Determining Determinism from State Space and Dynamics

Good to see you again. In the previous lesson, we separated determinism from predictability, stochasticity, causality, and reversibility. The working definition was: a theory is deterministic when its complete state, together with its laws, fixes one and only one later state.

This lesson turns that definition into a practical diagnostic. You will learn to inspect a toy theory’s proposed state space and its dynamical law, identify whether the law is genuinely well defined on that state space, and decide whether it permits a unique future, several futures, or only probabilities for futures. This is the basic method we will later apply to unitary quantum mechanics, collapse models, and hidden-variable theories.

State spaces, laws, and the question being asked

A toy physical theory has two indispensable parts:

a state space $S$ , whose elements represent complete instantaneous states according to the theory; and
a dynamical law, which says how such states change with time.

For a discrete-time theory, a deterministic law is a map

Φ : S \to S,

with evolution

s_{n + 1} = Φ (s_{n}) .

The crucial word is map: every admissible state $s_{n}$ must have one specified successor $Φ (s_{n})$ .

For a continuous-time theory, one often gives a differential equation such as

x ˙ = f (x),

or, more generally,

s ˙ = F (s, t) .

Here determinism requires more than merely writing down an equation. Given an admissible initial state $s (t_{0}) = s_{0}$ , the associated initial-value problem must have one unique solution over the time interval under consideration.

So use this precise test:

A theory is forward deterministic over its stated domain if every complete admissible state at an initial time yields exactly one physically allowed later state under its laws.

There are three ways this test can fail:

the alleged state is not actually complete;
the law permits more than one continuation from the same state;
the law does not provide any continuation for a state or time interval it claims to cover.

The first issue is often mistaken for indeterminism. The second is genuine non-uniqueness. The third is a failure of the theory to define an adequate evolution on its announced domain.

[PDF] Deterministic and Random Evolution

Read the opening parts of Jens Lorenz's text to make the mathematical state-space-and-law picture explicit. It treats both continuous-time initial-value problems and discrete-time maps, which are the two basic forms used in this lesson.

In Chapter 1, read Section 1, “Initial Value Problems for ODEs,” beginning where the text introduces vector-valued states and ending with its explanation that a specified state and law determine the evolution. Follow ODE state and law, paying particular attention to why uniqueness of a solution is the relevant condition. Then read Section 2, “Discrete-Time Dynamics,” from the paragraph beginning “Here, at first, M may be any set” through the logistic-map example. Focus on discrete maps: a map iterated from an initial state gives one forward orbit, even when it does not determine a unique past.

First diagnostic: is the proposed state complete?

Suppose someone proposes the following discrete model of a moving object:

x_{n + 1} = x_{n} + v_{n},

but declares that the state space is just the set of positions,

S = R,

with state $x_{n}$ .

This is not yet an indeterministic theory. It is not even a complete dynamical law on the stated space, because the next position depends on $v_{n}$ , which is absent from the state. One position can lead to many different next positions depending on the omitted velocity.

The natural repair is to enlarge the state:

s_{n} = (x_{n}, v_{n}) .

If the dynamics additionally specifies, for example,

x_{n + 1} = x_{n} + v_{n}, v_{n + 1} = v_{n},

then the theory is deterministic. Each pair $(x_{n}, v_{n})$ has exactly one successor:

Φ (x, v) = (x + v, v) .

This modest example contains a major methodological lesson:

You may call a theory indeterministic only after asking whether its own claimed complete state has been specified and whether its law is closed on that state space.

An observer who knows only $x_{n}$ faces uncertainty about $x_{n + 1}$ . But that uncertainty is due to an incomplete description by the observer, not multiple futures in the completed model.

This distinction is especially important in quantum foundations. A hidden-variable theory, if one were proposed, would not merely add unknown numbers to textbook quantum mechanics. It would claim that the theory’s complete state has a larger form than the quantum state alone. Whether that enlarged state evolves uniquely is a separate question from whether we happen to know its components.

A compact classification table

Proposed dynamical rule	Classification	Why
$s_{n + 1} = Φ (s_{n})$ , with one output for every $s_{n}$	Deterministic	The law is a single-valued evolution map.
$s_{n + 1} \in {Φ_{1} (s_{n}), Φ_{2} (s_{n})}$ with no selection rule	Indeterministic / under-specified	One complete state permits several future states.
$P (s_{n + 1} = a ∣ s_{n}) = 1/2$ , $P (s_{n + 1} = b ∣ s_{n}) = 1/2$	Stochastic at the stated level	The law assigns nontrivial probabilities rather than one successor.
$x_{n + 1} = x_{n} + v_{n}$ , but state is declared to be $x_{n}$ alone	Not a closed theory on the claimed state space	The next state depends on an omitted variable.
$x_{n + 1} = x_{n}^{2}$	Deterministic but non-reversible	Each present state has one future, but $x_{n + 1} = 4$ could have come from $x_{n} = 2$ or $x_{n} = - 2$ .

