a particular choice of the rotation angle, ,
which corresponds to
in the deterministic limit.
Deterministic trajectories wind clockwise and inward under these conditions,
as indicated by the inset to Fig. 3(b).
Other parameters were selected to mimic the experimental condition
in Sec. III.
The diffusion coefficient,
is appropriate for 1.5
diameter spheres diffusing in water
at room temperature, and corresponds
to a mobility of
.
The potential energy wells at each vertex were given a
depth of
and a width of
.
The results in Fig. 3(a)
for the radial and azimuthal components of the induced drift
velocity share much in common with results obtained from
one-dimensional thermal ratchet models (18).
Traces in this figure show the drift velocity normalized
by the natural velocity scale, , for a few representative
values of the inter-trap separation,
.
If the trapping pattern spends a long enough time
in each state,
the diffusing particles fall preferentially into the nearest
traps, and advance along the same trajectory as would have been
selected deterministically.
If, on the other hand, the pattern advances too rapidly, particles
cannot reach the nearest trap in one time step. Rather, they
preferentially find the nearest trap after two time steps, which
carries them backward along the same path (18).
The drift velocity vanishes for large
because the pattern
itself advances slowly.
It vanishes for small
because the traps do not exert enough
force to drag particles so quickly through the viscous medium.
Figure 3(b) shows how the radial and azimuthal components
of the drift velocity depend on the scale of the pattern for a particular
choice of . Smaller-scale patterns favor transport along the
deterministically selected direction. Larger inter-trap separations favor
flux reversal by requiring particles to diffuse further between
trapping events.
For a given state duration , the flux also depends on
the scale of the pattern,
.
Smaller values of
favor transport in the direction of
the deterministic solution.
Large values allow for flux reversal.
This behavior can be seen in Fig. 3(b) at a
fixed duration,
.
Both radial and azimuthal flux reversals resemble the single
flux reversal observed in one-dimensional models (18).
Because the scale of the inter-trap separation along a parastichy
depends on radius, however, flux may reverse in only
part of the pattern for a given cycle time .
In a pattern-averaged sense, then, flux reversal in the
radial and azimuthal direction need not occur at the same
values of
and
.
Each flux reversal in the Fibonacci spiral thermal ratchet therefore
can consist of two cross-overs.
This is one respect in which this model differs
from one-dimensional thermal ratchets.
Although computed for particular rotation angles, the results
in Fig. 3 also highlight another general feature
of transport in the Fibonacci spiral ratchet.
Regardless of the size scale and cycle
period, particles tend to follow the parastichy selected in the
deterministic limit for the specific rotation angle, .
§ III. Experimental Demonstration














We demonstrated both deterministic and stochastic modes of operation
with experiments on colloidal spheres moving through
holographically projected optical force landscapes.
Our implementation is shown schematically in
Fig. 4.
The sample consists of diameter colloidal silica spheres
(Duke Scientific Catalog #8150, Lot #30158) dispersed in a
30
thick layer of water between a glass microscope slide and
a glass coverslip.
The edges of the sample volume were sealed with UV-cured
adhesive (Norland Optical Adhesive Type 81) for mechanical
stability and to slow evaporation.
The sample was allowed to equilibrate to room temperature
(
) on the stage of an inverted optical
microscope (Nikon TE-2000U).
Patterns of 200 optical tweezers arranged in a Fibonacci spiral according to
Eqs. (1) through (4) were projected
into the sample using the holographic optical trapping technique
(33); (34); (35); (36); (37).
Computer-generated holograms (35) encoding the pattern of traps
were imprinted onto the wavefronts of the trapping laser (Coherent Verdi 5W,
532
) using a liquid-crystal spatial light modulator
(SLM; Hamamatsu X8267-16).
Powering the hologram with 1.8
provides each trap
with an estimated
, after accounting for the
hologram's diffraction efficiency and other losses in the optical train.
Each trap, therefore, has an estimated width of
(38) and a depth of
(39).
The microscope's objective lens (Nikon Plan-Apo , numerical
aperture 1.4, oil immersion) was used both to focus the traps
into the sample, and also to image the spheres through
conventional bright-field microscopy.
Images were acquired with a low-noise monochrome video camera
(NEC TI-324A II) at 30 frames per second with a spatial resolution
of 0.135
pixel.
