characteristic vectors for systems The question is motivated from a physics problem:

Let's first discuss the 1D infinitely long discrete system on a lattice, a system can look like:

system 1:

...(ABAC)(ABAC)(ABAC)...

this leads to its characteristic (or correlation) vector 

&lt;A&gt;=0.5 this is the appearing frequency of character A
&lt;B&gt;=0.25
&lt;C&gt;=0.25
&lt;AB&gt;=0.25 this is the appearing frequency of character AB


....

system 2:

...(AB)(AB)(AB)...

this leads to its characteristic (or correlation) vector

&lt;A&gt;=0.5
&lt;B&gt;=0.5
&lt;C&gt;=0
&lt;AB&gt;=0.5
...


Such infinite characteristic vectors may be able to uniquely correspond to any 1D system (I don't have proof)... And by using this types of vectors (its finite version), it is easier to build machine learning models based on vectors instead of the structures itself.

This idea can extend to 2D, 3D systems naturally and this is what we call cluster expansion or generalized ising model....

The problem is, in physics, system is not necessarily in a lattice:

a description of a system (for 1D) is usually:

System 3:
1, its periodicity vector
2, atomic positions inside its unit cell

for example, a 1D system can be:
1, periodicity vector (3)
2, atomic positions (A at 0, B at 1.5)


this is ---A    B     A     B---- system (atoms are only at specific coordinate, voids are everywhere else)

Is there any established theory/mathods in mathematics/computer science about its characters vectors/functions, that may be able to be used to build machine learning models? Thank you.
Yes.  You want to look at the bag-of-words model and n-gram models.

The bag-of-words model corresponds to the part of your characteristic vector: namely, the parts &lt;A&gt;, &lt;B&gt;, &lt;C&gt; of your characteristic vector, i.e., the frequencies of a single letter.  These are used as features for the ML algorithm.

n-gram models generalize this to a substring of length n.  For each substring of length n, we count the frequency of that substring.  These frequencies are again used as features.  For instance, we'd have a feature for the frequency of AB, the frequency of AC, and so on.

In real-world computer science, we don't deal with infinite systems; we only deal with finite systems.  (Our lifetime is finite, so a computer can only examine a finite portion of the input.)