The last row reinforces a distinction from the previous lesson: forward determinism does not require reversibility.

Deterministic maps and stochastic transition laws

Discrete-time models make the logical difference between deterministic and stochastic evolution particularly transparent.

Let

S = {A, B, C} .

If the law is

Φ (A) = B, Φ (B) = C, Φ (C) = C,

then this is deterministic. Starting at $A$ , the future is fixed:

A \mapsto B \mapsto C \mapsto C \mapsto \dots .

The fact that the system eventually becomes trapped at $C$ does not make it random or indeterministic. It simply makes the map many-to-one: both $B$ and $C$ lead to $C$ . That loss of information prevents unique backward reconstruction, but it leaves the future unique.

Now contrast a transition rule:

P (B ∣ A) = 0.7, P (C ∣ A) = 0.3.

If this probability rule is fundamental and $A$ is the complete state, then the theory is stochastic at $A$ : the present state does not fix whether the next state is $B$ or $C$ .

A convenient unifying notation is a transition kernel $K (s^{'} ∣ s)$ . A deterministic map can be represented by a special kernel:

K (s^{'} ∣ s) = {1, 0, s^{'} = Φ (s), otherwise .

A stochastic theory has at least one state $s$ for which two or more different successors have nonzero probability:

0 < K (s_{1}^{'} ∣ s) < 1, 0 < K (s_{2}^{'} ∣ s) < 1.

But retain the qualification at the stated level. A probability law for a coarse state may summarize unknown deterministic degrees of freedom. It follows that a stochastic-looking description does not, by itself, prove that no deterministic completion is possible. It establishes only that the presented state-plus-law pair is not deterministic.

[PDF] Deterministic and Random Evolution

Continue with Section 4, “Random Evolution,” for the contrast between a deterministic rule and a Markov transition matrix. The point is not the later long-run analysis of Markov chains, but the different kind of law a transition matrix supplies.

Read the opening of Section 4, where the text contrasts deterministic laws with probabilistic evolution, then follow deterministic and random evolution. Next read the finite-state transition-matrix example through the paragraph emphasizing the type of questions such a law can answer. Focus on the central contrast: a stochastic law specifies probabilities for alternatives, rather than selecting one next state.

Continuous dynamics: an equation is not automatically deterministic

In continuous time, the dynamical law often appears as an ordinary differential equation:

x ˙ = f (x) .

It is tempting to assume that because the right-hand side specifies a velocity at each point, the future must be fixed. Usually that is correct for physically well-behaved equations, but it is not true without a uniqueness theorem.

Consider first

x ˙ = - x, x (0) = x_{0} .

The unique solution is

x (t) = x_{0} e^{- t} .

Every allowed initial state has one future trajectory. This theory is deterministic.

Now consider the superficially similar equation

x ˙ = 2 ∣ x ∣, x (0) = 0.

One solution is plainly

x (t) = 0

for all $t \geq 0$ . Yet, for every waiting time $T \geq 0$ , another solution is

x_{T} (t) = {0, (t - T)^{2}, 0 \leq t \leq T, t > T .

For $t > T$ ,

x ˙_{T} (t) = 2 (t - T)

and

2 ∣ x_{T} (t) ∣ = 2 (t - T)^{2} = 2 (t - T),

so this later departure satisfies the same differential equation. The function is also continuously differentiable at $t = T$ : its velocity remains zero at the instant it “starts.”

Thus precisely the same initial state,

x (0) = 0,

has infinitely many allowed futures:

remain at $0$ forever;
remain there until $T = 1$ , then depart;
remain there until $T = 10$ , then depart;
and so on.

This is a genuine failure of determinism for the model as formulated. No omitted variable has been identified, and no probability rule selects a departure time. The law allows several distinct future trajectories from one state.

Mathematically, the issue is that $f (x) = 2 ∣ x ∣$ is not sufficiently regular near $x = 0$ to guarantee uniqueness. A common sufficient condition is Lipschitz continuity: roughly, the velocity field cannot change too abruptly between nearby states. It is a sufficient condition, not a necessary one; the key physical criterion remains the uniqueness of solutions, whatever method establishes it.

Norton’s dome: a classical non-uniqueness case

Norton’s dome makes this issue mechanically vivid. A point mass is placed at rest at the exact apex of a specially shaped frictionless dome. The shape is selected so that the effective radial equation of motion has the non-Lipschitz form associated with a square-root force.

A ball at rest at the apex of Norton's dome. The dome profile is chosen so that Newtonian dynamics allows the ball either to remain at the apex or, after an arbitrary waiting time, to begin sliding down in an arbitrary radial direction.

In suitable units, the radial motion can be represented by

r ¨ = r, r (0) = 0, r ˙ (0) = 0.

One solution is the static one:

r (t) = 0.

But for any $T \geq 0$ , another solution has the form

r_{T} (t) = ⎩ ⎨ ⎧ 0, \frac{( t - T ) ^{4}}{144}, t \leq T, t > T .

