If I understand correctly looking at the first law of thermodynamics we have a microscopic work term (heat: $Q$) and macroscopic work term ($W$)

$$ \Delta E = \Delta Q + \Delta W$$

The macroscopic work done can be nicely expressed as: $ \Delta W = F \Delta x= PA \Delta x = P\Delta V $

Heat is expressed as $$ \Delta Q = T \Delta S$$
Where the temperature and the entropy difference are quite abstract concepts. Temperature if I understand correctly is a function of average kinetic energy:

$$ T \propto <E\_k> =\frac{1}{2} \bar{mv^2}$$

Where $ <E\_k>$ is the expected value of the kinetic energy of a molecule, or average energy of a molecule.

My question is: is there a way of describing heat as microscopic work? Something along the line of:

$$ \Delta Q = \Delta \sum\_i E\_{k,i} = \sum\_i \Delta (\frac{1}{2}m\_i v\_i^2) $$
It seems that since heat 'flows' this can't really be it.