Laplace's Demon and Determinism: Is the Future Already Fixed?

0. Key points

• Laplace’s demon is the thought experiment of an intellect that knows the position and velocity of every particle in the universe together with all the forces acting on them; for such an intellect the future and the past would stand as plainly before its eyes as the present. Laplace wrote this in 1814.
• The claim is not merely a poetic image: it translates cleanly into a mathematical statement. The translation reads “the solution of the initial value problem for the equations of motion is unique”, and under suitable hypotheses this is provably true.
• Relax those hypotheses even slightly and uniqueness fails. We carry one such failure all the way through.
• In the twentieth century the demon was shot twice. The first shot was chaos, or sensitive dependence on initial conditions: if the demon is not perfect but only very slightly inaccurate, the error grows exponentially and prediction becomes meaningless within a finite time. For the logistic map we prove that the error doubles at every step exactly.
• The second shot was quantum mechanics. The uncertainty relation denies the demon’s very premise, that position and velocity can both be known exactly.
• Even so, determinism is not entirely dead: the time evolution of the quantum wave function is deterministic. What died was not “determinism” but predictability — that is the conclusion of this article.

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1. Motivation: why nineteenth-century thinkers believed the future was computable

To us the idea that everything in the universe could be pinned down by calculation sounds grandiose. To a scientist of the late eighteenth or early nineteenth century, however, it was a rather modest statement. The reason is simple: Newtonian mechanics was on an unbroken winning streak.

Edmond Halley argued that the comet of 1682 was the same object as comets recorded in earlier centuries, and predicted its return. Halley died before he could see the outcome, but Alexis Clairaut and his collaborators estimated by hand the delay caused by the gravitational pull of Jupiter and Saturn, and computed that the return would fall between the end of 1758 and the spring of 1759. The comet was spotted on Christmas Day, 1758. With paper and pen they had said where in the sky, when, and what would appear.

Neptune is even more dramatic. The observed positions of Uranus deviated slightly from the calculated ones. Assuming the deviation was caused by the gravity of an unknown planet, Urbain Le Verrier computed the direction in which that planet had to lie. In 1846 Johann Galle pointed a telescope in that direction at the Berlin Observatory and found a new planet within about one degree of the predicted position. A body no one had ever seen was found by the equations first.

After a run of successes like these, the next thought comes naturally.

If the wobble of Uranus can be explained by equations, then so can a marble rolling across a desk, a leaf tossed by the wind, and the atoms in my brain. In principle, then, all of it should be computable.

The most vivid formulation of that thought is a passage in Pierre-Simon Laplace’s A Philosophical Essay on Probabilities (1814). He wrote, in substance, that an intellect which at a given moment knew all the forces that set nature in motion and all the positions of the beings that compose it, and which was vast enough to analyse these data, would find nothing uncertain; the future, like the past, would be present to its eyes. This fictitious intellect later came to be called Laplace’s demon.

Note

Laplace himself never used the word “demon” (démon). What he wrote was simply “an intellect” (une intelligence). The nickname was attached by later writers, and probably stuck by analogy with Maxwell’s demon, the thermodynamic thought experiment. We treat that other demon in Maxwell’s Demon; that a successful Maxwell demon would violate the second law of thermodynamics is recorded in Proposition 3.2［Maxwell's Demon］.

What makes the passage interesting is that the book in which Laplace wrote it is a book on probability. His point was this: the universe is in truth completely determined; we use probability not because the universe is vague but because we are ignorant. Probability, in other words, is an instrument for measuring the amount of our ignorance. This position survives today as the philosophical starting point of Bayesian statistics.

In this article we rewrite the claim as a mathematical proposition and examine, step by step, how far it holds and where it breaks.

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2. Preliminaries: state, phase space, equations of motion

To keep the discussion sharp, we first replace the phrase “the future is fixed” with mathematical language.

