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The zero-field isothermal susceptibility of the one-dimensional Ising model with nearest-neighbor interactions and a finite number of spins is shown to have a relatively simple singularity as the temperature approaches zero, proportional only to the inverse temperature. This is in contrast to what is seen throughout the literature for the inifinite chain: an essential singularity that includes an exponential dependence on the inverse temperature. Assuming an arbitrary (but finite) number of spins and retaining terms that are usually considered ignorable in the thermodynamic limit, the analysis involves nothing beyond straightforward series expansions, starting either from the partition function for a closed chain in a magnetic field, obtained using the transfer-matrix approach; or from the expression for the zero-field susceptibility found via the fluctuation-dissipation theorem. In both cases, the exponential singularity is exactly removed. In addition, the susceptibility per spin is found to increase with the number of spins (except in the case of noninteracting spins), a result which is also at variance with what is normally reported for an infinite chain., Comment: 11 pages, LaTeX, mathematical details added in Secs. II and III, consideration of noninteracting case added, expanded conclusions, expanded references

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., Comment: 13 pages