Interaction between blackbodies in relation to the second law

[Shunyata]

I am trying to understand the behavior of blackbodies in interaction with each other.

Conditions as follows.

$1)$ There are two blackbodies, say $B_1$ and $B_2$, with corresponding temperature $T_1$ and $T_2$.

$2)$ Initially $T_1>T_2$.

$3)$ $B_1$ and $B_2$ have thermal contact only through radiation exchange.

And finally my question,

Can $B_2$ give energy to $B_1$ or is this prevented by the second law?

 

[cjl]

Sure it can. In fact, it always will be. However, B1 will always be giving more energy to B2 than the reciprocal effect, so the net energy exchange will always involve B1 giving energy to B2.

 

[Shunyata]

Is it as simple as

$dQ\Big( \frac{1}{T_2} - \frac{1}{T_1}\Big)=dQ\Big( \frac{T_1-T_2}{T_1T_2}\Big)>0,$

if $dQ=dQ_1 - dQ_2$ is the net-heat where positive direction is from $B_1$ to $B_2$ and $dQ_1>dQ_2$?