Particles distinguishable

Q is given by Equation (6.4) for a system of identical particles. We shall ignore any normalisation constants in our treatment here to enable us to concentrate on the basics, and so it does not matter whether the system consists of identical or distinguishable particles. We also replace the Hamiltonian by the energy, E. The internal energy is obtained via Equation (6.20) ... 

So far, we have treated the atoms as distinguishable particles, both in the general theory of Section II and in the application to H + H2 in Section III. Here, we explain how to incorporate the effects of particle exchange symmetry. First, we discuss how the symmetry of the system maps from the physical onto the double space, and then explain what effect the GP has on wave functions of reactions that (like H + H2) have identical reagents and products. 

For a system of distinguishable particles the total partition function of the system is the product of all the individual partition functions, i.e. 

For a system of N distinguishable particles in three-dimensional space, the classical Hamiltonian is... 

This discussion applies only to systems with distinguishable particles for example, systems where each particle has a different mass. The treatment of wave functions for systems with indistinguishable particles is more compli-... 

The probability density for a pair of distinguishable particles with particle 1 in state a and particle 2 in state b is a(l)p i/ /,(2)p. If the distinguishable particles are interchanged, the probability density is V a(2)p i/ 6(l)p. The probability density for one distinguishable particle (either one) being in state a and the other in state b is, then... 

Consider a system composed of n identical, but distinguishable, particles. The distinguishability of the particles may result, for example horn positions in space, e.g. their coordinates. It is useful in this simplified mo to assume, furthermore, that the energy of interaction between the partic... 

In some instances, we have prior knowledge of states of the system that are thermodynamically meaningful. Then we can take advantage of such information and generate the proper samples that allow, for instance, the calculation of the relative free energy of such states. Let us reconsider the partition function for the ensemble of states for N distinguishable particles in three dimensions,... 

The simplest situation, known as the quantized Maxwell-Boltzmann distribution occurs with a system of N distinguishable particles, with eigenfunction... 

Microscopic Identification Models. Many different optical and chemical properties of single aerosol particles can be measured by microscopic identification and classification in order to distinguish particles originating in one source type from those originating in another. The microscopic analysis receptor model takes the form of the chemical mass balance equations presented in Equation 1. 

The scattering amplitudes, on the other hand, depend strongly on the combination of isotopes. The total scattering amplitude for distinguishable particles (e.g., 4He + 3He) becomes... 

Figure 20. Energy dependence of total cross section for He (2 5) + He calculated from potentials of Fig. 14 and Table 111. Oscillations at low energies attributable to nuclear-symmetry. Glory effect is amplified in curve I, in which difference between cross sections for identical and distinguishable particles is plotted on an expanded scale. 

In principle, there is no way to directly measure the exchange process for 4He +4He scattering, as the particles are indistinguishable. But the cross section for metastability exchange can of course be calculated from the determined potentials assuming distinguishable particles.65 The expression for the total excitation transfer cross section is... 

Figure 24. Calculations for He (2 S) + He assuming distinguishable particles compared to experiment.

N. C. Petroni, C. Dewdney, P. Holland, A. Kyprianidis, and J. P. Vigier, Causal space-time paths of individual distinguishable particle motions in V-body quantum systems Elimination of negative probabilities, Lett. Nuovo Cimento 42(6) (Ser. 2), 285-294 (1985). 

Next, we follow the growth histories of each distinguished chain by following the time evolution of the distinguished particles. This permits us to specify the number concentration of the different types of distinguished particles at any time. 

Finally, the time of growth of each chain is found by determining how many distinguished latex particles stopped growing at any particular instant. This is obtained readily from the product of the number concentration of the different types of distinguished particles and the (known) rate coefficient for the appropriate kinetic event. Of course, the growth time of each chain determines the molecular weight of the polymer produced on termination. 

The overall time evolution of the singly distinguished particles is obtained by summing the changes that dissipate the distinguished character of the particles with those that conserve it ... 

The solution of equations (8) and (9) yields the number concentration of singly distinguished particles at any time t . 

First Order Chain Stoppage. The rate of stoppage of distinguished chains is obtained from the product of the populations of singly distinguished particles with the respective rate coefficient. Thus for stoppage in the absence of combination and disproportionation, we have from equations (4) and (5)... 

The Number Concentration of Doubly Distinguished Particles. In the 0-1-2 system, doubly distinguished particles (N 11) can only be formed from singly distinguished particles by entry of a free radical into a N -type particle or chain transfer (involving the nondistinguished chain) in an N -type particle. Therefore... 

Note that equation (13) shows that there is a hierarchy of differential equations solution of the Smith-Ewart equations provides the boundary conditions for the singly distinguished particle equations these in turn provide the boundary conditions for the doubly distinguished particle equations. 

The Time Evolution of the Doubly Distinguished Particles. In the 0-1-2 system, all events associated with the doubly distinguished particles lead to the loss of the particles. Entry, bimolecular combination, transfer (from either distinguished chain) and exit (again from either distinguished chain) all may occur ... 

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