Particle density, random number generators

The random selection in step (iii) is carried out by generating uniform random numbers U e [0, 1], For example, the index of a random particle selected from a set of N particles will be n = intup(//N) where intuP() rounds the argument up to the nearest integer. Note that for constant-density, statistically stationary flow, the effective flow rates will be constant. In this case, steps (i) and (ii) must be completed only once, and the MC simulation is advanced in time by repeating step (iii) and intra-cell processes. For variable-density flow, the mean density field ((p)) must be estimated from the notional particles and passed back to the FV code. In the FV code, the non-uniform density field is held constant when solving for the mean velocity field.15... 

To carry out this crude estimation one could use random numbers to choose for the particles of the system random positions uniformly distributed in the volume V of the system. That is easy. For each system configuration so generated one would calculate the energy I/(q ) [for the example (2)] and the probability density p(q ). The average of the product t/(q )p(q ) over many such configurations is evidently an unbiased estimate of / according to (2). 

---