Are living organisms physically deterministic at any given time? Since it's all physics and chemistry, it leads me to believe they are.
Determinism – Are Living Organisms Deterministic?
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The Gibbs Free energy is not defined as ∆G = ∆H - T∆S. That holds only at constant temperature. It can be defined as $\Delta G = \Delta H - \Delta (TS)$. For a reaction to be spontaneous at constant temperature and pressure, $\Delta G$ should be negative. However, all reactions go to a certain extent, even if only microscopically. Basically a $\Delta G$ of zero means that the equilibrium constant for that reaction is 1. A negative $\Delta G$ means that the equilibrium constant is greater than one, while a positive $\Delta G$ means that the equilibrium constant is less than one. This isn't magic, the relationship is $\Delta G = - RT\ln K$, where $R$ is the gas constant, $T$ the temperature, and $K$ the equilibrium constant.
One way to have a reaction take place to a significant extent with $\Delta G > 0$ is to couple it to another reaction with a large negative $\Delta G$. This is most often done by having one component in the desired reaction also a component in the "driving" reaction. And this can be done without applying work to the the reactants.
But a simple system if 3 particles interacting gravitationally fails to admit such a description.
No it doesn't. Given some set of initial conditions, the gravitational three-body problem can be solved for any time you like, to whatever degree of precision you like. It's true that the result can't be expressed in a neatly-packaged "closed form," but that's more a statement about how we write formulas down than it is a statement about the behavior of the system.
More interesting is the fact that the gravitational three-body problem is generically chaotic, meaning (among other things) that nearby initial conditions diverge from each other exponentially. From a practical point of view, this limits how far in the future we can reliably predict the motion of an observed three-body system. Because our measurements have finite resolution, we can't precisely pin down the true initial state of the system, and two initial conditions which are both compatible with our measurements will eventually evolve to completely different final states. Even if exact measurement of initial conditions were possible, our calculations necessarily have finite precision, which would give us the same problem.
However, this is not the same as non-deterministic evolution. Given exact initial conditions, the evolution of a chaotic system is completely deterministic - it's just that if the model is based on real-world observations, those predictions quickly become relatively meaningless because of uncertainty in the initial conditions and finite numerical precision.
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In the sense that most people mean, I'd say no because quantum mechanics isn't deterministic (at least the way most people think of "determinism").
Ultimately, to be more specific, I think this depends on your interpretation of quantum mechanics, and how reductionist you are.
The best you can do with reconciling quantum mechanics with determinism is taking a particular (admittedly relatively popular) interpretation of quantum mechanics (the many-worlds interpretation) which is "deterministic" but not in the sense you were probably thinking of, or in a sense which is very useful.
Reductionism, which is generally the working mode of thinking for scientists, implies that if quantum mechanics & physics is deterministic, then chemistry and biology are too because they are built only out of pieces of nature that are fully described by quantum mechanics and nothing else. If you want to be controversial, you can suppose that there exists "strongly emergent" phenomena which fundamentally can't be traced to quantum mechanics & physics (e.g. a soul which is not described by physics). This generally isn't very popular in experimental sciences because 1. it's not very useful and 2. these ideas generally don't have a great track record.
More on the quantum mechanics bit:
Quantum mechanics gives describes a system of particles and their evolution in time in terms of probabilities, not "deterministic" events, as classical mechanics would. And there's good reason (see Bell's inequalities and Bell tests) to believe that this probabilistic description is as fundamental as you can get for these things. The purest description of the universe is in terms of probabilities.
Probabilities seem manifestly non-deterministic. However, if you subscribe to the many-worlds theory then this is still in a sense deterministic, as quantum mechanics (or a similar theory) would correctly (and deterministically) describe the evolution of the state of the universe. Locally (i.e. in our part of the state which can't really observe other parts of the state) things look probabilistic, but if you could somehow observe the state of the whole universe, you'd deterministically predict how the whole thing evolves using quantum mechanics. If you don't know what I mean by this, I'd start by looking up the concepts of superposition and entanglement in quantum mechanics. As of yet I don't think there is anything testable about the differences between this and other interpretations of quantum mechanics (e.g. Copenhagen), so many people don't like reading into these things too much.