Systems with Identical Particles

The natural modeling of macro- and microscopy systems is made through the reduction of the components at stmctural units the elementary particles (or more precisely - fundamental), essentially the same (electrons, photons, protons, neutrons, nuclei, atoms, etc.). These are also known as ideal, if from the macroscopic point of view the systems that belong are isolated, while from the microscopic point of view they do not interact between them. 

The quality of isolated system ensures the conservation of particles numbers from the system. 

The quality of isolated system is accomplished if the particles do not interact it is considered that each particle is associated (quantified) with a given energy and around each thus energy there is a microscopic range 

Under these conditions, the system of particles is called ideal, being isolated and with particles considered as independent while the particles or the individual states of energy are considered in very ample numbers, then can be considered as forming a set of statistics, with a certain arrangement of the particles on the available energy levels, called microstate. Note that there may be several iso-states, which correspond at the same macroscopic 

The calculation of the number of microstates is made following a rigorous algorithm  

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It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

In an aqueous system with large particles weU-separated by a distance, s, (D and 5 > t ) the electrostatic repulsion energy between two identical charged spheres may be approximated (1) ... 

For the next step, we show how we consider the N-representability conditions for the 1-RDM y for a system with N particles that is all of its eigenvalues should be between zero and one [17]. In other words, this condition is equivalent to saying that y and / — y are positive semidefinite, where / is the identity matrix. Assuming that H is the one-body Hamiltonian, we have... 

The Eyring analysis does not explicity take chain structures into account, so its molecular picture is not obviously applicable to polymer systems. It also does not appear to predict normal stress differences in shear flow. Consequently, the mechanism of shear-rate dependence and the physical interpretation of the characteristic time t0 are unclear, as are their relationships to molecular structure and to cooperative configurational relaxation as reflected by the linear viscoelastic behavior. At the present time it is uncertain whether the agreement with experiment is simply fortuitous, or whether it signifies some kind of underlying unity in the shear rate dependence of concentrated systems of identical particles, regardless of their structure and the mechanism of interaction. 

The aim of this Chapter is the development of an uniform model for predicting diffusion coefficients in gases and condensed phases, including plastic materials. The starting point is a macroscopic system of identical particles (molecules or atoms) in the critical state. At and above the critical temperature, Tc, the system has a single phase which is, by definition, a gas or supercritical fluid. The critical temperature is a measure of the intensity of interactions between the particles of the system and consequently is a function of the mass and structure of a particle. The derivation of equations for self-diffusion coefficients begins with the gaseous state at pressures p below the critical pressure pc. A reference state of a hypothetical gas will be defined, for which the unit value D = 1 m2/s is obtained at p = 1 Pa and a reference temperature, Tr. Only two specific parameters, Tc, and the critical molar volume, VL, of the mono-... 

It is more important in this context that one should never rely on a single measurement. A dust sample is never fully identical with the next, because a solid substance can never be homogenized in such a way that an ideally mixed system with identical humidity and particle size distribution in each sample is achieved. If possible, up to 20 measurements should be performed. 

Equation [A9] is valid for our system of identical particles with unit mass. The equation can be generalized to the case of particles with arbitrary masses. 

We have shown that there are two possible cases for the wave function of a system of identical particles, the symmetric and the antisymmetric cases. Experimental evidence (such as the periodic table of the elements to be discussed later) shows that for electrons only the antisymmetric case occurs. Thus we have an additional postulate of quantum mechanics, which states that the wave function of a system of electrons must be antisymmetric with respect to interchange of any two electrons. Was important postulate is called the Pauli principle, after the physicist Wolfgang Pauli. 