Definition 2.1（State and phase space）

Consider a classical mechanical system of $N$ point masses. Write $\boldsymbol{r}_i(t) \in \mathbb{R}^3$ for the position and $\boldsymbol{p}_i(t) \in \mathbb{R}^3$ for the momentum of the $i$-th particle at time $t$, and collect them into the $6N$-dimensional vector

$$\boldsymbol{x}(t) = \bigl(\boldsymbol{r}_1(t), \ldots, \boldsymbol{r}_N(t),\ \boldsymbol{p}_1(t), \ldots, \boldsymbol{p}_N(t)\bigr) \in \mathbb{R}^{6N} .$$

We call $\boldsymbol{x}(t)$ the state of the system at time $t$, and the space $\mathbb{R}^{6N}$ in which the state moves the phase space.

The essential point is that we pair positions with momenta rather than using positions alone. Positions alone do not determine the future: two balls at the same place behave differently in the next instant depending on whether one is moving right and the other left. Newton’s equation of motion involves the acceleration, a second derivative in time, so the initial data must include both position and velocity (momentum). Laplace’s mention of both the “positions” and the “forces” corresponds precisely to this fact.

Definition 2.2（Deterministic system）

Suppose a vector field $\boldsymbol{F} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ on phase space is given and the time evolution of the system is written as the first-order ordinary differential equation

$$\frac{d\boldsymbol{x}}{dt} = \boldsymbol{F}(\boldsymbol{x}) .$$

If for every initial state $\boldsymbol{x}(0) = \boldsymbol{x}_0$ this equation has exactly one solution, we call the system deterministic.

Newton’s equation of motion $m_i \ddot{\boldsymbol{r}}_i = \boldsymbol{F}_i(\boldsymbol{r}_1,\ldots,\boldsymbol{r}_N)$ becomes, on setting $\boldsymbol{p}_i = m_i \dot{\boldsymbol{r}}_i$, the first-order system

$$\dot{\boldsymbol{r}}_i = \frac{\boldsymbol{p}_i}{m_i}, \qquad \dot{\boldsymbol{p}}_i = \boldsymbol{F}_i(\boldsymbol{r}_1,\ldots,\boldsymbol{r}_N) ,$$

which fits exactly into the form of Definition 2.2. All we have done is trade the second order for twice as many variables.

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3. Translating Laplace’s demon into mathematics

Definition 3.1（Laplace's demon）

We call Laplace’s demon a hypothetical being with the following three capabilities.

1. It knows the state $\boldsymbol{x}(t_0)$ of the universe at some time $t_0$ with zero error.
2. It knows the governing field $\boldsymbol{F}$, that is, all the laws of force.
3. For any $t$ it can carry the solution $\boldsymbol{x}(t)$ of the initial value problem to completion in finite time.

Do these three conditions in fact fix the future? That is a purely mathematical question, and the following theorem answers it.

Theorem 3.2（Uniqueness for the initial value problem）

Let $D \subset \mathbb{R}^n$ be open and $\boldsymbol{F} : D \to \mathbb{R}^n$ continuous, and suppose there is a constant $L > 0$ with

$$\lVert \boldsymbol{F}(\boldsymbol{u}) - \boldsymbol{F}(\boldsymbol{v}) \rVert \le L \lVert \boldsymbol{u} - \boldsymbol{v} \rVert \qquad (\forall \boldsymbol{u}, \boldsymbol{v} \in D)$$

(this hypothesis is called the Lipschitz condition). Then if two $C^1$ solutions $\boldsymbol{x}(t), \boldsymbol{y}(t)$ on the interval $[0, T]$ both take values in $D$ and satisfy $\dot{\boldsymbol{x}} = \boldsymbol{F}(\boldsymbol{x})$, $\dot{\boldsymbol{y}} = \boldsymbol{F}(\boldsymbol{y})$ and $\boldsymbol{x}(0) = \boldsymbol{y}(0)$, we have $\boldsymbol{x}(t) = \boldsymbol{y}(t)$ on all of $[0,T]$.

