Non-interacting particles

In this section we consider a many-partiele system in which the particles act independently of each other. For such a system of N identical particles, the Hamiltonian operator. (1, 2, N) may be written as the sum of one-particle Hamiltonian operators H i) for i = 1, 2. A 

In this case, the operator FT(1, 2,. .., A) is obviously symmetric with respect to particle interchanges. For the N particles to be identical, the operators H i) must all have the same form, the same set of orthonormal eigenfunctions and the same set of eigenvalues where 

The A-particle eigenfunctions I v(l, 2,. .., A) in equation (8.47) are not properly symmetrized. For bosons, the wave function (1, 2,. .., N) must be symmetric with respect to particle interchange and for fermions it must be antisymmetric. Properly symmetrized wave functions may be readily con- 

The expression (8.49a) for two bosons is not quite right, however, if states tpa and tpb are the same state (a = b), for then the normalization constant is rather than 2 /2 go that 

From equation (8.49b), we see that the wavefunction vanishes for two identical fermions in the same single-particle state 

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The kinetic theory of gases has been used so far, the assumption being that gas molecules are non-interacting particles in a state of random motion. This... 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2,... and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

In Pauli s model, we still envisage a core of rigid cations (metal atoms that have lost electrons), surrounded by a sea of electrons. The electrons are treated as non-interacting particles just as in the Drude model, but the analysis is done according to the rules of quantum mechanics. 

Although the concept of non-interacting particles is an idealization, the model may be applied to real systems as an approximation when the interactions between particles are small. Such an approximation is often useful as a starting point for more extensive calculations, such as those discussed in Chapter 9. 

Consider two identical non-interacting particles, each of mass m, in a onedimensional box of length a. Suppose that they are in the same spin state so that spin may be ignored. 

In the most simple case of ideal, energy homogeneous surface the adsorption equilibrium of non-interacting particles is described by the Langmuir isotherm [33] ... 

Systems of chemical interest typically contain particles in molar quantity. Mathematical modelling of all interactions in such a system is clearly impossible. Even in a system of non-interacting particles, such as an ideal gas, it would be equally impossible to keep track of all individual trajectories. It is consequently not a trivial matter to extend the mechanical description (either classical or non-classical) of single molecules to macrosystems. It would be required in the first place to define the initial state of each particle in terms of an equation... 

Generally, these behave as Newtonian fluids and, for the case of an extremely dilute suspension of spherical non-interacting particles having a density equal to that of the continuous medium, we can apply the Einstein formula for a suspension of spheres ... 

In the case of dynamical interaction the pair potentials U r), (7BB(r) ar d Uab(t) should be incorporated into equation (2.3.45). It could be done using the Smoluchowski equation [27, 83, 84] for a particle drift in the external potential W (r) and expressed in terms of single particle DF (or concentration of such non-interacting particles)... 

Spatial structure of a system of mobile non-interacting particles... 

Up to now we have been discussing in this Chapter many-particle effects in bimolecular reactions between non-interacting particles. However, it is well known that point defects in solids interact with each other even if they are not charged with respect to the crystalline lattice, as it was discussed in Section 3.1. It should be reminded here that this elastic interaction arises due to overlap of displacement fields of the two close defects and falls off with a distance r between them as U(r) = — Ar 6 for two symmetric (isotropic) defects in an isotropic crystal or as U(r) = -Afaqjr-3, if the crystal is weakly anisotropic [50, 51] ([0 4] is an angular dependent cubic harmonic with l = 4). In the latter case, due to the presence of the cubic harmonic 0 4 an interaction is attractive in some directions but turns out to be repulsive in other directions. Finally, if one or both defects are anisotropic, the angular dependence of U(f) cannot be presented in an analytic form [52]. The role of the elastic interaction within pairs of the complementary radiation the Frenkel defects in metals (vacancy-interstitial atom) was studied in [53-55] it was shown to have considerable impact on the kinetics of their recombination, A + B -> 0. 

