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Thermodynamics Far from the Thermodynamic Limit.

Understanding how small systems exchange energy with a heat bath is important to describe how their unique properties can be affected by the environment. In this contribution, we apply Landsberg's theory of temperature-dependent energy levels to describe the progressive thermalization of small systems as their spectrum is perturbed by a heat bath. We propose a mechanism whereby the small system undergoes a discrete series of excitations and isentropic spectrum adjustments leading to a final state of thermal equilibrium. This produces standard thermodynamic results without invoking system size. The thermal relaxation of a single harmonic oscillator is analyzed as a model example of a system with a quantized spectrum than can be embedded in a thermal environment. A description of how the thermal environment affects the spectrum of a small system can be the first step in using environmental factors, such as temperature, as parameters in the design and operation of nanosystem properties.