I dimly remember this topic came up before: random number generators without internal state.

Here is a talk given at GDC about using noise pattern generators instead of traditional random number generators:

The basic idea is to replace the usual
rnd_seed(some_seed);
number = rnd_get();
number = rnd_get();
...

with
number = noise_get(1, some_seed);
number = noise_get(2, some_seed);
number = noise_get(3, some_seed);
...

The talk explains in good detail how this can give better randomness with more control, while being faster to compute. I reference it here because this kind of noise generators should be much better suited for GPU programming.

Linkback: https://fractalforums.org/programming/11/state-less-random-numbers/3936/