Are continuous mathematical models of discrete physical phenomena messy because of a disconnect between "continuous" and "discontinuous"? Examples from statistical mechanics and continuum mechanics abound: a discrete phenomenon (e.g. kinetic energy of molecules) is "averaged" out over the constituents of the system to which it applied (continuing the example: temperature, or pressure), and then calculations are simplified for certain special cases (e.g. ideal gases), at the cost of increasing "messiness" as increasingly general cases are considered (e.g. non-ideal gases).
I am not familiar with combinatorics and discrete mathematics, so please be gentle: are there trains of thought being explored in modern research that wonder if discrete physical phenomena might be better modelled by discrete mathematics?
If such a direction is not being seriously looked into because discrete math doesn't model discrete phenomenon in a neater way, why are discrete math models of discrete physical phenomena as messy as continuous math models of discrete physical phenomena?
P.S. What resources could I use to further read about related subjects?
 A: I would not say that "increasing 'messiness' as increasingly general cases are considered" is a "cost" of the continuum approximation or averaging procedure.  The more general cases would be far messier still without the continuum approximation/averaging.
There are branches of statistical mechanics in which discrete phenomena are modeled by discrete mathematics, although temperature is still continuous in the models I'm thinking of, and the Boltzmann average over all system configurations is still done.  (After all, the point of statistical mechanics is to derive macroscopic behavior by averaging over microscopic degrees of freedom.)
The primary example I'm thinking of is the Ising model, where space is a lattice and the variables associated with each lattice site also take discrete values.  There are many approaches to the study of the Ising model, some of which make use of developments in discrete mathematics.  In particular, the method of dimers, developed by Michael Fisher, is related to the theory of perfect matchings in graph theory, which is one of the main branches of discrete mathematics.
This example proves the point, however.  The model has only been solved exactly in two dimensions with the external magnetic field set to zero.  In that case, the free energy, spontaneous magnetization, and correlation functions have all been computed exactly.  But the computations are not simple, and quantities such as the magnetic susceptibility still cannot be computed exactly.  Furthermore, the ideas of scaling theory, renormalization group, and conformal field theory eliminate some of the complexity of these discrete computations by looking at the system close to its critical point, where a continuum limit applies.
Discrete mathematics is a young field compared with analysis, and a big part of the reason for this appears to be the inherent difficulty of discrete mathematics.
