Is there a formula of heat that expresses it in microscopic work done? 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.
 A: Your last equation is an expression for the change in internal energy, specifically, the change in the molecular kinteic energy component of internal energy, not heat. There is no such thing as a "change" in heat ($\Delta Q$). Heat is transfer of energy due solely to temperature difference.
You can think of heat as the transfer of molecular kinetic energy (KE) from a substance where the average KE of its molecules is higher (as measured by higher temperature) to a substance where the  average KE of its molecules is lower (as measured by a lower temperature).
In the case of heat transfer by conduction, that transfer of kinetic energy is the result of, on average, high kinetic energy molecules colliding with lower kinetic energy molecules such that the higher kinetic energy molecules lose energy and the lower kinetic energy molecules. In effect, the transfer is the result of microscopic work being done by the high KE molecules on the low KE molecules, transferring energy from the high KE molecules to the low KE molecules.
Hope this helps.
A: Thinking of heat as “microscopic work” invites deep confusion, I suggest.
Heat is a very special type of energy transfer—that driven by a temperature difference.
Work is every other type of energy transfer not involving mass transfer.
If you wish to distinguish heat and work microscopically, consider McQuarrie’s dichotomy: work elevates energy states in concert while maintaining their population, whereas heating expands the population of energy states without altering them (p. 44, Statistical Mechanics).
