Particle creation

In a dense system, the acceptance rate of particle creation and deletion moves will decrease, and the number of attempts must be correspondingly increased eventually, there will come a point at which grand canonical simulations are not practicable, without some tricks to enliance the sampling. 

In the simplest version, a one-component system is simulated at a given temperature T in both boxes particles in different boxes do not interact directly with each other however, volume moves and particle creation and deletion... 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

The terms in delta functions will be integrated to get the invariant 5 , which is still space- and time-independent. Hence, these two Green s functions can be combined as they can have the same value at the moment of particle creation (t = 0) and at this time only these terms are non-zero. This expression needs a little further manipulation. Following Lebedev et al. [506] on the variational principle, drop the term V ZJe- 17 VGG and include a new term of the form V (Ae fJUG G — e u4>G — e u4tG )r where f is a unit radial vector and

scalar functions of r which are related directly to the boundary conditions which both G and G satisfy. Multiply the delta function term by 2. 

The existence of the (quasi) steady-state in the model of particle accumulation (particle creation corresponds to the reaction reversibility) makes its analogy with dense gases or liquids quite convincing. However, it is also useful to treat the possibility of the pattern formation in the A + B —> 0 reaction without particle source. Indeed, the formation of the domain structure here in the diffusion-controlled regime was also clearly demonstrated [17]. Similar patterns of the spatial distributions were observed for the irreversible reactions between immobile particles - Fig. 1.20 [25] and Fig. 1.21 [26] when the long range (tunnelling) recombination takes place (recombination rate a(r) exponentially depends on the relative distance r and could... 

Let us apply general stochastic equations (2.2.15) to the simple A+B —> C reaction with particle creation - the model problem discussed more than once ([84] to [93]). A relevant set of kinetic equations reads... 

The potential m, is defined similarly. As it is seen from equation (2.3.50), the mean force acting on a particle A at coordinate fhas both the contribution from direct interactions within a group of (m + m/)-particles and indirect interactions (integral terms). If the particle creation and recombination terms were absent, the steady-state solution of equation (2.3.47) would correspond to putting the fluxes and J m, equal zero. The corresponding set of... 

Summing up, note that the direct statistical (computer) simulation does not demonstrate serious errors of the superposition approximation for equal reactant concentrations. Divergence begins first of all for unequal concentrations for not very large reaction depths T 2 it is almost negligible, at T > 2 and especially asymptotically (as t —> oo) it becomes important, but the complete quantitative analysis cannot be done due to unreliable statistics of results. In this Section we have restricted ourselves to the A + B —> 0 reaction without particle generation. Testing of the superposition approximation accuracy for the case of particle creation will be done in Chapter 7. 

As earlier in the case of diffusion-controlled concentration decay, one-species reactions, A +A -> A and A + A —> 0, in one-dimension, could be used as a proving ground for the testing theory of particle accumulation, when random particle creation is added. For the former reaction the master equation could be easily derived in the form [95]... 

These ideas have been applied to gravitational particle creation at the end of inflation by Chung, Kolb Riotto (1998) and Kuzmin Tkachev (1998). Particles with masses M of the order of the Hubble parameter at the end of inflation, Hi 10-6Mpi 1013 GeV, may have been created with a density... 

Equation may be used to model a CSTR by modifying the vector c to allow for the removal of latex particles in the effluent stream while incorporating the particle creation terms of the type discussed above. The form of the term describing latex particle removal is given simply for all i by... 

Equation (22) follows from Eq. (5) if the vector c is given the form appropriate to CSTRs [see Eq. (18)j for removal of latex particles in the effluent stream and if the particle creation terms are considered as boundary cooditions operating at Smith-Ewart coupling matrix vanishes on summation because of the conservation of panicle numbers. The value of is given by... 

The shape of a zeolite sorption uptake isotherm, a quantitation of the amount of a given sorbate taken up as a function of its partial pressure in the gas phase in equilibiitun with the zeolite sorbent, depends both on the zeolite sorbate interaction and on the sorbate - sorbate interactions. Simulation of such isotherms for one or more sorbates is accomplished by the Grand Canonical Monte Carlo method. Additional to the molecular reorientation and movement attempts is a particle creation or annihilation, the probability of which scales with the partial pressure [100,101]. This procedure thus simulates the eqmlibrium between the sorbed phase in the zeolite and an infinite gas / vapor bath. Reasonable reproduction of uptake isotherms for simple gases has been achieved for a small number of systems (e.g. [100,101]), and the molecular simulations have, for example, explained at a molecular level the discontinuity observed in the Ar - VPI-5 isotherm. 

This new definition of normal ordering changes our analysis of the Wick s theorem contractions only slightly. Whereas before, the only nonzero pairwise contraction required the annihilation operator to be to the left of the creation operator (cf. Eq. [84]), now the only nonzero contractions place the q -particle operator to the left of the -particle creation operator. There are only two ways this can occur, namely. 

The analogous contractions that place the -particle annihilation operator to the right of the -particle creation operators are zero ... 

The algebra of the composite particle creation operators ipf is more complicated. Denoting the adjoint operators by ipf, the commutation rules, in the general case, read ... 

A proper definition of (quasi-)particle-creation and (quasi-)particle-annihilation operators an and a is provided by diagonalization of the (time-independent) unperturbed part Ho = //ext+Z/e-e of the total Hamiltonian. After the iteration is performed (e.g. on the Dirac-Fock level) the latter may be cast into the form... 

While evaluating the matrix elements of H exp(T ) between cind (j> , it becomes convenient to rewrite H in normal order with respect to d>v as vacuum. This simplification of computation of matrix-elements was noted earlier by Hose and Kaldor [17] and has since been exploited by many workers [14,15,25,26]. Since T" has only excitations out of it has only hole-particle creation operators defined with respect to d>v and consequently exp(T ) is in normal order with respect to (jx. Using Wick s theorem, we then find... 

The field (x), l (a ) is classical in the sense that no particle creation and annihilation is yet introduced. 

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