Circular security notions for PKE schemes capture the security of (PKE) schemes when encrypting the secret decryption key.
Is there an analogous notion but for encrypting the randomness used for the encryption? i.e. what if one first fixes the random coins $r$ to be used for for the encryption, and then computes the ciphertext for this $r$: $c := E_{pk}(m; r)$, with $m = r$?