Perhaps let me try to address this question based on a discussion with Prof. Frank Wilczek. This post is not going to be complete or anything.
The punch line question I discussed with him: is there a set of mathematical equations from physical principles to distinguish life and lifeless beings? (say, hand in a system as an input, one can check its live or dead, by running algorithms/numerics/experiments in computer, which outputs the answer.)
Take virus for example may be hard to tell the "phase" difference between life or lifeless, as Brandon suggested. However, I believe, there is certainly a distinction between life and lifeless beings. An extreme example is simply the moment when life beings becoming dead (due to natural process, illness or external factors). Live or dead all concerns the same "organic" system. However, some properties changes between it is live or it is dead. So the virus example is confusing because it is about the complexity of its composition - whether it is organic enough. It is not exactly what Idear is pushing after, I thought.
There may be this distinction between life and lifeless, because the former(life) has "too many degree of freedoms" than the later(lifeless), especially in a time dependent dynamical process. The former (life one) is difficult to reconcile with Determinism.
So let me share with you, Prof. Frank Wilczek's mail response after a conversation, (I wished his generous understanding on sharing his thought to the community here, every copyright on his thought goes to his):
I enjoyed talking with you about this. It's an interesting question. As we discussed, I think the common notion of "life" is a vague around the edges, but of course that was (and still is) true of concepts like "symmetry", "energy", "force", "temperature" -- the scientific notions don't necessarily completely coincide with the everyday notions that inspired them, but capture what can be captured consistent with precision. Also as we touched on, I think that life may probably can only be viewed usefully in terms of how a system responds to an appropriate environment. Thus a living system, roughly, is one that grows and at some level reproduces in an appropriate class of environments. (If we want to exclude crystals, the last phrase is important.) One could attempt to make that definition more "physical" with some notion that life is connected with creating, maintaining, and amplifying entropy gradients.
All the best,