In the philosophy of thermal and statistical
physics, the Brownian ratchet or Feynman-Smoluchowski
ratchet is a thought experiment about an apparent
perpetual motion machine first analysed in
1912 by Polish physicist Marian Smoluchowski
and popularised by American Nobel laureate
physicist Richard Feynman in a physics lecture
at the California Institute of Technology
on May 11, 1962, during his Messenger Lectures
series The Character of Physical Law in Cornell
University in 1964 and in his text The Feynman
Lectures on Physics as an illustration of
the laws of thermodynamics.
The simple machine, consisting of a tiny paddle
wheel and a ratchet, appears to be an example
of a Maxwell's demon, able to extract useful
work from random fluctuations (heat) in a
system at thermal equilibrium in violation
of the second law of thermodynamics.
Detailed analysis by Feynman and others showed
why it cannot actually do this.
== The machine ==
The device consists of a gear known as a ratchet
that rotates freely in one direction but is
prevented from rotating in the opposite direction
by a pawl.
The ratchet is connected by an axle to a paddle
wheel that is immersed in a fluid of molecules
at temperature
T
1
{\displaystyle T_{1}}
. The molecules constitute a heat bath in
that they undergo random Brownian motion with
a mean kinetic energy that is determined by
the temperature.
The device is imagined as being small enough
that the impulse from a single molecular collision
can turn the paddle.
Although such collisions would tend to turn
the rod in either direction with equal probability,
the pawl allows the ratchet to rotate in one
direction only.
The net effect of many such random collisions
would seem to be that the ratchet rotates
continuously in that direction.
The ratchet's motion then can be used to do
work on other systems, for example lifting
a weight (m) against gravity.
The energy necessary to do this work apparently
would come from the heat bath, without any
heat gradient.
Were such a machine to work successfully,
its operation would violate the second law
of thermodynamics, one form of which states:
"It is impossible for any device that operates
on a cycle to receive heat from a single reservoir
and produce a net amount of work."
== 
Why it fails ==
Although at first sight the Brownian ratchet
seems to extract useful work from Brownian
motion, Feynman demonstrated that if the entire
device is at the same temperature, the ratchet
will not rotate continuously in one direction
but will move randomly back and forth, and
therefore will not produce any useful work.
The reason is that since the pawl is at the
same temperature as the paddle, it will also
undergo Brownian motion, "bouncing" up and
down.
It therefore will intermittently fail by allowing
a ratchet tooth to slip backward under the
pawl while it is up.
Another issue is that when the pawl rests
on the sloping face of the tooth, the spring
which returns the pawl exerts a sideways force
on the tooth which tends to rotate the ratchet
in a backwards direction.
Feynman demonstrated that if the temperature
T
2
{\displaystyle T_{2}}
of the ratchet and pawl is the same as the
temperature
T
1
{\displaystyle T_{1}}
of the paddle, then the failure rate must
equal the rate at which the ratchet ratchets
forward, so that no net motion results over
long enough periods or in an ensemble averaged
sense.
A simple but rigorous proof that no net motion
occurs no matter what shape the teeth are
was given by Magnasco.If, on the other hand,
T
2
{\displaystyle T_{2}}
is smaller than
T
1
{\displaystyle T_{1}}
, the ratchet will indeed move forward, and
produce useful work.
In this case, though, the energy is extracted
from the temperature gradient between the
two thermal reservoirs, and some waste heat
is exhausted into the lower temperature reservoir
by the pawl.
In other words, the device functions as a
miniature heat engine, in compliance with
the second law of thermodynamics.
Conversely, if
T
2
{\displaystyle T_{2}}
is greater than
T
1
{\displaystyle T_{1}}
, the device will rotate in the opposite direction.
The Feynman ratchet model led to the similar
concept of Brownian motors, nanomachines which
can extract useful work not from thermal noise
but from chemical potentials and other microscopic
nonequilibrium sources, in compliance with
the laws of thermodynamics.
