Noninteracting particles

As an example, let us consider now the simplest case of non-interacting particles, U(r) = 0, all particles are created with the same separation I (3.2.14). The solution of equation (3.2.7) is [66]  

The integration of equation (3.2.9) yields the time development of the conditional probability to find a pair at time t  

For a short reaction time, 2 Dty (( — tq), the asymptotic expansion of equation (3.2.20) gives the estimate 

This expression turns out to be surprisingly close to the asymptotic survival probability seen from equation (3.2.20) 

In the two-dimensional case (e.g., on a surface or for highly anisotropic layered solids), we get in equation (3.2.7) the modified Laplace operator (3.2.8) and instead equation (3.2.18) we have 

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If we consider a shear flow of a diluted suspension of noninteracting particles, then substitution of spheres by particles of ellipsoidal form leads only to a variation of... 

It is, finally, fairly obvious that the method of Green s function and the inhomogeneous Schrodinger equation can also be written down for a system of 3N noninteracting particles, in analogy to the last paragraph in Section 8.9. 

The formulation of the theory outlined above is particularly well-suited for the description of scattering processes, i.e., experiments consisting of the preparation of a number of physical, free noninteracting particles at t — oo, allowing these particles to interact (with one another and/or any external field present), and finally measuring the state of these particles and whatever other particles are present at time t = + co when they once again move freely. The infinite time involved... 

Fig. 6.16 Mossbauer spectra of 8 nm a-Fe203 particles at the indicated temperatures, (a) Data for phosphate coated (noninteracting) particles, and (b) data for the uncoated (interacting) particles. (Reprinted with permission from [77] copyright 2006 by the American Physical Society)...

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

For a classical system of N point particles enclosed in a volume V,at a temperature T, the canonical partition function can be decomposed in two factors. The first one (Qt) comes from the integration over the space of momenta of the kinetic term of the classical Hamiltonian, which represents the free motion of noninteracting particles. The second one, which introduces the interactions between the particles and involves integration over the positions, is the configuration integral. This way, equation (30)... 

From the discussion so far, it is clear that the mapping to a system of noninteracting particles under the action of suitable effective potentials provides an efficient means for the calculation of the density and current density variables of the actual system of interacting electrons. The question that often arises is whether there are effective ways to obtain other properties of the interacting system from the calculation of the noninteracting model system. Examples of such properties are the one-particle reduced density matrix, response functions, etc. An excellent overview of response theory within TDDFT has been provided by Casida [15] and also more recently by van Leeuwen [17]. A recent formulation of density matrix-based TD density functional response theory has been provided by Furche [22]. 

In this short review, a brief overview of the underlying principles of TDDFT has been presented. The formal aspects for TDDFT in the presence of scalar potentials with periodic time dependence as well as TD electric and magnetic fields with arbitrary time dependence are discussed. This formalism is suitable for treatment of interaction with radiation in atomic and molecular systems. The Kohn-Sham-like TD equations are derived, and it is shown that the basic picture of the original Kohn-Sham theory in terms of a fictitious system of noninteracting particles is retained and a suitable expression for the effective potential is derived. 

Pick s laws describe the interactions or encounters between noninteracting particles experiencing random, Brownian motion. Collisions in solution are diffusion-controlled. As is discussed in most physical chemistry texts , by applying Pick s Pirst Law and the Einstein diffusion relation, the upper limit of the bimolecular rate constant k would be equal to... 

Using a Langevin dynamics approach, the stochastic LLG equation [Eq. (3.46)] can be integrated numerically, in the context of the Stratonovich stochastic calculus, by choosing an appropriate numerical integration scheme [51]. This method was first applied to the dynamics of noninteracting particles [51] and later also to interacting particle systems [13] (see Fig. 3.5). 

Thus the wave function of a system of noninteracting particles is the product of wave functions for each particle, and the energy is the sum of energies for each particle. 

The theory of irreversible diffusion-controlled reactions is discussed in Chapter 6 the effects of particle Coulomb and elastic interactions are analyzed in detail. The many-particle effects (which in principle cannot be explained in terms of the linear theory) are demonstrated. Special attention is paid to the pattern formation and similar particle aggregation in systems of interacting and noninteracting particles. 

The Fermi energy (level) is derived from the Fermi-Dirac statistics which describe the distribution of indistinguishable, noninteracting particles in n available... 

The most familiar example is the energy spectrum of a nuclear hamiltonian that can be described by a gaussian distribution. It must be remarked that for a system of many, say N, noninteracting particles the gaussian distribution of the eigenvalues is a direct consequence of the central limit theorem for large values of N. 

When these matrices are applied in an N-particle configuration space (N k) the moments of the hamiltonian behave on ensemble averaging in the limit of large dimensionality as in the case of noninteracting particles without averaging. This reflects the dominance of binary correlations in the operator products HP when the ensemble averaging is performed in the dilute system (k N). 

In fact, the initial conditions are not important because of dissipation (the memory about the initial state is completely lost after the relaxation time). However, in some pathological cases, for example for free noninteracting particles, the initial state determines the state at all times. Note also, that the initial conditions can be more convenient formulated for Green functions itself, instead of corresponding initial conditions for operators or wave functions. 

On the basis of Assumption (3), the drag force of a single noninteracting particle Fpo is written as... 

Thermodynamics deals with relations among bulk (macroscopic) properties of matter. Bulk matter, however, is comprised of atoms and molecules and, therefore, its properties must result from the nature and behavior of these microscopic particles. An explanation of a bulk property based on molecular behavior is a theory for the behavior. Today, we know that the behavior of atoms and molecules is described by quantum mechanics. However, theories for gas properties predate the development of quantum mechanics. An early model of gases found to be very successftd in explaining their equation of state at low pressures was the kinetic model of noninteracting particles, attributed to Bernoulli. In this model, the pressure exerted by n moles of gas confined to a container of volume V at temperature T is explained as due to the incessant collisions of the gas molecules with the walls of the container. Only the translational motion of gas particles contributes to the pressure, and for translational motion Newtonian mechanics is an excellent approximation to quantum mechanics. We will see that ideal gas behavior results when interactions between gas molecules are completely neglected. 

Problem 12). If the noninteracting particles comprising the system are identical, then they have the same molecular partition function and... 

The other components, namely, b f and b j, may be constructed straightforwardly using their relations with the given ones [see Eqs. (4.192)]. For a random system, that is, for an assembly of noninteracting particles with a chaotic distribution of the anisotropy axes, the average of any Legendre polynomial is zero, so that b[1> = b, and the linear dynamic susceptibility reduces to... 

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