THERMAL OSCILLATIONS OF A PISTON

According to classical theorem about uniform distribution of kinetic energy between the degrees of freedom the mean kinetic energy in one degree of freedom of any molecular structure in thermal equilibrium equals to kT/2, where k is Boltzman constant, and T is the temperature. Let us consider the mechanical oscillator, which consists of a piston and spring. Such mechanical system possesses one degree of freedom because its state can be described by a single coordinate x of a piston.  Under the action of unbalanced  collisions of the molecules the piston will be chaotically displaced from its position of equilibrium. If the system is in thermodynamic equilibrium, i.e. there is no macroscopic thermal conductivity processes, then the mean kinetic energy of the piston during its motion along the axis of the cylinder equals to the mean kinetic energy kT/2 of the molecule motion in the same direction. During chaotic motion the kinetic energy of piston will be transferred to the potential energy of spring and back. It is not difficult to prove that the mean values of kinetic and potential energies are equal. The mean potential energy equals to a<x²>/2, where a is the rigidity of the spring, x is the displacement of the piston from the position of equilibrium,  < > means averaging of a value over considerable period of time. Equating the mean potential energy of the spring to kT/2 we can find the mean-square value of the thermal oscillations of the piston:

<x²>^1/2 = (kT/a)^1/2

The thermal oscillation amplitude <x²>^1/2 is very small for real mechanical systems. Nevertheless, it imposes the principal (theoretical) limitation for sensitivity in highly precision measurement systems based on mechanical resonators.