Motion Captured – Speed Demons

March 1, 2015


Below we have a box filled with a gas and a partition in the middle made of a very special material. This allows particles of greater than average speed to pass left to right but not right to left and particles of less than average speed to pass right to left but not the other way. As a guide to the speed of a molecule more blue means slower, more red means faster. You can click to restart.

This week we look at some nice paradoxes in classical thermodynamics. If that doesn’t get your pulses racing I don’t know what you are doing here. “Maxwell’s demon” is a thought experiment designed to break the second law of thermodynamics. This is no mean feat, to quote Eddington:

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

For our purposes the Clausius form of this law is most useful

Heat can never pass from a colder to a warmer body without some other change occurring at the same time

which is equivalent, via various clever manipulations to the statement

In a thermodynamic process, there is an increase in the sum of the entropies of the participating systems.

The idea that the quantity called `entropy’ always increases in any process is one of the the most profound discoveries of 19th century physics. Boltzmann’s relation of `entropy’ – a quantity defined in terms of heat added over temperature – to the amount of disorder in a system was similarly revolutionary. The statistical flavour of this argument, that physics proceeds in the most probable direction, is fundamentally different than the Newtonian picture of nature advancing according to some fundamental equation. We might have hoped that we would discover some strange time asymmetric law that explains why coffee left on the windowsill tends to cool down. Boltzmann and others showed that, by thinking about the constituent coffee atoms bouncing around and colliding according to Newtonian mechanics, time reversibly, the macroscopic result was overwhelmingly likely to be cooling down to match the temperature of the surrounding environment. These statistical considerations set the stage and shaped the thinking of the early pioneers of quantum mechanics, particularly Einstein and Plank, though in the end the classical probabilities of statistical thermodynamics were insufficient to describe our quantum world.

Nevertheless statistical mechanics underpins and explains much of thermodynamics and we will follow the early thinkers by considering a box containing many molecules. These molecules are of the `billiard ball’ type, undeformable spheres that collide perfectly elastically. We now add, as in most thermodynamics examples, a partition to the box. This partition comes with a small trap door that can be opened and closed very quickly, practically instantaneously from the point of view of the colliding atoms. We then hire, at considerable expense, “a being whose faculties are so sharpened that he can follow every molecule in its course” to act as our demon. We instruct this demon to open the trapdoor to let the faster molecules escape from the left into the right compartment and allow the slower molecules to move right to left. In this way we can, without the expenditure of work raise the temperature of the right half (temperature essentially being the sum of the squared velocities of the molecules) and lower the left. This is in contradiction to the Second Law.

This paradox is understood by considering the whole system of demon + gas, instead of just the gas. The standard argument is that in observing the molecules the demon himself generates an increase in entropy to compensate the reduction in entropy of the gas. If it occurred to you that the demon must do work to open and close the trapdoor, we can in principle make the opening and closing as easy as possible, so very little work has to be done to open and shut it, too little to generate enough entropy in the demon to compensate. The energy expenditure from making the observations is also not enough. It is possible in principle to make thermodynamically reversible measurements. For example imagine very slowly (but still fast on the molecular timescale) sliding a marker along a scale using the reversible expansion of a different gas. This is a thought experiment and a matter of principle so we are allowed these contrivances.

Giving all this advanced technology to the demon how can we save the 2nd law? One way is to evoke Landauer’s Principle. This is the statement that the increase in entropy comes when a measurement is erased, expelling that amount of entropy into the environment. Thus the advanced reversible measuring apparatus we gave to the demon earlier must have a lot of storage space, otherwise he will have to obey the second law after all. Since Maxwell’s original proposal real demons have been constructed and the second law holds in all cases, with the entropy increase of the demon in each case compensating the entropy reduction elsewhere

Indeed if any second law violating processes could be created we could construct a perpetual motion machine (of the second kind). For example we could use the heat differential set up by the demon to drive a heat engine, getting mechanical work out of thermal equilibrium. This is a better class of perpetual motion machine typically proposed by a higher quality of crank. John Wilkins, Bishop of Chester in 1670 gave three principle types of perpetual motion machine designs based on “Chymical Extractions”, “Magnetical Virtues” and “the Natural Affection of Gravity”. All still in vogue among the thermodynamically backward inventors of today.

Stevens 008
A Brownian ratchet is a very well oiled, molecular sized one of these.

Another interesting machine of this kind is the so-called Brownian ratchet. It consists of a cog that rotates freely in one direction and is blocked from rotating in the other. This is connected to a paddle wheel which, for example, raises weights against gravity or stirs tea. If the cog is sufficiently light and frictionless then collisions of gas molecules with the teeth will make it turn in one direction while the block (called a pawl) will stop rotation in the opposite direction. We have violated the second law again. But not really. The pawl is itself subject to Brownian motion and bounces up and down, allowing teeth to slip and the net effect is no motion, on average.

The relation of a black hole’s surface area to its entropy, the so called ‘black hole information paradox’, the holographic principle and many others mean that the ideas of reversibility and irreversibility, information deletion and Maxwell demons remains relevant even in modern fundamental physics. To see if you are suitable to work as a demon try the exercise below (yes it is possible to win).