Brownian ratchet
Encyclopedia
Philosophy of thermal and statistical physics
The philosophy of thermal and statistical physics is that part of the philosophy of physics whose subject matter is classical thermodynamics, statistical mechanics, and related theories...
, the Brownian ratchet, or Feynman-Smoluchowski ratchet is a thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
about an apparent perpetual motion
Perpetual motion
Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....
machine first analysed in 1912 by Polish physicist Marian Smoluchowski
Marian Smoluchowski
Marian Smoluchowski was an ethnic Polish scientist in the Austro-Hungarian Empire. He was a pioneer of statistical physics and an avid mountaineer.-Life:...
and popularised by American Nobel laureate physicist Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
in a physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
lecture at the California Institute of Technology
California Institute of Technology
The California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
on May 11, 1962, and his text The Feynman Lectures on Physics
The Feynman Lectures on Physics
The Feynman Lectures on Physics is a 1964 physics textbook by Richard P. Feynman, Robert B. Leighton and Matthew Sands, based upon the lectures given by Feynman to undergraduate students at the California Institute of Technology in 1961–63. It includes lectures on mathematics, electromagnetism,...
as an illustration of the laws of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
. The simple machine, consisting of a tiny paddle wheel
Paddle wheel
A paddle wheel is a waterwheel in which a number of scoops are set around the periphery of the wheel. It has several usages.* Very low lift water pumping, such as flooding paddy fields at no more than about height above the water source....
and a ratchet
Ratchet (device)
A ratchet is a device that allows continuous linear or rotary motion in only one direction while preventing motion in the opposite direction. Because most socket wrenches today use ratcheting handles, the term "ratchet" alone is often used to refer to a ratcheting wrench, and the terms "ratchet"...
, appears to be an example of a Maxwell's demon
Maxwell's demon
In the philosophy of thermal and statistical physics, Maxwell's demon is a thought experiment created by the Scottish physicist James Clerk Maxwell to "show that the Second Law of Thermodynamics has only a statistical certainty." It demonstrates Maxwell's point by hypothetically describing how to...
, able to extract useful work from random fluctuations
Thermal fluctuations
In statistical mechanics, thermal fluctuations are random deviations of a system from its equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they disappear altogether as temperature approaches absolute zero.Thermal fluctuations are a...
(heat) in a system at thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
in violation of the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
. Detailed analysis by Feynman and others showed why it cannot actually do this.
The machine
The device consists of a gear known as a ratchetRatchet (device)
A ratchet is a device that allows continuous linear or rotary motion in only one direction while preventing motion in the opposite direction. Because most socket wrenches today use ratcheting handles, the term "ratchet" alone is often used to refer to a ratcheting wrench, and the terms "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
Paddle wheel
A paddle wheel is a waterwheel in which a number of scoops are set around the periphery of the wheel. It has several usages.* Very low lift water pumping, such as flooding paddy fields at no more than about height above the water source....
that is immersed in a fluid of molecules at temperature
. The molecules constitute a heat bath in that they undergo random Brownian motionBrownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
with a mean kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
that is determined by the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. 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 should be for the ratchet to rotate continuously in that direction. The ratchet's motion then can be used to do work on other systems, for example lifting a weight 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
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
, 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. A simple way to visualize how the machine might fail is to remember that the pawl itself will undergo Brownian motion. The pawl therefore will intermittently fail by allowing the ratchet to slip backward or not allowing it to slip forward. Feynman demonstrated that if the temperature
of the ratchet and pawl is the same as the temperature 
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,
is smaller than 
, the ratchet can 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 engineHeat engine
In thermodynamics, a heat engine is a system that performs the conversion of heat or thermal energy to mechanical work. It does this by bringing a working substance from a high temperature state to a lower temperature state. A heat "source" generates thermal energy that brings the working substance...