The time course of orientational changes induced by electric fields contains information on the orientation mechanism, and on the electrical and geometrical properties (main dipole axis, length) of the aligning and deorienting molecules. For instance, permanent dipole orientation of a given particle type in the presence of a constant electric field builds up with zero slope and has two modes, whereas the build-up of induced dipole orientation starts with maximum slope and is characterized by only one time constant. The deorientation relaxation of a system of identical particles, after termination of the step pulse, is monophasic, independently of the presence of permanent or induced dipoles. Table 3 summarizes the characteristic features of the rotational kinetics indicated by electric dichroism and birefringence for small perturbations. We see that there are a number of specific relationships to differentiate between permanent and induced dipole mechanism. In particular, the technique of field-reversal is a sensitive indicator for the relative contributions of permanent or induced dipoles. 

Any solid material that consists of a combination of two or more types retaining their separate identity. In polymer technology, the term is reserved for these polymeric systems in which additions of solid particles result in reinforcing effect. The composites are divided into reinforced filled systems (with a particle size ... 

In statistical thermodynamics, a system with interacting particles is depicted with the canonical ensemble that describes a collection of a large number of macroscopic systems under identical conditions (for instance, N particles in a volume V at temperature T). In each system, laws that describe interactions between molecules are identical. They differ by the coordinates of each particular molecule corresponding to a microstate. The static picture of the canonical ensemble is equivalent to the development of a system over time [10,14]. In other words, the measurement of a macroscopic property reflects a succession of microstates. Thus, the measured property corresponds to a time-averaged mean value and thermodynamic equilibrium corresponds to the most probable macroscopic state. 

The wavefimction of a system must be antisynnnetric with respect to interchange of the coordinates of identical particles y and 8 if they are fermions, and symmetric with respect to interchange of y and 5 if they are bosons. 

The total Hamiltonian operator H must commute with any pemiutations Px among identical particles (X) due to then indistinguishability. For example, for a system including three types of distinct identical particles (including electrons) like Li2 Li2 with a conformation, one must satisfy the following commutative laws ... 

Particulate systems composed of identical particles are extremely rare. It is therefore usefiil to represent a polydispersion of particles as sets of successive size intervals, containing information on the number of particle, length, surface area, or mass. The entire size range, which can span up to several orders of magnitude, can be covered with a relatively small number of intervals. This data set is usually tabulated and transformed into a graphical representation. 

Temperature becomes a quantity definable either in terms of macroscopic thermodynamic quantities, such as heat and work, or, with equal validity and identical results, in terms of a quantity, which characterized the energy distribution among the particles in a system. With this understanding of the concept of temperature, it is possible to explain how heat (thermal energy) flows from one body to another. 

There are two possible cases for the wavefunction of a system of identical fundamental particles such as electrons and photons. These are the symmetric and the antisymmetric cases. Experimental evidence shows that for fermions such as electrons and other particles of half integer spin the wavefunction must be anti-symmetric with respect to the interchange of particle labels. This... 

We have thus far only considered the relativistic quantum mechanical description of a single spin 0, mass m particle. We next turn to the problem of describing a system of n such noninteracting spin 0, mass m, particles. The most concise description of a system of such identical particles is in terms of an operator formalism known as second quantization. It is described in Chapter 8, The Mathematical Formalism of Quantum Statistics, and Hie reader is referred to that chapter for detailed exposition of the formalism. We here shall assume familiarity with it. 

In quantum theory, identical particles must be indistinguishable in order for the theory to predict results that agree with experimental observations. Consequently, as shown in Section 8.1, the wave functions for a multi-particle system must be symmetric or antisymmetric with respect to the interchange of any pair of particles. If the wave functions are not either symmetric or antisymmetric, then the probability densities for the distribution of the particles over space are dependent on how the particles are labeled, a property that is inconsistent with indistinguishability. It turns out that these wave functions must be further restricted to be either symmetric or antisymmetric, but not both, depending on the identity of the particles. 

The wave fiinetion for a system of N identical particles is either symmetric or antisymmetric with respect to the interchange of any pair of the N particles. Elementary or eomposite particles with integral spins (s = 0, 1,2,. ..) possess symmetrie wave functions, while those with half-integral spins (s = 1. .)... 

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