These results could be complemented well with the curve slopes in the double logarithmic coordinates as plotted in Fig. 6.33(a) using idea of the intermediate critical exponent a(t), equation (4.1.68). In the traditional chemical kinetics its asymptotic limit ao = a(oo) = 1 is achieved already during the presented dimensionless time interval, t 104. For non-interacting particles and if one of two kinds is immobile, Da = 0, it was earlier calculated analytically [11] that the critical exponent is additionally reduced down to ao = 0-5. However, for a weak interaction (curve 1) it is observed that in the time interval t 104 amax 0.8 is achieved only for a given n(0) = 0.1, i.e., the... 

Of greater interest is the behaviour of the joint correlation functions presented in Fig. 6.35(b). At any reaction time X (r, f) > X (r,t) holds now an increase of the maximum of X (r,t) in time is very slow. According to the above-given estimates for neutral non-interacting particles, it has a logarithmic character ... 

At longer times, when re , the effect of the statistical aggregation of similar particles begins to dominate, which takes also place for neutral non-interacting particles as it was discussed in Chapter 5. At this stage the reaction leads to the formation of A- or B-rich domains with the linear size in turn, these domains are structured inside themselves into smaller blocks having the typical size of re, which however no longer affects the kinetics. 

As it was noted in [77], reduction of the reaction rate with time observed for non-interacting particles at high concentrations/long reaction times, Section 6.2, is unlikely to occur for charged particles since spatial fluctuations in particle densities are now governed not by (i) but the screening radius in other words, the Coulomb repulsion of similar particles prevents their aggregation. 

No such thing as photon-photon collision has ever been observed, and to all practical purposes photons must be considered as non-interacting particles the collisional cross-section of the photon has been estimated from theory to be less than 10 70 cm2. This means that electromagnetic radiation (even cavity radiation) cannot be compared with a gas of molecules that can reach thermodynamic equilibrium through collisions which result in exchange of momentum and energy. 

Let us simply mention two other textbooks Elements of Applied Mathematics (with A. D. Myshkis) [72] and Elements of Mathematical Physics. Vol. 1. Non-Interacting Particle Medium (with A. D. Myshkis) [73] which are a continuation of Higher Mathematics for Beginners and its Applications to Physics, published in 1960, and are written with the same pedagogical goals. 

Zeldovich Ya. B., Myshkis A. D. Elementy matematicheskoi fiziki. T. 1 Sre-da iz nevzaimodeistvuiushchikh chastits [Elements of Mathematical Physics. V. 1 Non-Interacting Particle Medium]. Moscow Nauka, 351 p. (1973). 

For gases this term is usually small compared with P. An ideal gas is a gas consisting of non-interacting particles, and hence this term is zero. 

If the same minimization procedure is applied to a system of non-interacting particles moving in an effective potential, Ve[, the corresponding Euler equation would be ... 

If we identify the last three terms of eq. (2.26) with Veff, then clearly the same Euler equations will result since T0 was defined as the kinetic energy of a system of non-interacting particles. For non-interacting particles, the Schrodinger equation can be written... 

The set of magnetic susceptibilities of an assembly of non-interacting particles is defined by relation (4.85), which refers to the magnetization of the system in the direction of H = Hh. Therefore, of all the components of the corresponding susceptibility tensors, we retain only the combinations that determine the... 

One says that the above results are valid for a chain with non-interacting particles. However, the monomers in a real macromolecule interact with each another, and this ensures, above all, that parts of the molecule cannot occupy the place already occupied by other parts i.e. the probabilities of successive steps are no longer statistically independent, as was assumed in the derivation of the above probability distribution functions and mean end-to-end distance (Flory 1953). So, considering the coarse-grained model, one has to introduce lateral forces of attractive and repulsive interactions. The potential energy of lateral interactions U depends on the differences of the position vectors of all particles of the chain and, in the simplest case, can be written as a sum of pair interactions... 

The last term on the right-hand side is readily identified with the inverse Xs1 (r>r ) °f the density response function of a system of non-interacting particles... 

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