, in compliance with the second law of thermodynamics. Conversely, if
is greater than 
, the device will rotate in the opposite direction, again functioning as a heat engine.The Feynman ratchet model led to the similar concept of Brownian motor
Brownian motor
Brownian motors are nano-scale or molecular devices by which thermally activated processes are controlled and used to generate directed motion in space and to do mechanical or electrical work...
s, nanomachines which can extract useful work not from thermal noise but from chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
s and other microscopic nonequilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
sources, in compliance with the laws of thermodynamics. Diode
Diode
In electronics, a diode is a type of two-terminal electronic component with a nonlinear current–voltage characteristic. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material connected to two electrical terminals...
s are an electrical analog of the ratchet and pawl, and for the same reason cannot produce useful work by rectifying thermal fluctuations
Thermal fluctuations
In statistical mechanics, thermal fluctuations are random deviations of a system from its equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they disappear altogether as temperature approaches absolute zero.Thermal fluctuations are a...
in a circuit at uniform temperature.
History
The ratchet and pawl was first discussed as a Second Law violating device by Gabriel LippmannGabriel Lippmann
Jonas Ferdinand Gabriel Lippmann was a Franco-Luxembourgish physicist and inventor, and Nobel laureate in physics for his method of reproducing colours photographically based on the phenomenon of interference....
in 1900. In 1912 Polish physicist Marian Smoluchowski
Marian Smoluchowski
Marian Smoluchowski was an ethnic Polish scientist in the Austro-Hungarian Empire. He was a pioneer of statistical physics and an avid mountaineer.-Life:...
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 if T1, the temperature of the paddle, was greater than T2, the temperature of the ratchet, it would function as a heat engine
Heat engine
In thermodynamics, a heat engine is a system that performs the conversion of heat or thermal energy to mechanical work. It does this by bringing a working substance from a high temperature state to a lower temperature state. A heat "source" generates thermal energy that brings the working substance...
, 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
Quasistatic process
In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. However, it is very important of note that no real process is quasistatic...
approximation, resulting in incorrect equations for efficiency. Magnasco
Marcelo Osvaldo Magnasco
Marcelo Osvaldo Magnasco is a biophysicist, currently a professor and head of laboratory at the Rockefeller University.He is known for his work on thermal ratchets as models of biological motors, auditory biophysics, neural coding , and other studies of biological networks such as leaf...
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
Derek Abbott
Derek Abbott is a physicist and electronic engineer. He is a Professor of Electrical and Electronic Engineering at the University of Adelaide, Australia...
, Bruce R. Davis
Bruce R. Davis
Bruce Raymond Davis was born in Adelaide, 1939, and is an electronic engineer notable for his research in mobile communication systems, satellite communications, and high frequency data communication systems.-Education:...
and Juan Parrondo, reanalyzed the problem and extended it to the case of multiple ratchets, showing a link with Parrondo's paradox
Parrondo's paradox
Parrondo's paradox, a paradox in game theory, has been described as: A losing strategy that wins. It is named after its creator, Spanish physicist Juan Parrondo, who discovered the paradox in 1996...
.
Granular gas
Whilst being theoretically proven to fail for the thermal equilibrium case, 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 motionBrownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
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 axis to only move in one direction, random collisions with the moving beads resulted in a rotating vane. 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
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, 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
- Brownian motionBrownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
- Quantum stirring, ratchets, and pumpingQuantum stirring, ratchets, and pumpingA pump is an alternating current-driven device that generates a direct current . In the simplest configuration a pump has two leads connected to two reservoirs. In such open geometry the pump takes particles from one reservoir and emits them into the other...
- Geometric phaseGeometric phaseIn classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase , Pancharatnam phase or most commonly Berry phase, is a phase acquired over...
(section Stochastic Pump Effect) - Parrondo's paradoxParrondo's paradoxParrondo's paradox, a paradox in game theory, has been described as: A losing strategy that wins. It is named after its creator, Spanish physicist Juan Parrondo, who discovered the paradox in 1996...
- Beverly ClockBeverly ClockThe Beverly Clock is a clock situated in the foyer of the Department of Physics at the University of Otago, Dunedin, New Zealand. The clock is still running despite never having been manually wound since its construction in 1864 by Arthur Beverly....
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)
Articles
- Lukasz Machura: Performance of Brownian Motors. University of Augsburg, 2006 (PDF)