Adiabatic invariant

An adiabatic invariant is a property of a physical system that stays constant when changes occur slowly.

In thermodynamics, an adiabatic process is a change that occurs without heat flow, and slowly compared to the time to reach equilibrium. In an adiabatic process, the system is in equilibrium at all stages. Under these conditions, the entropy is constant.

In mechanics, an adiabatic change is a slow deformation of the Hamiltonian, where the fractional rate of change of the energy is much slower than the orbital frequency. The area enclosed by the different motions in phase space are the adiabatic invariants.

In quantum mechanics, an adiabatic change is one that occurs at a rate much slower than the difference in frequency between energy eigenstates. In this case, the energy states of the system do not make transitions, so that the quantum number is an adiabatic invariant.

The old quantum theory was formulated by equating the quantum number of a system with its classical adiabatic invariant. This determined the form of the Bohr–Sommerfeld quantization rule: the quantum number is the area in phase space of the classical orbit.

In thermodynamics, adiabatic changes are those that do not increase the entropy. They occur slowly, and allow heat flow only between objects at the same temperature. For isolated systems, an adiabatic change allows no heat to flow in or out.

If a container with an ideal gas is expanded instantaneously, the temperature of the gas doesn't change at all, because none of the molecules slow down. The molecules keep their kinetic energy, but now the gas occupies a bigger volume. If the container expands slowly, however, so that the ideal gas pressure law holds at any time, gas molecules lose energy at the rate that they do work on the expanding wall. The amount of work they do is the pressure times the area of the wall times the outward displacement, which is the pressure times the change in the volume of the gas:

If no heat enters the gas, the energy in the gas molecules is decreasing by the same amount. By definition, a gas is ideal when its temperature is only a function of the internal energy per particle, not the volume. So

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