Continuum assumption: validity & motivation I still can't grasp why the so-called continuum assumption can be taken as reasonable under the proper conditions.
Let's consider the space-time microscopic distribution of a generic tensor field. If I average the field over space-time regions which are:


*

*small enough with respect to the macroscopic space-time scales under investigation

*still big enough to contain a huge amount of samples
Than I know that, no matter the randomness of the underlying process, the dispersion of my average result will be close to zero (because of central limit theorem of statistics). So far so good.
But here is my question: how can this be a valid justification to ignore the microscopic dynamics while retaining only an average value? In other words, how can I be sure that the microscopic fluctuations (which could be of high amplitude both in space and in time) do not affect significatively the result of my macroscopic problem?
 A: If I'm understanding your question correctly, you're wondering about "outlier states"—that is, states of the tensor field that are unlikely, but would make a large dent in the macroscopic properties of the continuum in question. An example would be having every free molecule in a liquid suddenly align their microscopic motion rightwards; the kinetic energy of the liquid would spike quickly and massively.
If events like these were likely, then of course we'd see the effect of these states on the averaged (a.k.a. continuum) model. If events like these do not perturb the continuum properties very much, then the continuum model would represent the microscopic physics fairly well. So we only need to consider the "low probability—high impact" scenario.
The key to deciphering why these states aren't considered in most continuum models is that the underlying probability density function for the microscopic states usually evolves continuously with respect to space and time. 
Consider the case of a molecule interacting with a neighboring molecule. Although we might imagine the interaction behavior to be like a collision, where the molecules' momentum suddenly changes from one thing to another, it actually changes sharply but continuously—the molecule is repelled by whatever sharp but continuous intermolecular force the molecules share with until they reach a "post-interaction" value.
As a result, the probability density function for most many-body systems obeys a transport equation in state space. Roughly this means that for the continuum to "experience" such an outlier state, it needs to "traverse" a bunch of improbable, low-impact states, a process which is far more unlikely than simply popping into an outlier state.
That doesn't mean microscopic, zero-average phenomena can't influence macrodynamics—consider diffusion, for example—it just guarantees that there are nice macro-analogues to them that can be readily incorporated into a continuum model.
Hope this helps!