Proof（Theorem 3.2）

Integrating both sides of the differential equation from $0$ to $t$, the fundamental theorem of calculus gives

$$\boldsymbol{x}(t) = \boldsymbol{x}(0) + \int_0^t \boldsymbol{F}(\boldsymbol{x}(s))\, ds, \qquad \boldsymbol{y}(t) = \boldsymbol{y}(0) + \int_0^t \boldsymbol{F}(\boldsymbol{y}(s))\, ds .$$

Subtracting, the initial terms cancel by the hypothesis $\boldsymbol{x}(0) = \boldsymbol{y}(0)$, leaving

$$\boldsymbol{x}(t) - \boldsymbol{y}(t) = \int_0^t \bigl( \boldsymbol{F}(\boldsymbol{x}(s)) - \boldsymbol{F}(\boldsymbol{y}(s)) \bigr) ds .$$

Put $u(t) := \lVert \boldsymbol{x}(t) - \boldsymbol{y}(t) \rVert \ge 0$. Since the norm of an integral is at most the integral of the norm (the integral form of the triangle inequality), the Lipschitz condition yields

$$u(t) \le \int_0^t \lVert \boldsymbol{F}(\boldsymbol{x}(s)) - \boldsymbol{F}(\boldsymbol{y}(s)) \rVert\, ds \le L \int_0^t u(s)\, ds .$$

Now set $U(t) := \int_0^t u(s)\, ds$. As $u$ is continuous, $U$ is $C^1$ with $U'(t) = u(t)$, and the inequality just obtained reads $U'(t) \le L\,U(t)$. Multiplying by the positive number $e^{-Lt}$ and rearranging,

$$\frac{d}{dt}\Bigl( e^{-Lt} U(t) \Bigr) = e^{-Lt}\bigl( U'(t) - L\,U(t) \bigr) \le 0 ,$$

so the function $e^{-Lt}U(t)$ is non-increasing on $[0,T]$. Since $U(0) = 0$, we get $e^{-Lt}U(t) \le 0$ and hence $U(t) \le 0$ for all $t \in [0,T]$. On the other hand $u \ge 0$ forces $U(t) \ge 0$, so $U(t) = 0$. Therefore $u(t) = U'(t) = 0$, that is, $\boldsymbol{x}(t) = \boldsymbol{y}(t)$ throughout $[0,T]$.

∎

In other words, as long as the Lipschitz condition holds, the demon’s claim is mathematically correct. Under this hypothesis a universe starting from a given state cannot branch into two futures. In this sense Laplace’s assertion was not speculation but a theorem.

Conversely, what must we give up in order to destroy the conclusion? The proof used the Lipschitz condition in exactly one place. Let us remove it there.

Example 3.3（An equation whose solution is not unique）

Consider the one-dimensional equation

$$\frac{dx}{dt} = 3\,x^{2/3}, \qquad x(0) = 0 .$$

The function $F(x) = 3x^{2/3}$ is continuous but fails the Lipschitz condition near $x = 0$. Indeed, taking $v = 0$ and $u = h > 0$,

$$\frac{|F(h) - F(0)|}{|h - 0|} = \frac{3h^{2/3}}{h} = \frac{3}{h^{1/3}} \xrightarrow{h \to +0} \infty ,$$

so whatever constant $L$ we propose, there is an $h$ for which the ratio exceeds $L$.

The solution is then not unique. First, $x(t) \equiv 0$ is obviously a solution (left side $= 0$, right side $= 3\cdot 0 = 0$). But for every constant $c \ge 0$ the function

$$x_c(t) = \begin{cases} 0 & (0 \le t \le c) \\ (t - c)^3 & (t > c) \end{cases}$$

is also a solution. Let us check the range $t > c$: the left side is $\dfrac{d}{dt}(t-c)^3 = 3(t-c)^2$ and the right side is $3\bigl((t-c)^3\bigr)^{2/3} = 3(t-c)^2$, which agree. The junction at $t = c$ is $C^1$ as well, since the derivative equals $0$ from both sides.

As $c$ ranges over all non-negative reals, the initial condition $x(0)=0$ admits infinitely many solutions. Read physically: a particle at rest at the origin is not told by the equation how long it stays at rest. Solutions in which it starts moving “of its own accord” at some arbitrary moment are permitted.

Remark 3.4

Example 3.3 may look like a mathematical fabrication, but the “Norton dome” proposed by the philosopher of science John Norton realises a mechanical system of essentially the same structure as a point mass sliding on a dome whose apex has a carefully chosen shape. A mass placed at the apex admits solutions in which, untouched by anything external, it spontaneously slides off at an arbitrary time — and this within Newtonian mechanics. It should be said that whether such a dome can be built from real material, and whether the point-mass approximation is legitimate, are debated, so the example is not universally accepted as a counterexample to the commonplace that classical mechanics is deterministic. What can be asserted with confidence is that determinism is not a premise of physics but a consequence of the smoothness of the equations.

