@compositehiggs#17785

That's a pretty interesting proposition for how to think about randomness in games. I hope you don't mind if I use it as a jumping-off point to talk a bit about quantum mechanics and classical statistical mechanics. As you mentioned the former, it inspired me to write about a topic of research that is closely related to what I work on and since I already wrote it, it would be a waste not to post it.

To everyone in the thread: I tried to write for a mix of a general audience and people with at least some technical knowledge of quantum mechanics, but it may be really bad for both audiences for that very reason. But hopefully someone will find it comprehensible and interesting.

One of the more interesting research directions in recent years in my (somewhat biased) opinion is the relation between classical statistical physics/thermodynamics and quantum mechanics. In quantum mechanics, the time-evolution of the quantum state is unitary and entirely deterministic determined by a linear equation (the Schrödinger equation) which means that a priori even pseudo-randomness associated with chaos is not possible (this requires nonlinear equations). The “randomness” of quantum mechanics is related to measurement outcomes for which probabilities can be assigned based on the quantum state. A measurement is therefore a dynamical process which is seemingly very different from the one described by the linear Schrödinger equation. This is a somewhat hairy issue and not what I want to talk about.

Instead, I'd just like to point out that the unitary quantum dynamics of pure quantum states can lead to dynamical behavior of observables that is seemingly chaotic and more importantly equilibrium expectation values described by a classical thermodynamic probabilistic ensemble. The ontology of the quantum state (wavefunction) is still a topic which is up for debate and in some sense the real objects of interest are physical observables. Nonetheless, we generally describe a quantum system by its quantum state and derive values of observables from this quantum state. What we can measure are the observables, however, and these can behave in ways that are not obvious by simply considering the quantum state (for example they can be chaotic). In quantum mechanics, an observable can be assigned an expectation value which is essentially the average of that observable with respect to the quantum state. This is generally different from a classical ensemble and arises due to the superposition of quantum states, not lack of knowledge about the state. In fact, it is possible to include the latter within the quantum formalism and simply combine classical statistical physics with quantum theory, but we are concerned with a so-called pure quantum-state (isolated and undergoing unitary evolution) and whether classical equilibrium statistical mechanics can emerge from the quantum “probabilities”.

In general most physical systems consists of many interacting particles. However, many quantities of interest are few-body observables. This includes quantities such as particle momentum, position and energy. This means their expectation values can be determined by “averaging out” the effect of the large system on the few-body observable (tracing out the rest of the system) and can be effectively described with respect to a reduced few-body quantum state. In such a process information about the full system is thrown away, but the full system still has an effect on the reduced few-body system, with correlations in the many-body system giving rise to so-called mixed reduced states (states that are described by classical probabilities in addition to the quantum “probabilities”). Note that the full system is pure, the introduction of statistical probabilities is due to information loss when averaging over the rest of the many-body system. So already we see that it is possible to obtain classical probabilities from a pure quantum state if we are only looking at a subsystem. This might seem obvious, if we don't have the full knowledge of the system we describe it probabilistically. The reason this is actually interesting is that we very rarely have access to the full state, most of the processes we are interested in concern observables of reduced subsystems. Classical probabilities are therefore related to correlations with an environment. In fact, some people think that environment-induced decoherence in which the system end up being described by purely classical probabilities solves the measurement problem, but I am not currently convinced. It does, however at least show how interference associated with quantum superposition can be destroyed by environmental interactions and that quantum systems can look classical when larger numbers of particles are considered.

So far this is relatively old news in physics, what I think is really cool and more recent goes one step further and asks whether or not these classical probabilities can be related to those of statistical mechanics. In essence, it is concerned with the old question of how probabilistic laws emerge from seemingly deterministic microscopic processes. Complicated many-body quantum systems are often chaotic. As I mentioned initially, this statement makes no sense on the level of the quantum state, but looking at few-body observables, behavior commensurate with classical notions of chaos can be observed. It has been shown that for such systems and observables and an arbitrary initially pure quantum state the expectation values of these observables will in fact thermalize!

That is, despite the unitary and reversible nature of the time-evolution the system will equilibrate to an equilibrium value (seemingly an irreversible process) determined by a set of classical probabilities and this equilibrium value is equivalent to the classical microcanonical ensemble. Note that this ensemble value is obtained from the quantum coefficients and the emergent description in terms of classical probabilities is entirely due to unitary quantum evolution with the Schrödinger equation. A priori there is no reason to expect it to be equivalent to the microcanoncial ensemble, indeed for so-called integrable systems (which have many conserved quantities, in addition to the energy) it is not.

The point is that for few-body observables the many-body system behaves as its own environment and we observe equilibration with equilibrium expectation values that are entirely equivalent to those obtained by statistical equilibrium ensembles and the systems can therefore be described by the latter, despite the underlying description being quantum mechanical. This helps bridge the gap between quantum “probabilities” and classical probabilities and shows the emergence of statistical physics from unitary time-evolution.

To take it back to the initial point, I think that a statistical description of the universe is more fundamental than some make it out to be as it is a natural consequence of environmental correlations that cannot be taken fully into account in a description of a given system. Many processes are therefore effectively statistical as they concern few-body physical quantities in a many-body correlated world. This is a different notion than the pseudorandomness associated with classical chaos which is related to the fact that one can never measure initial conditions with 100% accuracy. Note that this is from a purely physical point of view about how processes work in the physical world, I don't think this brings anything particularly new to a more fundamental philosophical understanding of randomness.

To bring it back to the original topic, I agree that it might be interesting to think about implementing probabilities in games in terms of correlations and players not having access to the full system, but I’m not really sure what such a system would look like in practice and whether it would be meaningfully different. If it were, it would mean that the player has to somehow be able to gain access to more information about the full system and thereby figure out the deterministic full process or at least get a better approximation.