Particles conservation

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

Thus this operator must satisfy the necessary demands of the particle conservation law and detailed balance principle as was the case for its simpler analogue in Eq. (4.23) and Eq. (4.24). They are... 

Eq. (5.8) formulates the same particle conservation law that was expected to hold for any f(0) in Eq. (4.65). The meaning of Eq. (5.9) becomes clear if one looks for rotational energy relaxation, which obeys the equation... 

It is noteworthy that, having assumed Eq. (5.26), we no longer can keep the diagonal elements unchanged. They must be corrected to prevent breakdown of the particle conservation law. The requirement expressed in Eq. (4.65) has to be used to define diagonal elements via other already corrected elements ... 

In analytical investigations it is often desirable to leave the particle number free and consider operators that fix only the parity, but in applications to electronic structure theory one deals with fixed particle number and one may restrict A to have a definite action on the particle number N, so that A+A is particle conserving. There are then two cases for the one-body operator A consideration of A = with undetermined coefficients gives rise to the... 

One may now proceed to write out y+y and w+w and, to make the connection to the present work, retain only the terms that are particle conserving. The result are representability conditions, and they include terms quadratic in/I and terms quadratic in/2, but no mixed terms. It will be clear then that the extreme conditions—and they are all that matter—involve either /I or /2, but not both moreover, one will observe that 0 < y+y and 0 < w+w lead to the same conditions, which are the real cases of the T1 and the strengthened T2 conditions. 

Imposing the particle conservation condition, which introduces a new Fermi level A, we can define the so-called generalized occupation numbers... 

Let us denote with E rii ) the stationary value of the functional (32) with respect to arbitrary occupation numbers rii = (ni,rz2,...) = n for the orbitals 4>i, satisfying the particle conservation condition, i.e. ... 

For the treatment of a mixture of para-H2 and ortho-H2, we next compute reven> the fraction of H2 pairs associated with even (, and r0dd> the fraction associated with odd f, and rno, the fraction without exchange symmetry (para-H2-ortho-H2 pairs). Particle conservation requires... 

Here V(m ) is the probability distribution for the generalized mean size in the first phase, taken over partitions with fixed and N with equal a priori probabilities. Note that given m, irP is fixed in the second phase by the moment equivalent of particle conservation iV W1) + N mPl = Nm(° The integral in (17) can be replaced by the maximum of the integrand in the thermodynamic limit, because In V(m ) is an extensive quantity. Introducing a Lagrange multiplier pm for the above moment constraint then shows that the quantity pm has the same status as the density p = p0 itself Both are thermodynamic density variables. This reinforces the discussion in the introduction, where we showed that moment densities can be regarded as densities of quasi-species of particles. 

Bearing in mind that only this element of M differs from zero, we obtain from the matrix IET equation (3.469) and the particle conservation law (3.460) a single integral equation ... 

Utilizing the particle conservation equation, the left-hand side terms can be expressed as the time rate of momentum change plus the particle source contribution (if any)... 

The previous section considered the derivation of second quantized Hamiltonians that can be used in post-DHF calculations. From now on we will regard the matrix elements of h and g as (complex) numbers and direct the attention to the associated operators. By applying the no-pair approximation we retained only particle conserving operators in the Hamiltonian. Such operators can concisely be expressed using the replacement operators Eq = a p Q and... 

We should remark that periodic collisions of a particle with walls of the chosen potential well, in which a particle oscillates, are not regarded as strong collisions, since energy of a particle conserves after such collisions. 

Insofar as two-body scattering among similarly charged particles conserves momentum, hence current, r is unrelated to the electrical conductivity. Nevertheless, x may figure in connection with other relaxation processes, and is experimentally observable. 

In this article, we shall discuss fluid systems. The collective (Na = TV, where N is the total number of particles) conserved variables are the number density. 

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