4.2 Generation process

The value of the wavevector for newly generated particles using a single random number sampling has been discussed in Section 3.7.5. A problem in this approach arises, if the finite range of the multi-index q is considered.

In (4.1) the variable s is bounded by the finite coherence length L. To ensure that the Wigner transform and its inverse are unitary, the transformed variable q must also be bounded. The range of (any component of) the multi-index q is restricted by

The momentum offsets of the newly generated particles (ℓ₁ and ℓ₂) should be such that their momenta remain within the set bounds, i.e. (q + ℓ₁,q + ℓ₂) ∈. The problem arises if particles are generated in pairs with momenta q ±ℓ, using a single sampling of the distribution function (or ). To maintain a balance of positive- and negative-signed particles, it seems reasonable to reject both particles and sample the distribution again, until a momentum offset appears which renders both momenta valid simultaneously. Such a rejection technique clearly influences the statistics. Moreover, the probability for a valid particle pair to be generated decreases, the closer the momentum of the generating particle is to the limit ±K, as illustrated in Figure 4.11. In such a case, a small momentum offset is more likely to produce a pair of generated particles with valid momenta, thereby unfairly promoting the generation of particles with small offsets in momenta and influencing the momentum distribution of the particle ensemble as a whole. Due to this biasing, particles with a high momentum persist much longer, since it becomes impossible for particles to be generated with significantly different momenta.

(a) (b)

Figure 4.11: A particle can generate a particle pair only with momenta in the valid range (a). If a symmetric offset (q′) is use for both generated particles the statistics are biased towards smaller offsets such that the new momenta remain in the valid range (b).

To avoid this systematic biasing of the statistics, the generated particles should not be rejected in pairs, if one is assigned a momentum which is out of bounds, but rather only a single momentum offset for the invalid particle should be regenerated until a valid momentum (inside the finite bounds) is obtained. In practice, if no valid momentum can be obtained after a set number of attempts, both of the generated particles are ’destroyed’. This ensures that the balance between positive and negative particles is maintained at all times.

This optimized particle generation algorithm enables particles with a high momentum to again return to a lower momentum; the persistence of high-momentum particles is no longer promoted by the generation statistics.