They're different kinds of storage, with principal differences in access speed, sharing between cores, physical location and so on. No mathematics needed for a working understanding.
The differences in access speed, density, and power between different types of caches are due to how our physical world works.
Therefore,uUsing a physical analogy is totally appropriate since we see the same constraints in different contexts; namely, there's an energetic cost to be paid for increased physical locality, and this is intuitively obvious to most people.
The specific details (e.g. eviction policy, cache topology, etc.) are mostly irrelevant for understanding this key point.
Analogies are unhelpful here. I don't need an analogy to help people understand that different kinds of memory have different characteristics. I can just say it.
It is easier to understand if I simply state that there are different kinds of memory, and that they have different constraints in terms of speed, density and power. That is really, really easy to understand and from there I can accurately reason about it.
I don't need an analogy to understand any of this. Just saying that different kinds of memory have different speed, densities and power consumptions explains everything and provides all the intuition needed.
They're different kinds of storage, with principal differences in access speed, sharing between cores, physical location and so on. No mathematics needed for a working understanding.