Diodes are an electrical analog of the ratchet
and pawl, and for the same reason cannot produce
useful work by rectifying Johnson noise in
a circuit at uniform temperature.
Millonas
as well as Mahato
extended the same notion to correlation ratchets
driven by mean-zero (unbiased) nonequilibrium
noise with a
nonvanishing correlation function of odd order
greater than one.
== History ==
The ratchet and pawl was first discussed as
a Second Law-violating device by Gabriel Lippmann
in 1900.
In 1912, Polish physicist Marian Smoluchowski
gave the first correct qualitative explanation
of why the device fails; thermal motion of
the pawl allows the ratchet's teeth to slip
backwards.
Feynman did the first quantitative analysis
of the device in 1962 using the Maxwell–Boltzmann
distribution, showing that if the temperature
of the paddle T1 was greater than the temperature
of the ratchet T2, it would function as a
heat engine, but if T1 = T2 there would be
no net motion of the paddle.
In 1996, Juan Parrondo and Pep Español used
a variation of the above device in which no
ratchet is present, only two paddles, to show
that the axle connecting the paddles and ratchet
conducts heat between reservoirs; they argued
that although Feynman's conclusion was correct,
his analysis was flawed because of his erroneous
use of the quasistatic approximation, resulting
in incorrect equations for efficiency.
Magnasco and Stolovitzky (1998) extended this
analysis to consider the full ratchet device,
and showed that the power output of the device
is far smaller than the Carnot efficiency
claimed by Feynman.
A paper in 2000 by Derek Abbott, Bruce R.
Davis and Juan Parrondo, reanalyzed the problem
and extended it to the case of multiple ratchets,
showing a link with Parrondo's paradox.
Léon Brillouin in 1950 discussed an electrical
circuit analogue that uses a rectifier (such
as a diode) instead of a ratchet.
The idea was the diode would rectify the Johnson
noise thermal current fluctuations produced
by the resistor, generating a direct current
which could be used to perform work.
In the detailed analysis it was shown that
the thermal fluctuations within the diode
generate an electromotive force that cancels
the voltage from rectified current fluctuations.
Therefore, just as with the ratchet, the circuit
will produce no useful energy if all the components
are at thermal equilibrium (at the same temperature);
a DC current will be produced only when the
diode is at a lower temperature than the resistor.
== Granular gas ==
Researchers from the University of Twente,
the University of Patras in Greece, and the
Foundation for Fundamental Research on Matter
have constructed a Feynman-Smoluchowski engine
which, when not in thermal equilibrium, converts
pseudo-Brownian motion into work by means
of a granular gas, which is a conglomeration
of solid particles vibrated with such vigour
that the system assumes a gas-like state.
The constructed engine consisted of four vanes
which were allowed to rotate freely in a vibrofluidized
granular gas.
Because the ratchet's gear and pawl mechanism,
as described above, permitted the axle to
rotate only in one direction, random collisions
with the moving beads caused the vane to rotate.
This seems to contradict Feynman's hypothesis.
However, this system is not in perfect thermal
equilibrium: energy is constantly being supplied
to maintain the fluid motion of the beads.
Vigorous vibrations on top of a shaking device
mimic the nature of a molecular gas.
Unlike an ideal gas, though, in which tiny
particles move constantly, stopping the shaking
would simply cause the beads to drop.
In the experiment, this necessary out-of-equilibrium
environment was thus maintained.
Work was not immediately being done, though;
the ratchet effect only commenced beyond a
critical shaking strength.
For very strong shaking, the vanes of the
paddle wheel interacted with the gas, forming
a convection roll, sustaining their rotation.
The experiment was filmed.
== See also ==
Quantum stirring, ratchets, and pumping
Geometric phase (section Stochastic Pump Effect)
Hawking radiation
== 
Notes ==
== 
External links ==
Why is a Brownian motor not a perpetuum mobile
of the second kind?
Coupled Brownian Motors - Can we get work
out of unbiased fluctuation?
Experiment finally proves 100-year-old thought
experiment is possible (w/ Video)
Richard Feynman: Messenger Series lectures
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