Coupled Brownian Motors
In collaboration with:
Peter Reimann (Augsburg, Germany), Christian Van den Broeck (Limburg,
Belgium), Peter Hänggi (Augsburg, Germany)
Publication
P. Reimann, R. Kawai, C. Van den Broeck, and P. Hanggi
"Coupled Brownian motors: Anomalous hysteresis
and zerobias negative conductance"
Europhys. Lett. 45, (1999) 545
C. Van den Broeck, P. Reimann, R. Kawai, and P. Hänggi
"Coupled Brownian Motors"
in Lecture Notes in Physics (in press, 1999)
Can we get work out of unbiased fluctuation?
It is possible for macroscopic fluctuations. However, the situation
is more subtle for microscopic fluctuations. We know from the second
law of thermodynamics that is is impossible to get work in a repetitive
cyclic manner from a single heat bath at equilibrium. But nothing
forbids the appearance of Maxwell demon in the case of nonequilibrium
fluctuations. Then, it is possible to construct Brownian motors which
are driven by microscopic nonequilibrium fluctuations. Such motors
may be found in biological systems operating far from equilibrium.
Recently, various mathematical models of Brownian motors have been
investigated. Some of them show remarkable properties such as
negative mobility. In the following we introduce our own model.
See References for other models.
Consider a globally coupled system of N particles with
statedependent noise (multiplicative noise):

dx_{i}
dt

= f(x_{i}) + g(x_{i})x_{i} + 
1
N


N å
j = 1

k(x_{i},x_{j}) 

where f(x), g(x_{i}), and k(x,x¢) are deterministic force, fluctuating force, and
coupling between particles, respectively.
We assume that the fluctuation is represented by independent Gaussian white noises x:
< x_{i} > = 0,
<
x_{i}(t)
x_{j}(t
¢) > =
d_{ij}
d(tt
¢) 

To illustrate the characteristic feature of our Brownian motor, we use
the following specific model:
f(x) =  sinx  2A sin2x

g(x) = Ö
(2T+2Q (sinx + 2A sin2x)^{2} )

k(x,y) = K
sin(xy) 
where T and Q represent strength of equilibrium
and nonequilibrium fluctuations, respectively.
In order to produce flux, other models necessarily use a
asymmetric potential (ratchet potential) or potential fluctuation
with a finite correlation in time. Our model has neither spacial
nor temporal bias but nevertheless it produces flux. How can it
happen? A key point is that the system undergoes noiseinduced
nonequilibrium phase transition and reflection symmetry of the
system is spontaneously broken. (See Figure 1 and movie)
As a result the particles experiences
effectively a ratchetlike potential. Figure 2 illustrates a
phase diagram.
The response properties of this model is very interesting. If
uniform force applied to the system in the symmetrybroken phase,
anomalous hysteresis is observed (See Figure 3).
Even more interestingly, if the
force is applied to the system just outside the symmetrybroken
region, the current flows in the opposite direction to the force.
(negative zerobias mobility, See Figure 4).
When many Brownian particles driven by multiplicative noise are coupled,
many interesting phenomenon are induced through
nonequilibrium phase transition and breaking of ergodicity. The
coupled Brownian motor is one example. More intersting
phenomenon are to be discovered.


Figure 1. Snapshot of a steady state.
Quicktime movie(1Mb) of a steady state.
Figure 2.
Phase Diagram (T=2 and A=0.15)
Figure
3. Anomalous Hysteresis Loops of the steady state current vs. force
F. (T=2, A=0.15, Q=4, and K=10)
Figure 4. Negative zerobias mobility for global(circle)
and 2D nearest neighbor (plus) couplings. (Q=2
and K=8 for global coupling ans Q=6 and K=15 for 2D
nearest neighbor coupling)

