It's not uncommon to use with to introduce a clause like this:
A particle's energy state jumps about randomly, with such transitions governed by the temperature of the system.
What's the grammar rule validating this expression and the use of with with the past participle?
If there is grammatical terminology to describe this, that would be preferred.
With is a preposition. The prepositional object is such transitions governed by the temperature of the system. You can use with to form an adverbial phrase of condition or circumstance, as in this case. A simple example:
The balloon popped with a loud bang.
Within the prepositional object in your example, you could say the participle or participial phrase is predicative, which means that it "actively" assigns a property to its head (the noun) the way a subject or object complement does; this is opposed to attributive adjectives, which merely "passively" mention a property of a noun. The "fact" that it is predicative can be seen by the fact that it is clearly linked to a specific head-noun (such transitions) but appears after it rather than before it; attributive modifiers normally appear before their heads, while predicative modifiers normally appear after them.
This predicative use sounds very slightly informal to me when you use it after with.
The following is the same kind of adverbial constituent with internal predication, but in a simpler sentence:
She left the area, with her daughter in the back seat.
Here the adverbial phrase/constituent with her daughter in the back seat has an internal predication: the adverbial phrase in the back seat assigns a property (location) to its head, her daughter. So the adverbial phrase in the back seat serves the same function as the participle governed does in your example.
