# Brownian Ratchet Plausibility

Alright I'm going to throw whatever reputation I have on the line here. And yes this is a serious question.

Apologies for the shoddy imagery. I had a couple ideas to get the Brownian Ratchet to work. I know that the Second Law of Thermodynamics says such things are impossible. But as far as I understand it the second law is there not because of some hard mathematical argument but as an underlying axiom. Mostly because its obviously true in almost every theoretical and practical experiment you can imagine. Then we come up with oddballs like the Brownian Ratchet and it gets murky.

So I had two ideas for getting our ratchet to work. Now I haven't read up on the argument Feynman gives directly. From what I gather from around Stack-exchange and Wikipedia there are two immediate issues.

The first is that that pawl has a lot of energy and jiggles with its heat. Occasionally it slips and any of these random time events can be modeled as uniform pressure in the long run instead of the Brownian motion they are, since it all evens out anyway.

This seems like a logical argument until you consider the effect of torque. If the heat energy is everywhere equal then if you make the paddle large enough, but still light enough to turn, then the force exerted on the outer edge should exceed the force from heat on the pawl.

This is the same principle we use everywhere in gears and levers. It takes less force to move but a larger distance and a larger time. Since the required force to accomplish the turn in one direction is not the same as to go in reverse you should have net energy over large spans of time.

Besides the large paddle, small pawl idea I also considered getting rid of the problematic pawl for a design that was unidirectional. Asymmetrical gears I've heard were suggested at one point. I personally have considered a complicated “caterpillar ratchet” that grips on and moves around the ratchet inchworm style and a simpler “spider” ratchet. The spider one has a track that a set of teeth mesh with. The teeth are like leaf springs that can handle a compressive load. They push up against the outer wall and are prevented from rolling backward both by this compression as well as a staggered set of teeth that are offset.

Now the reason for the staggered set of teeth is not immediately obvious. If you have a single set of symmetric teeth then much like a single pawl there is a moment when the position allows one to slide backwards. If improbably it tried to move backwards at the speed of sound it could make more than a full revolution in reverse with a symmetric or pawl setup. With an offset set of teeth there will always be a tooth in contact with the lock. You will get at most a half-tooth reversal.

Now our random jitters should supply an equally lucky reversal for each forward jolt we experience. Except the probabilities are different because the teeth cater to their own statistics. The forward motion is a long-drawn-out process by virtue of the large paddle and torque trade-off. You take the same energy (thermodynamic equilibrium) over a longer period of time than the short jolts required to move backwards. Since no matter how powerful the jolt it can only slip a half-tooth back there will be a paddle size where the frequency trade off that favors the larger slower paddle.

Also, I could be wrong but thermodynamic equilibrium means "even temperature" and does not imply the same pressure across the system. Couldn't you manipulate a pressure gradient to your advantage?

• Alright well I hasn't even been a minute and I'm already regretting posting this. And just in case, the perpetual-motion tag is a joke. Although if you had "free" energy from heat I suppose you could give Carnot's absurdity argument a stab and start making heat flow slowly up hill. Jan 25, 2015 at 10:16
• So go ahead and delete it. No harm no fowl [sic] Jan 25, 2015 at 13:13
• @CarlWitthoft Thanks for the vote of confidence :P. After biting the bullet, I really want to know why the new and improved ratchet doesn't work though, as it's almost certain not to, even with that darned good torque argument. I'm really just doing warm up whimpers before physics starts making me cry. ^_^; Jan 25, 2015 at 13:33
• Are we allowed to edit this so that there isn't this huge block of text? Sep 21, 2018 at 14:22
• Can I suggst reading Feynamn's original lecture. It is freely avalible online and famously well written Sep 23, 2019 at 13:15

It is not going to work in the way you describe. At equilibrium the fluctuations have enough energy to move the ratchet in one direction than in the other. Forwards transitions are as likely as backwards transitions.

Assume the ratchet moves, at equilibrium, in a net counterclockwise direction. The system is in equilibrium, the microscopic laws are reversible, so a movie of the ratchet played backwards should give a plausible physical process too. In this last case you will have a system in equilibrium that moves clockwise. Thus, at equilibrium, both clockwise and counterclockwise directions are equally probable, which contradicts the assumption that there is only one privileged counterclockwise direction of motion.

• do you mean the microscopic laws a reversible in the second paragraph? Sep 23, 2019 at 13:13
• yes, exactly that
– user65081
Sep 23, 2019 at 13:17
• Not sure it entirely addresses the question but a) apparently I never explicitly stated it anywhere. b) this is a very good way of intuiting this situation. c) I found what I missed so this should probably "close"... Sep 23, 2019 at 17:23
• I am sorry you feel there are issues I had not addressed. Feel free to ask me specific things and I will try to answer them.
– user65081
Sep 24, 2019 at 2:10

I'm not convinced there is anything wrong with this, or that it violates the second law at all.

You have a paddle, and random brownian-motion forces (in, say, water) on the paddle result in a ball rotating at random rates along its axis clockwise and not counterclockwise. This motion affects the molecules of water that hit it in a subtle way, I'm guessing that they will wind up with more rotational energy and less kinetic energy than they would have if the ball did not rotate.

