A Complementary Relation of Fluctuations between Energy and Temperature in a Small Specimen

Abstract

Research on fluctuations in energy and temperature is presented for a small specimen. The small specimen in contact with a heat bath shows energy fluctuations, [Formula: see text], at the constant temperature. On the other hand, when this small specimen is isolated from the reservoir and adiabatic isolation is kept, it exhibits temperature fluctuations, [Formula: see text], at the constant energy. This means that the temperature is unsharp if a sharp energy is assigned. A complementary relation between [Formula: see text] and [Formula: see text] is proposed in a simple formula. The connection between [Formula: see text] and [Formula: see text] is mediated by the heat capacity [Formula: see text]. This complementary relation is valid in general and it does not depend on the amount of substance. If the constituent number[Formula: see text] of the system is of the order of Avogadro’s number, then the fluctuations have been masked by large [Formula: see text]and we cannot see the influence of the fluctuations. However, when the number [Formula: see text] decreases, the intrinsic features of fluctuations come out gradually. This paper presents the quantitative analyses of the fluctuations in the energy and temperature for several physical models. Typical characteristics in the fluctuations can be clearly seen only in a small specimen, which are shown in the graphical representations. It is stressed that the values of [Formula: see text] and [Formula: see text] are defined for the different prescribed conditions specified above.