"Differential sail" seems to be a term for a surface that when put in some uniform (meaning homogeneous and isotropic) bath of particles (including light, which you can think of as being composed of particles) will indefinitely produce a force in one direction without using additional energy. It seems this is generally discussed as a possible form of propulsion for a spacecraft. This idea might seem clever (if a bit weird) at first, but actually such a device is impossible, because it violates the laws of thermodynamics. Below I'll explain how we know it violates the laws of thermodynamics, and then I'll look at a specific example to show why the device would not work.
Young lady, in this house we obey the laws of thermodynamics! — Homer Simpson
We can say the "differential sail" is impossible, because it violates the second law of thermodynamics. The 2nd law may be stated in many different ways, but let me use the following:
It is impossible for any cyclic process to occur whose sole effect is the extraction of heat from a reservoir and the performance of an equivalent amount of work. The differential sail is supposed to be able to sit in a uniform reservoir of particles, such as a gas of particles (or a light field) at a fixed temperature, and provide a force. Since it produces a force as it moves, it can do work, so you can extract heat from the reservoir and turn it into work, directly in contradiction of the 2nd law.
Another way of looking at it is that the 2nd law tells you that it's not possible to build a perpetual motion machine, and this is exactly what the differential sail would let you do. For example, imagine you have a rigid rod and you attach a differential sail to each end, so that the two sails produce a forces perpendicular to the rod and in opposite directions, causing the rod to spin.
Sail 1 | --> Force
<-- | Sail 2
If you had two of these setup so that they spin in opposite directions, you could then connect an electrical generator between them and have a source of unlimited energy that goes on forever, a perpetual motion machine.
Since the existence of a differential sail would break the 2nd law of thermodynamics, we can safely say that it's impossible. If there is a design for such a device that seems to work as a differential sail and operates only on the known laws of Physics, then we can be confident that the explanation of how this would work contains a mistake.
I've said that any design for a differential sail based on the known laws of physics must contain a mistake (because it violates one of those laws), and that argument is enough logically, but I think it will be more satisfying to demonstrate with a particular example.
In 1963 Penzias and Wilson discovered that space is filled with microwave radiation, now known as the cosmic microwave background. This radiation is very close to being the same in every direction (i.e. isotropic) and is essentially thermal blackbody radiation with a temperature of 2.7 K.
Let's do a thought experiment where we have a large surface, which we'll call the "sail", with one side, we'll call the right side, that is perfectly absorbing and one side, we'll call the left side, that is perfectly reflecting. Recall that electromagnetic waves like microwaves (and light) carry momentum, so they can exert pressure on objects. The same light will exert twice as much pressure on a reflecting surface as on an absorbing one.* This means there is twice as much pressure on the reflecting side as the absorbing side, and the sail feels a net force pointing to the right. This certainly sounds like a functional version of the differential sail as described above.
Remember, in the form of the 2nd law I quoted, it specified the law would be violated if the sole effect of was turning heat from the reservoir into work, but there is another effect I haven't yet considered here: the sail heats up! As the right side absorbs microwaves it gets hotter. As it heats up, the right side of the sail starts to emit blackbody radiation of its own†, and since that radiation carries momentum, the light that the sail emits off to the right produces a force that points to the left (like a rocket). The hotter the sail gets, the larger this force becomes, until eventually the sail has reached thermodynamic equilibrium and is at the same temperature as the cosmic microwave background. At that point, the amount of light leaving the absorbing side of the sail is exactly equal to the amount of light hitting it, and the momentum exchange for the absorbing side is no different than for the reflecting side. As a result, the net force on the sail is zero; it no longer provides any thrust.
The scenario I just described is a little contrived. Besides the perfect absorption and reflection, there's also the point that I implicitly assumed the sail was initially at zero temperature by assuming it initially was not emitting any blackbody radiation. In fact, unless you took measures to cool it down, the sail would be at or above the temperature of the cosmic microwave background radiation. As I've just said, if the sail is already at the same temperature as the microwave background, then it produces no thrust at all. If the sail is at a different temperature than the background radiation it will get some thrust for a little while until it reaches equilibrium (which is consistent with the 2nd law), but it will not provide thrust indefinitely as in the definition for a "differential sail" I made at the beginning (which would violate the 2nd law).