No particular work is being done. If you put a hundred million of these into a milliliter of water, maybe at equilibrium the molecules of water might have a bigger rotational energy and a smaller kinetic energy.

So what? It is not necessary that those energies come out the same under all circumstances. Molecules of water have a different mix of energies if you add a little bit of gelatin and let it turn into Jello too.

It's a novelty to have something which rotates in only one direction, but that doesn't violate the second law.

Imagine this -- suppose you had tiny paddles that were frozen in place so their shafts couldn't rotate at all relative to the other end. How is it less entropy to have paddles that rotate in one direction but not the other, compared to paddles that don't rotate?

If you could stick those axles through a membrane, so your gears stuck out the other side, and if you had a way to collect energy from their random turning in only one direction, then you would have something interesting!

Bacteria with flagella already have part of that. The flagella shaft does poke through the cell wall, and it has some sort of friction-reducing bushing. It has a gear on the inside, which is powered by an engine which pumps hydrogen ions from a place where they have high concentration to a place where the concentration is low, and this turns the gear. All you need is to change the flagellum to a paddle, and reverse the engine so it pumps from low concentration to high concentration, and you can test whether it collects energy.

Edit --

Try these two systems. One is just like yours, but the case that the ratchet works against has a definite mass, say rather more than the paddle.

For the second, the case is not particularly massive, but it has its own fixed paddle attached to it.

In the second case, the device doesn't pick up much momentum, its own motion keeps getting trumped by that of the molecules that hit it. Small forces -- not enough to turn the ratchet -- balance out. Large forces either turn the ratchet one direction, or turn the whole device the other direction. The forces which do that are the ones which turn one paddle much more than the other. The net result is -- not much different from a device with both paddles fused, or one that could rotate both ways.

In the first case, the heavy case gets hit by molecules too. The faster it turns on average, the more angular momentum molecules will get when they hit it headon. So they will tend to keep it from turning too fast, they will pick up its momentum and carry it off.

I think.

• So what? As far as I understand it... The very definition of the 2nd law prevents you giving order to chaos for free, it's the problem with Maxwell's Demon. This device taking a random chaotic motion and converting it to rotation in a single direction would violate it by itself. Adding something like a winch would, of course, make it a more interesting thought experiment as you would now have free energy and is the usual logical jump for the Brownian ratchet. Sep 21, 2018 at 23:09
• interesting bit about the flagella though. I didn't know that's how they worked, always thought they worked like the cilium. Sep 21, 2018 at 23:09
• It's been years since I've looked at this and there might be other kinds of bacterial flagella known now. But with the kind I know, the cell will have two or more flagella, and when they turn they kind of wrap up together or something and push the cell in a consistent direction. When the gear shifts into reverse (each flagella has its independent gears, of course) and they rotate backward, they come apart and start rotating the cell into a random direction. So when it shifts into forward again, its direction is different. Sep 22, 2018 at 9:13
• "The very definition of the 2nd law prevents you giving order to chaos for free, it's the problem with Maxwell's Demon." You don't have order from chaos. You only have a different distribution of chaotic stuff. Entropy is not at all reduced. If you change the ratio of translational velocity to rotational velocity, but the entropy stays the same, you haven't violated the second law. Sep 22, 2018 at 9:20
• That makes sense to me on some level, but bends my mind a little bit. Reducing degrees of freedom should require that the remaining df have increased entropy to compensate. If the rotation that remains has a forward movement of 1 vs a backwards movement of .5 doesn't that mean it has reduced entropy? We've ordered a 1 vs -1 on some other df to 1 vs -.5 on our new df. Assuming my thoughts on the staggered teeth were correct of course. Sep 22, 2018 at 22:43

Given my understanding of the processes involved my question can be boiled down to a simple statistical question, here we'll take the stat redistribution in energy instead of time: Given perfect thermodynamic equilibrium we'll have an energy N on both the pawl and vane side. Let's discretize the units and by extending the vane let's call this a torque of energy 2 on the vane side, this discretized energy can act on one or both directions at once which will result in the following:

• 25% chance clockwise
• 25% chance counterclockwise.
• 50% no movement

Now let's consider the pawl side: we can have a random set of events on the pawls' side with energy 2. We'll have a pawl and an offset pawl. If the energy required to lift a pawl is 1 then by discretely distributing our 2 energy we have the following scenarios (taking into account pressing down as an option):

• 16.6% chance both pawls lift, rotation can happen either direction.
• 33.3% chance only one pawl lifts, rotation can happen one direction (let's say Counterclockwise).
• 50% one or both pawls press down with enough force to bind movement

If both sets of events are happening at random then the movement odds are as follows:

• Counterclockwise: 12.5%
• Clockwise: 4.16%
• No Movement: 83.33%

Given these simple statistics what is the factor I'm missing that redistributes the 4.16% Counterclockwise to Clockwise? It would seem that not considering the movement caused by the "binding of movement" (this would be half steps backwards in some cases) is probably the source of error.

• Just a lite version of my thought process for anybody passing by who thinks this would work but doesn't see the part I missed (or didn't consider that pressing down is an option for the pawls to use the random energy, that's actually important...). Sep 23, 2019 at 17:33