For the moving portion,

r ¨_{T} (t) = \frac{( t - T ) ^{2}}{12} = \frac{( t - T ) ^{4}}{144} = r_{T} (t) .

Therefore the ball may remain at rest indefinitely, or begin moving after any chosen waiting time. In the full two-dimensional dome, it may also depart in different downhill directions. Newton’s second law does not select one possibility.

The point of the dome is not that ordinary balls can spontaneously roll off ordinary hilltops. Its physical idealizations are controversial, and its special surface is highly non-generic. Its value for us is diagnostic: even a classical-looking force law may fail to be deterministic if the relevant initial-value problem lacks unique solutions.

Causal Determinism - Stanford Encyclopedia of Philosophy

Read the Stanford Encyclopedia of Philosophy discussion of determinism in physical theories, concentrating on its treatment of classical counterexamples. It shows why the association “classical mechanics equals automatic determinism” is too quick.

In Section 4.1, “Classical mechanics,” first skim the examples of escape to infinity and multiple-particle collisions to see different ways an evolution law can fail to fix later states. Then read the Norton’s-dome discussion carefully, beginning the dome argument. Read through the end of that paragraph as well, noting the important qualification that philosophers dispute whether the dome should count as an acceptable Newtonian system. Treat it as a lesson about uniqueness conditions, not as a typical classical prediction.

A reusable procedure for classifying toy theories

When a proposed theory arrives in words, equations, or a diagram, do not start by asking whether it looks random. Work through the following sequence instead.

1. Identify the state space

Write down exactly what counts as a state:

s \in S .

For a classical particle, this might be $(x, p)$ , not position $x$ alone. For a finite automaton, it may be one element of a finite set. For a quantum system later in this course, we will need to ask whether the state is a wavefunction, density operator, wavefunction plus additional variables, or something else.

Also identify the state space’s domain. Are all real values permitted? Is the state constrained to a circle, a finite set, or a region defined by an energy condition?

2. Put the law in its actual mathematical form

Common forms include:

s_{n + 1} = Φ (s_{n}),

s ˙ = F (s, t),

and

K (s^{'} ∣ s) = P (s_{t + Δ t} = s^{'} ∣ s_{t} = s) .

A time-dependent law is not thereby indeterministic. One may write

s (t_{1}) = Φ_{t_{1}, t_{0}} (s (t_{0})),

and ask whether this map assigns one state at $t_{1}$ to each initial state at $t_{0}$ .

3. Ask whether the law is closed on the announced state

If the formula for the future uses a quantity missing from $s$ , the theory needs a larger state space or an additional law.

For example, the statement “the next output depends on a hidden coin flip” is not a complete deterministic law or a complete stochastic law until the role of that coin flip is formally specified. Is the flip an unmodeled physical variable? Is it an irreducibly random event? Does it have a distribution? The answer determines the theory’s classification.

4. Test forward uniqueness

For a discrete map, check whether each $s$ has exactly one $Φ (s)$ .
For a differential equation, check whether the initial-value problem has exactly one solution from each admissible initial state.
For a rule stated as alternatives, ask whether something in the law selects one alternative.

One counterexample state is enough to show that the theory is not deterministic globally over its stated state space. A theory might still be deterministic on a restricted subset, so always state the domain of your conclusion.

5. Distinguish probabilistic laws from distributions over states

A distribution over initial states,

P (s_{0}),

does not make deterministic dynamics stochastic. One may be uncertain which initial state obtains while each possible state evolves uniquely.

By contrast, a nontrivial conditional transition law,

K (s^{'} ∣ s),

makes the dynamics stochastic at that description level if the present state $s$ has several possible successors with nonzero probability.

6. Report the result with its scope

A good conclusion has this structure:

Given the stated complete state space $S$ and the law $L$ , the model is / is not forward deterministic because [every state has one successor / this state has multiple solutions / the purported law is not closed on $S$ ]. This conclusion concerns the model as formulated and does not, by itself, establish whether a larger deterministic completion exists.

That last sentence is not philosophical hedging. It prevents an important slide from “this operational model has stochastic transitions” to “no deeper deterministic theory is possible.” The later Bell and contextuality modules will show why that further question has stringent constraints—but it is not answered merely by observing probabilities in a toy model.

The central takeaways are these:

Determinism is assessed relative to a theory’s complete state space and dynamical law.
A deterministic discrete-time law is a single-valued map from each state to one successor state.
A continuous-time equation is deterministic only when its initial-value problems have unique solutions.
An incomplete state description creates apparent uncertainty, not necessarily indeterminism.
A probability kernel with several nonzero possible transitions is stochastic at the stated level.
Non-unique solutions, as in Norton-style examples, are a direct failure of determinism even when the equations resemble ordinary classical mechanics.

Next, we will apply this exact uniqueness criterion to the Schrödinger equation and show why closed-system unitary evolution fixes a unique quantum state at later times.

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