Statistical Mechanics For Finite Systems And Temperature Fluctuations

With the technological advances allowing us to make measurement on small samples, e.g., the measurement of thermodynamic quantities like temperature with a spatial resolution on the nanometer scale, there is a growing demand for better understanding of thermal properties of nanoscale devices, individual nanostructures, and nanostructured materials. All these reveal rich non-thermodynamical limit effects. Besides nano-physics, the same effects present in nuclear physics, Bose-Einstein condensation, molecular physics etc. Mentioned explicitly or implicitly, a so-called statistical mechanics for small system or finite number of particles is in progress, in which the requirement of thermodynamical limit is not met. This dissertation first reviews the physical phenomena and the theories dealing with them.After carefully examining the progresses in exploring the non-thermodynamical limit effects, no full statistical mechanics has been found that can deal with discrete energy systems. The fist part of the main body of this dissertation is dedicated to develop a formalism of statistical mechanics for multi-polynomial distribution. In this distribution, there are N particles distributed among r states and each state possesses a discrete energy. Since no particle has a peculiar mechanics property, i.e., every particle occupies any state with equal probability. We develop a statistical mechanics for this multi-polynomial distribution based on the equal a priori probabilities and the probability theory. All results given by the formalism turns out to be the usual ones in thermodynamic limit.The second part of the main body of the dissertation is using the above established statistical mechanics for multi-polynomial distribution to resolve a theoretical difficulty. The usual statistical mechanics for all paramagnetic materials, and phonon gas, etc. gives divergent temperature fluctuations when the temperature itself approaches to the zero Kelven because their heat capacities goes to zero faster than the temperature square. We explicitly calculate the simplest paramagnetic material: finite N independent spin-1/2 paramagnets in a constant magnetic field. In our generalized canonical ensemble for finite number of particles, all fundamental thermodynamic quantities fluctuate, including the entropy and temperature, and all fluctuations are gradually vanishing as the temperature itself approaches zero. Our approach presents a remedy of infinite temperature fluctuations and reproduces other reasonable results that are given by standard theory, which includes an intensive temperature fluctuation that is introduced in this dissertation. Since the finite sized system does not satisfy the additivity, and different statistical ensembles are not equivalent to each other, the microcanonical ensemble treatment yields the identical result. So, our conclusion has universality. By universality we mean that the conclusion the temperature fluctuation for this system is not divergent is ensemble independent, and moreover this conclusion holds for other systems that in usual treatment temperature fluctuations appear divergent as temperature approaches zero.The last part of the main body of the dissertation presents a universal criterion for the existence of an equilibrium state at low temperatures, which is established on the base of the requirement that the temperature fluctuations be small and the third law of thermodynamics. The criterion implies that at sufficiently low temperatures the minimum number of particles increases as the temperature decreases. The application of the criterion to the phonon gas, ideal Bose gas, and the ideal Fermi gas gives quantitative results that are compatible with recent results for nanoscale systems which have been given in literature with numerical methods. In addition, the Ising models are also treated and reasonable results are obtained.The main body of this dissertation has been published in Annals of Physics (N. Y.) and American Journal of Physics, etc. Especially, the paper on the universal criterion for the existence of an equilibrium state at low temperatures has been selected into the"Virtual Journal of Nanoscale Science & Technology"established by both AIP and APS.