  Information ratchets extract work from heat baths using low entropy
information reservoirs. We find that the structure of the reservoirs affects
both their performance and phase diagram. By deriving exact probabilities of
recurrence in d-dimensional reservoirs we calculate the net extraction of work.
Performance is characterised by two critical dimensions (i) For d < 3 driven
exploration of the reservoir is essential to extract work (ii) For d > 4 purely
diffusive exploration is optimal. This has important consequences for
computation where sequential operations constitute a 1D path.
