One example of non-living matter tending towards lower entropy is the formation of mineral crystals in void spaces in rocks. This does require a high-energy in

One example of non-living matter tending towards lower entropy is the formation of mineral crystals in void spaces in rocks. This does require a high-energy input, i.e. heated water dissolving large volumes of silica which later slowly crystallize. The entropy of the silicon and oxygen atoms in the quartz crystal is lower than that in the dissolved aqueous state. Similar effects are postulated to be involved in the origin of life, with energy sources like oceanic hydrothermal vents providing the energy sources driving the synthesis of complex organic molecules which eventually developed the capability of self-replication, aka decreasing randomness, increasing order, lowering local entropy. A mechanistic 'non-living' model of life, entropy-wise, could be a waterwheel driven by a river (of sunlight and geothermal energy) which operates a sawmill, a steel mill, a chip fab, a robot factory, a paper mill and a printing press - with each generation of robots building more waterwheel-based units based on the instructions (DNA) provided by the printing press. This all relies on a robust source of energy, since the Gibbs equation (dg = dh - tds) says that for a process to move forward, the energy release (dh) must be greater than the entropy reduction (tds) that it is coupled to. Such a system meets all mechanistic definitions of life without being alive... a philosophical conundrum I suppose.