One-particle interactions

As the concentration is increased above q> 0.01, hydrodynamic interactions between particles become important. In a flowing suspension, particles move at the velocity of the streamline corresponding to the particle centre. Hence particles will come close to particles on nearby streamlines and the disturbance of the fluid around one particle interacts with that around passing particles. The details of the interactions were analysed by Batchelor17 and we may write the viscosity in shear flow as... 

Obviously, the one-particle interactions can be subdivided into those involving a nucleus and those involving an electron. We consider each type in turn. 

Figure 8. Second-order diagrams involving a one-particle interaction in the perturbation theory of nuclei and electrons.

In particular, the Hartree-Fock Hamiltonian is of this form. We are interested in improving such an independent-particle description of a many-electron system by means of perturbation theory. For the sake of simplicity, we will first consider the case where the perturbation is also a sum of one-particle interactions. 

In the special case that the perturbation is a sum of one-particle interactions, the total Hamiltonian of the system... 

Flocculation proceeds by biparticle collision when the surface, covered by the polymer, of one particle interacts with the uncovered surface of another particle. The change in the number of particles in a xmit volume of dispersion is proportional to the surface area of the particle covered by the polymer, S i, the number of these particles, Nj, and the surface area of particles not eovered by the polymer (Sf — S i) ... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

No more than one particle may occupy a cell, and only nearest-neighbour cells that are both occupied mteract with energy -c. Otherwise the energy of interactions between cells is zero. The total energy for a given set of occupation numbers ] = (n, of the cells is then... 

In many colloidal systems, both in practice and in model studies, soluble polymers are used to control the particle interactions and the suspension stability. Here we distinguish tliree scenarios interactions between particles bearing a grafted polymer layer, forces due to the presence of non-adsorbing polymers in solution, and finally the interactions due to adsorbing polymer chains. Although these cases are discussed separately here, in practice more than one mechanism may be in operation for a given sample. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

This example shows the round particle in cell B,B with two possible nonbonded cutoffs. With the outer cutoff, the round particle interacts with both the rectangle and its periodic image. By reducing the nonbonded cutoff to an appropriate radius (the inner circle), the round particle can interact with only one rectangle—in this case, the rectangle also in cell B,B. ... 

The assumption that the probability of simultaneous occurrence of two particles, of velocities vt and v2 in a differential space volume around r, is equal to the product of the probabilities of their occurrence individually in this volume, is known as the assumption of molecular chaos. In a dense gas, there would be collisions in rapid succession among particles in any small region of the gas the velocity of any one particle would be expected to become closely related to the velocity of its neighboring particles. These effects of correlation are assumed to be absent in the derivation of the Boltzmann equation since mean free paths in a rarefied gas are of the order of 10 5 cm, particles that interact in a collision have come from quite different regions of gas, and would be expected not to interact again with each other over a time involving many collisions. 

If H is the one-particle energy operator, the total energy operator for non-interacting systems can be written 2a HafaA or 2a The... 

The appropriate expression for the operator H in the above equations is that appearing in Eq. (8-160). Eor a first example, consider an ideal gas without interactions. Assuming that the one-particle wave functions used in the population density operators are the energy eigenfunctions, then the matrix H0llA is diagonal, and we can write... 

It is a characteristic feature of all these relativistic equations that in addition to positive energy solutions, they admit of negative energy solutions. The clarification of the problems connected with the interpretation of these negative energy solutions led to the realization that in the presence of interaction, a one particle interpretation of these equations is difficult and that in a consistent quantum mechanical formulation of the dynamics of relativistic systems it is convenient to deal from the start with an indefinite number of particles. In technical language this is the statement that one is to deal with quantized fields. 

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