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Excluded-volume effects of radial oscillations in disks confined to a circular box

The effect of radial vibrations on the properties of one or two disks confined to a circular trap in contact with a thermal reservoir are investigated. The vibrational amplitudes and energies are assumed to be quantized, with the motions corresponding roughly to certain modes for ringlike or tetrahedral molecules (such as benzene or methane, respectively). The calculation of the partition function requires integrations over the internal phases describing the oscillations, as well as the disks' center-of-mass positions and momenta; while an exact result is obtained for a single disk, for two disks the position-space integration can only be approximated. In spite of the small number of disks considered, various "thermodynamic" quantities are evaluated from the partition function. It is found that the average energy of the system is increased---compared to that for rigid disks---as are the entropy and compressibility; the pressure, however, is decreased, and there is a variable effect on the heat capacity, depending on the ratio of the vibrational energy to the temperature. These changes can be traced either directly or indirectly to the oscillations causing an effective increase in the area available to the molecules within the circle, which in turn leads to an increase in the size of the accessible phase space, with the increase being larger for higher energy states.