If the problem that keeps our thought experiment from actually acting like a differential sail is that it heats up, one thing you might try to do is cool it down. You could do this either by putting in contact with a colder reservoir or by doing work to pump heat out (as a refrigerator or air conditioner does). Neither of these methods breaks the laws of thermodynamics. The first method won't be much use in the middle of space, because there is no available reservoir in space cooler than the microwave background. The second method can work, but then the extra kinetic energy you gain will just be equal to the amount of heat you pump away, and it would be more efficient just to heat the sail up above the background radiation temperature and use the blackbody radiation the non-reflective side emits to drive the sail. What you'd have then might be called a photon rocket but not a differential sail.
The setup I described to create a differential sail with the cosmic microwave background simply does not work. It can't provide continuous thrust without additional work being done, so it doesn't satisfy my definition for a differential sail. Another version that has been suggested is to use a one-way mirror that lets light pass through in one direction but absorbs or reflects light traveling the opposite direction. Again this fails. A "one-way" mirror is not actually one-way; it is just a partially silvered mirror that passes light just as well in each direction. It only gives the appearance of being one-way when it is dark on the other side of the mirror. So, while the details of the failure in this second version of the differential sail are different, it fails all the same.
[Thermodynamics] is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown. — Albert Einstein
In some ideas for a differential sail, like the one based on a misunderstanding of how one-way glass works, the flaws in the reasoning might be easy to see, but one can come up with much more complicated schemes (such as ones involving vacuum fluctuations and the Casimir force) where finding what exactly the error is might be very difficult. The point to remember is that we showed in the first section that any differential sail (as I've defined the term) must violate the 2nd law of thermodynamics and is, therefore, impossible according to the known laws of physics, completely independent of any other details about how the device works.
Now, someone could propose such a device that operates on some new, hypothesized law of Physics that would break the 2nd law; however, a violation of the second law would allow perpetual motion machines and all sorts of phenomena that defy common sense and would turn everything we know about the world on it's head. The laws of thermodynamics are widely considered some of the most well grounded principles in Physics, because without them the world would not make much sense, so without extraordinary proof we can safely dismiss any hypothesis that violates them. Moreover, debating the design of devices that violate the known laws of physics is largely futile, because at that point one can just as well claim any design as a possibility. If we aren't even constrained by the laws of physics, how can we even say how a device will behave or which designs are good ones?
If you're not a scientist or engineer, I would not expect it to be obvious at first why a device like the differential sail will not work (otherwise I wouldn't have bothered explaining it!). Indeed, even for many scientists and engineers the problem with the idea might not be apparent at first, and it's not hard to come up with examples where finding the flaw in the reasoning is difficult. As a result, it's not surprising that this idea has gotten a lot of circulation on the Internet. What is disturbing, though, is that I found references to this sort of device on a NASA website  without any mention of the flaws I've talked about here. I have some ideas about how it might have ended up there but no facts, so it's probably better not to speculate. Suffice it to say that even the people at NASA can make mistakes, and not everyone there has expertise in Physics. I hope that I've made it plain why such a device will not work.
* If a photon carries momentum p and is absorbed then that momentum is transferred to the sail, and it gains momentum p. If the photon is reflected, then it goes back the way it came with momentum -p and the sail must have gained momentum 2p according to conservation of momentum.
† You might wonder, "won't the reflecting side give off blackbody radiation too?" Well, the reason it's called blackbody radiation is that it is really for something that is a perfect absorber, a black body. The better an absorber something is, the better an emitter it is (due to time reversal invariance). A perfectly reflecting surface does not absorb anything at all, hence it also emits no blackbody radiation.
The preceding was my own analysis, which obviously has not been peer reviewed or anything similar, but I believe it to be correct. I hope it speaks for itself.
- Charles Kittel and Herber Kroemer. Thermal Physics. New York: W.H. Freeman and Company, 1980.
- Ideas Based On What We’d Like To Achieve. http://www.nasa.gov/centers/glenn/research/warp/ideachev.html. NASA Glenn Research Center. January 8, 2006.