Individual particles were located in each snapshot to within 20
using standard methods of digital video microscopy (40).
Their locations were linked into time-resolved trajectories with a
maximum likelihood algorithm (40).
These trajectories, in turn, were used to estimate the
mean ratchet-induced flux using non-parametric density estimators
(41).
The image of the focused traps inset into Fig. 4 was
obtained by replacing the sample with a front-surface mirror.








The data in Fig. 5 show the radial and angular drift
velocities measured for an optical Fibonacci spiral ratchet operating in
the deterministic regime.
The scale for the trapping pattern, , was selected
to be small enough that traps overlap in consecutive states.
The pattern was rotated according to Eq. (6) through
an angle of
by projecting a sequence of holograms
with the SLM.
The data in Fig. 5 were obtained by varying the
duration
over which each hologram was projected.
Each value represents the average of 1000 particles' trajectories
measured over 80 cycles each.
Any possible influence of out-of-plane fluctuations
was minimized by projecting the traps at the spheres' equilibrium height
above the wall (42).
The solid curves in Fig. 5
show results from numerical solutions of the Fokker-Planck equation,
averaged over the entire pattern.
Excellent agreement between simulation and experiment is obtained
with no adjustable parameters.
As expected, no flux reversal occurs in this range of conditions.
The induced drift vanishes as in the long-time limit.
It also vanishes in the short-time limit because the traps do not exert
enough force to move the particles so rapidly through the water.
Over the entire range from
to
, the
particles follow the trajectories predicted by the deterministic map
in Fig. 3.
Increasing the scale of the trapping pattern to
moves the system into the stochastic regime.
Whereas the deterministic ratchet advected particles clockwise and
inward along the
parastichy,
the thermal ratchet admits flux reversal in both radial
and azimuthal coordinates, as revealed in Fig. 6.
The data for each
value in this plot were obtained from
particles' trajectories each consisting of 150 cycles.
Substantially greater statistics are required in this case because
the particles' drift is a comparatively small bias on their otherwise
random trajectories.
Even so, the ratchet-induced drift follows the deterministic map in the
long-time limit.
At shorter times, the drift velocity reverses direction,
and particles run backward along the deterministically selected parastichy,
The solid curves in Fig. 6(a) and (b) represent the
numerical solutions of Eq. (12) and (15)
for this set of conditions, again with no adjustable parameters.
As in the deterministic limit, the optical Fibonacci spiral ratchet
acts in quantitative agreement with theory when operated as a thermal ratchet.
Flux reversal occurs when the characteristic distance,
, that a particle diffuses during one step
of the three-step cycle is smaller than one third of the distance
between traps on the deterministically selected parastichy.
This distance, however, is obtained from Eq. (9)
and depends nontrivially on position within the spiral.
Unlike a one-dimensional thermal ratchet, therefore, flux reversal
can arise at different cycle times
at different radii within a Fibonacci spiral ratchet.
These experiments were carried out with 40 or fewer particles interacting with the trap array, or roughly 1 particle for every 5 traps. This occupation number appears to be small enough for quantitative agreement with the single-particle theory in Eqs. (11) through (15). Collisions arising in more highly occupied patterns could give rise to additional transport phenomena such as cooperative flux reversal of the type that is observed in ratchets for magnetic flux quanta (43); (44) and bacterial swarms (45). This is a matter for future study.
§ IV. Conclusions
We have demonstrated that the Fibonacci spiral can serve as the
basis for a two-dimensional thermal ratchet model, and have
implemented this model experimentally using colloidal spheres
and holographic optical traps.
Periodically rotating a spiral trapping pattern through a three-state
sequence causes diffusing particles to drift both radially and
azimuthally. The speed and direction of the ratchet-induced
motion have a very rich dependence on the scale, , of the pattern
and on the angle
of the rotation. Remarkably, this
seemingly complex dynamical system
follows comparatively simple rules, with particles flowing along
well-defined paths through the pattern that are selected principally
by
for a given radius within the pattern.
Varying
and the cycle time
affords control over the rate
and direction of motion along these paths.
The transition to flux-reversed transport need not occur uniformly within the Fibonacci spiral. Rather, some regions may follow the deterministic route while others flow in a retrograde direction. This creates the possibility that the system-averaged flux may undergo radial flux reversal separately from angular flux reversal.
This work was supported by the National Science Foundation under grant DMR-0855741.
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