A Statistical Physics Study On Spatial Inhomogeneous Systems With Small Number Of Particles

Equal a priori probability is a fundamental principle and even a unique principlein statistical physics. This principle can describe the properties of microworldprofoundly. Identity principle in quantum mechanics requires the quantum stateentering the system’s quantum state with identical weight, which has consistencyspirit of quantum mechanics. The main scientific discovery of this dissertation is theenergy inversal partition phenomenon in classical statistics, which is bijective withthe negative specific heat.This dissertation is mainly divided by two parts. Part one is the major component,which investigated the relation between the negative specific heat and energypartition of systems with small number particles in framework of Boltzmann statistics,and find that negative specific heat is a process that energy focus on a single particleand we call this phenomenon apparent energy inversal partition. Part two is thesecondary component. The quantum statistical methods are applied to investigate thecomputation on microstate numbers of harmonic wells and thermodynamic limit.Volume is not an extension variable because the harmonic well is not a homogeneoussystem in space, and the ensemble inequivalence is still maintained and manyprevious conclusions should be considered carefully again. We find the quantum andclassical conclusions approaches consistence in classical limit once we consideredindistinguishable properties of particles.Chapter one is introduction, at first the importance of the investigation on thefinite systems is illustrated, subsequently introduced the basic theory related to thisarea, then summarized the progress of this area in theory and experiment, at last themotive and constitution of this dissertation is introduced briefly.Part one is divided by four chapters, which is from chapter two to chapter five.Mainly inspired by the observing conclusion on negative specific heat of147sodiumatoms cluster made by M. Schmidt group, we investigated the particle-spilling-from-wells model. The particle-spilling-from-wells denoted that there is a potentialarea in a box, and particles constrained in the potential area at first but will spill outto the box with the increasing of energy. This model can be solved exactly instatistical physics. We can get the variational curves of particle’s mean kinetic energyversus mean internal energy-Caloric curve by computation then obtained the energy area where the negative specific heat exists. In chapter two the situation that there isonly one wells in box and find negative specific heat is a phenomenon which relatedto the apparent energy inversal partition. The detailed procedure is given and weinvestigated the significant influence of well’s volume ratio to the Caloric curve. Theexact and reasonable results in physics on particle numbers in well were also obtained.At last we investigated the apparent energy inversal phenomenon and find that it canbe eliminated once we consider the hypothesis of identity in quantum mechanics.The situations where there are two wells in box is investigated in chapter three,moreover we confirm that negative specific heat is a phenomenon related to apparentenergy inversal partition. The computation is more complicated and we give thecomputing detail and discussed it carefully. The volume ratio occurrence at first isreplaced by two parameters including volume ratio and the relative height of the twowells, and the exact and reasonable results in physics were obtained too. The situationthat includes more wells can be dealed with the same as double wells.In chapter four, the thermodynamic properties of a typical system withinteraction, that is Lennard-Jones system is investigated under single potential wells.The interaction is supposed very weak, with the force acting only in short range inthis article. The mean kinetic energy is derived approximately by series expansion forpoint density in phase space under this condition, and the validity and shortage of thisapproach was discussed too.In chapter five we computed system’s fluctuation curve of tempreture andparticle numbers in wells with various total particle numbers and volume ratios. Wefind the spilling of particles always follows the huge fluctuations of system’stempreture by comparing with the Caloric curve. The smaller the well’s volume is themore significant of this effect.The chapter six is the second part of this dissertation. The problem on thecounting of particle’s microstates number is investigated under microcanonicalconditions in this chapter. The microstate number is obtained in classical statistical atfirst, then computed the microstate numbers of bosons and fermions partly. Onceconsidered the identity of particles, we find the solutions of bosons and fermions arerigorously equal when the particle number is very large. This problem seems trivial,but it does not. Our investigation showed that many results of statistical physics inthermodynamic limit are still valid for inhomogeneous systems once the particlenumber changes very large.In the end, we summarized the main conclusion of this article, and remarked the present situation and the possible research directions in the future.