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There are several steps you can take to speed up decomposition of organic material. Decomposition of organic debris is both a physical and chemical process. Some of the steps listed below are needed to counteract some of the side effects of mulching that slow the mulching process.
The amount of heat generated by your mulch pile is an indicator of how fast breakdown is occurring. More steam from the pile interior when you mix it indicates a faster chemical breakdown but this higher heat also means you will need to perform regular physical mixing of the pile and keep adding water to keep the pile from getting too hot and killing the fungus and bacteria inside (slowing breakdown). It is entirely possible if you don't keep the pile wet enough it may get hot enough to come close to ignition, so keep a close eye on the pile during early mulching.
Some of the other answers here mention adding worms to the pile. The problem with this is that if you are trying for a fast breakdown, the heat inside will tend either cook any worms inside or drive them out of the pile. Worms are great as natural aerators in slow decomposition mulch piles or in normal soil, since neither of these create the large amounts of heat that "fast" decomposition does.
1) First, use mechanical force to break the contents of your mulch pile into as many small pieces as possible. Set your wood chipper to "liquefy". The smaller the starting mulch pieces, the faster the mulching breakdown process. Thin organic matter like tree leaves and grass break down faster than wood chips and twigs which break down faster than large wood pieces like branches.
2) Add water as needed to keep your pile wet. Moisture encourages faster breakdown by encouraging bacterial and fungal growth. Usually keeping the pile covered by a plastic cover sheet keeps too much water from escaping the pile and keeps the surface of the pile moist.
3) Mix your mulch pile on a daily basis using a garden fork. If your pile is breaking down quickly, you will notice steam coming from inside of the pile due to the heat from chemical reactions during breakdown. Mixing allows heat to escape from the pile as well as spreading bacteria and fungus around and aerates (allows air to mix into) the pile. Add more water as needed to help keep things moist as well as to cool the mulch.
4) Mix some crushed lime into the pile using a garden fork a few days after starting the mulching process. The acid level of your pile will increase over time as your mulch pile breaks down, and this slows bacterial growth. Adding lime will decrease acidity, encouraging bacterial growth and speeding breakdown.
The steps listed above apply to any mulch pile, but if you added a lot of wood to your pile when you created it, there are two further steps you might want to use. Wood in a mulch pile changes the chemical breakdown process.
5) Mix nitrogen fertilizer (such as ammonium nitrate) into the mulch pile. Nitrogen is used up during chemical breakdown of wood fiber, and insufficient nitrogen slows this breakdown.
6) Sprinkle/mix fungi into the pile. Fungus (such as mushrooms) breaks down the physical structure of wood pieces so the bacteria can get into the interior of the wood chips.
Eventually your mulch pile will no longer be warm and you will no longer see steam escaping when you mix it. This means that the breakdown of the mulch is slowing and is mostly complete. There may still be some wood chips mixed into the black organic crumbly mass of the decomposed mulch. Once the mulching breakdown slows like this, you can choose to age the pile for a few months longer to allow the mulch to completely break down, or if you are in a hurry you can use the mulch as is.
If you choose to age the mulch further, you won't need to mix the pile as often as you did during the early mulching process. Continue to mix the pile once a week or so and add water when needed to keep it moist. Continued aging will allow any remaining un-decomposed matter such as wood chips to completely decompose.
If you decide to use the mulch without aging it completely and wood still remains in the mulch, you should add nitrogen fertilizer to the mulch before using it on plants. If you don't add nitrogen, you risk starving the plants if when the wood decomposes it uses up all the nitrogen out of the soil.