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From this review on Brownian motors, there is such a statement without detailed explanation:

(I think this statement is general enough so that one does not need to read the article)

"Apart from transients, directed transport in a spatially periodic system in contact with a single dissipation- and noise-generating thermal heat bath is ruled out by the second law of thermodynamics."

I don't have any idea how to relate the second law thermodynamics to the absence of directed transport.

The first obstacle, I think, is which description of the 2nd law should we use as the starting point?

And then, how to relate that particular description to the absence of directed transport? How does the nature of the heat bath relate to the problem?

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  • $\begingroup$ Note that a non-paywalled version of the report in question can be found here. $\endgroup$ – Nathaniel Jun 16 '13 at 6:27
  • $\begingroup$ Also note that the statement you quote is from the very beginning of the introduction - I would expect that a detailed explanation of it will be found later in the paper. $\endgroup$ – Nathaniel Jun 16 '13 at 6:30
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The basic idea is illustrated on page 64 of the paper you cited. In this case, "directed transport" refers to a bias in the direction the wheel can turn, which could then be used to lift the weight and do work. A full explanation of this example can be found in section 2.1, beginning on page 64.

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This can be generalised to other situations. Let's imagine a thought experiment where you start with a system in equilibrium, and then you add a small device that transports something from one part of the system to another, without interacting with any other system. We must imagine this device not to contain its own power supply, so it ends up in statistically the same set of states as it starts in.

Initially the system was in equilibrium, and hence its entropy was at its maximum possible value. Now that the directed transport has taken place, the system is out of equilibrium, and hence its entropy must be lower. This implies that it would be possible to extract work from the system while returning the system to its equilibrium state. This cycle could then be repeated, creating a perpetual motion machine.

But the second law says that an isolated system's entropy cannot increase (on average), so this situation must be impossible. As a result, we have to conclude that directed transport is impossible without using some other system as a source of energy (or more precisely, negative entropy) to drive the transport process.

Another example should help to clarify. Let's imagine I have two containers containing salt dissolved in water. If they are in equilibrium then the salt concentration in the two containers will be the same, and if we connect them by a tube there will be no net transport of salt from one container to the other. Individual ions will move between the containers, but over long time scales the flow of ions from left to right will exactly balance the flow of ions of the same species from right to left. On short time scales there can be fluctuations, but in the long run they must always cancel out.

But now let's imagine we have a special kind of asymmetrical membrane that will let sodium and chloride ions flow in one direction more readily than in the other. It might seem at first that such a thing could exist, but it turns out it couldn't, and the reason is as follows. Let's use our imaginary membrane to connect our two containers of salt water. Our asymmetrical membrane will let salt flow from left to right, but it will tend to block the reverse flow from right to left. As a result the salt concentration will increase in the right-hand container while decreasing in the left-hand one. But now we have a difference in osmotic pressure, which can be used to extract work. (There are working power stations that use this principle to extract power from the osmotic pressure difference between river water and sea water.) In our case this would give rise to a perpetual motion engine of the second kind, and it must therefore be impossible.

This argument prevents any kind of physical system that would allow you to construct such a device. This means it has to be impossible for any kind of directed transport to take place in any system in thermal equilibrium, and this is fundamentally what the statement you quoted is referring to.

A tiny caveat regarding the membrane example: there is a type of asymmetric membrane that can exist, but its asymmetric properties only manifest themselves when the system is already out of equilibrium. It's a little like a diode or a one-way valve in a pipe: it can block flow in one direction, but it can't cause stuff to flow when it's already in equilibrium.

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  • $\begingroup$ I understand the ratchet and pawl quite well, but I can't follow the general case. What do you mean by "adding a process"? How do you define a "process" in this scenario? How do you "add" it? $\endgroup$ – wdg Jun 17 '13 at 7:00
  • $\begingroup$ Sorry, I agree that talking about "adding a process" was confusing. I've changed it so that it talks about putting a small "device" in the system instead, and I've added another example that should help make it clearer. $\endgroup$ – Nathaniel Jun 17 '13 at 8:18
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I think it simply means that the system will eventually reach a thermodynamic equilibrium which is by definition free of any kind of flux.

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  • $\begingroup$ I wouldn't say that thermodynamic equilibrium is free of fluxes "by definition." Rather, the absence of fluxes is a somewhat non-trivial consequence of the second law. $\endgroup$ – Nathaniel Jun 16 '13 at 8:07
  • $\begingroup$ It depends on the adopted point of view. One can define equilibrium this way and invoke the second principle as a reason why any system would tend toward such a state. $\endgroup$ – gatsu Jun 16 '13 at 20:50

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