It's quite easy. The money you "inject" needs to have demurrage.
You can then give all of the $2 trillion demurrage dollars directly to the most corrupt individual on the planet and it won't matter.
There are two scenarios:
1. The recipient keeps the money: It will automatically be returned.
2. The recipient spends the money: It has flown from the most corrupt individual on the planet to a less corrupt individual.
If we assume that each transaction allows the next recipient to make a decision on how to best spend the money, it would automatically spread out, since it cannot stagnate and pool up at any given individual.
Short term corruption doesn't pay, because accumulating excess money that you don't need via corruption just means your demurrage fee is higher.
Investing or lending money means giving it to someone that needs it more than you do and since the demurrage defeats the zero lower bound, it is rational to do so even at 0% interest since it allows you to avoid the demurrage.
And here is something that isn't intuitive for people whose mental model assumes permanent capital scarcity. If supply and demand on the money capital markets are balanced out, the expected interest rate is 0%.
Lower capital costs mean that products can be cheaper now. It is also much easier to invest into long term projects, e.g. against climate change, since the investment only needs to break even rather than growing the economy even further.
Oh yeah, also you now get full employment, like classical economic theory predicts. No more automation scare and feeling like the economy is trying to get rid of you or justify removing working class humans or being hostile to them.
The absence of unemployment makes welfare unattractive, which in turn means that governments will simultaneously have more tax payments and less welfare expenses simultaneously. All welfare should be paid out as demurrage currency in all countries. It's completely logical.
Since capital markets are now fair, it is possible to reduce total debt by paying off loans, whereas with permanent money the holder of money can always decide "I'll spend it later" thereby delaying the possibility of paying off a debt using that money, which is implicitly an extension of the duration of the debt. Given an infinite holding duration, the expected duration of the corresponding debt is obviously infinite too (eternal growing debt woooo).
Oh and it could have saved pension systems if it had been adopted ahead of time. Pension systems collapse because shrinking populations yield less output in the future but permanent money tells you that you can build a time machine to take labor from today and just teleport it straight into the future. Who needs young people working for you, if you can let the money work for you instead! Everyone knows the value of a dollar bill is backed by a humanoid robot inside the dollar bill that can perform exactly one dollar of work.