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When the process changes, cr3 change to point to the page table of the new current process.
This creates a problem: the TLB is now filled with a bunch of cached entries for the old process.
A simple and naive solution would be to completely invalidate the TLB whenever the cr3 changes.
However, this is would not be very efficient, because it often happens that we switch back to process 1 before process 2 has completely used up the entire TLB cache entries.
Basically, the OS assigns a different ASID for each process, and then TLB entries are automatically also tagged with that ASID. This way when the process makes an access, the TLB can determine if a hit is actually for the current process, or if it is an old address coincidence with another process.
The x86 also offers the invlpg instruction which explicitly invalidates a single TLB entry. Other architectures offer even more instructions to invalidated TLB entries, such as invalidating all entries on a given range.