  2. Fast or slow? What is an adiabatic process? Most of the readers would probably answer that this is a process with a gas which is so fast that there is no heat exchange with the surroundings. However, this is only a half of the truth, and actually the less important half. In fact, it is quite easy to understand that this is not entirely correct: consider a cylinder, which is divided by a thin wall into two halves; one half is filled with a gas at a pressure p, and the other one is empty. Now, let us remove momentarily the wall: the gas from one half fills the entire cylinder. Since no external work is done (the wall can be removed without performing a work), the energy of the gas is preserved, hence, the temperature remains the same as it was at the beginning. Meanwhile, for an adiabatic process we would expect a decrease of temperature by a factor of 2γ-1: part of the internal energy is supposed to be spent on a mechanical work performed by the expanding gas. However, if the piston moves faster than the speed of sound, the gas will be unable to catch up and push the piston. So, the adiabatic law was not followed because the process was too fast! It appears that the adiabatic law for thermodynamics has also a counterpart in classical mechanics – the conservation of the adiabatic invariant. For mechanical systems (oscillators) performing periodic motion, the adiabatic invariant is defined as the area of the closed curve drawn by the system in phase space (which is a graph where the momentum p is plotted as a function of the respective coordinate x), and is (approximately) conserved when the parameters of the system are changed adiabatically, ie. slowly as compared with the oscillation frequency. For typical applications, the accuracy of the conservation of the adiabatic invariant is exponentially good and can be estimated as e–fτ, where f is the eigenfrequency of the oscillator, and τ is the characteristic period of the variation of the system parameters. How are related to each other (a) adiabatic invariant and (b) adiabatic process with a gas? The easiest way to understand this is to consider a one-dimensional motion of a molecule between two walls, which depart slowly from each other (Figure 1). Let us use the system of reference where one of the walls is at rest, and the other moves with a velocity u << v, where v is the velocity of the molecule (the interaction of the molecule with the walls is assumed to be absolutely elastic). One can say that such a molecule represents an oscillator with a slowly changing potential:  the potential energy U(x) = 0 for 0 