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4. The first counterattack: chaos — deterministic yet unpredictable

Theorem 3.2 says that if the initial states are exactly equal, so are the futures. That is why condition 1 of the demon demanded zero error. What happens if the demon is only slightly inaccurate? Nineteenth-century common sense said that slightly perturbed input yields slightly perturbed output. In general this is false.

Sensitive dependence on initial conditions. Two trajectories that nearly coincide at the left separate exponentially as time goes on

Definition 4.1（Lyapunov exponent）

Consider the discrete-time dynamical system $x_{n+1} = f(x_n)$ generated by a map $f$. Let $\delta_n$ be the separation after $n$ steps between the orbits starting at $x_0$ and at the nearby point $x_0 + \delta_0$. When the limit

$$\lambda = \lim_{n \to \infty} \lim_{\delta_0 \to 0} \frac{1}{n} \ln \left| \frac{\delta_n}{\delta_0} \right|$$

exists, it is called the Lyapunov exponent. When $\lambda > 0$ the separation grows exponentially, roughly as $\delta_n \approx \delta_0 e^{\lambda n}$. We then say the system has sensitive dependence on initial conditions, and a motion confined to a bounded region that has sensitive dependence on initial conditions is called chaotic.

This is abstract, so let us take one example in which the Lyapunov exponent can be computed exactly, and carry it through.

Proposition 4.2（Closed form and Lyapunov exponent of the logistic map）

Consider the map $f(x) = 4x(1-x)$ on the interval $[0,1]$. For $x_0 \in [0,1]$ define $\theta_0 \in [0, \pi/2]$ by $x_0 = \sin^2 \theta_0$. Then the sequence given by $x_{n+1} = f(x_n)$ satisfies

$$x_n = \sin^2\bigl(2^n \theta_0\bigr) .$$

Moreover, for almost every $x_0$ such that $\theta_0/\pi$ is irrational, the Lyapunov exponent of this system is $\lambda = \ln 2$.

Proof（Proposition 4.2）

We first prove the closed form by induction. For $n = 0$ the statement $x_0 = \sin^2\theta_0$ is the definition. Assume $x_n = \sin^2(2^n\theta_0)$ and write $\theta_n := 2^n \theta_0$. Using $1 - \sin^2\theta_n = \cos^2\theta_n$,

$$x_{n+1} = 4 x_n (1 - x_n) = 4 \sin^2\theta_n \cos^2\theta_n = \bigl(2\sin\theta_n\cos\theta_n\bigr)^2 = \sin^2(2\theta_n) = \sin^2\bigl(2^{n+1}\theta_0\bigr) ,$$

so the claim follows from the double-angle formula $\sin 2\theta = 2\sin\theta\cos\theta$ alone.

Next we examine the growth of the error. Shifting the initial value from $\theta_0$ to $\theta_0 + \epsilon$ shifts the angle after $n$ steps from $2^n\theta_0$ to $2^n(\theta_0 + \epsilon) = 2^n\theta_0 + 2^n\epsilon$. That is, the error in the angle doubles exactly at every step. Since the derivative of $x = \sin^2\theta$ is $\dfrac{dx}{d\theta} = 2\sin\theta\cos\theta = \sin 2\theta$, the error in $x$ is

$$\delta_n \approx \bigl| \sin(2\theta_n) \bigr| \cdot 2^n \epsilon .$$

Since $\delta_0 \approx |\sin(2\theta_0)|\,\epsilon$, we obtain

$$\frac{1}{n}\ln\left|\frac{\delta_n}{\delta_0}\right| = \frac{1}{n}\left( n \ln 2 + \ln\left|\frac{\sin 2\theta_n}{\sin 2\theta_0}\right| \right) = \ln 2 + \frac{1}{n}\ln\left|\frac{\sin 2\theta_n}{\sin 2\theta_0}\right| .$$

The second term $\ln|\sin 2\theta_n / \sin 2\theta_0|$ oscillates with $n$, but when $\theta_0/\pi$ is irrational, $2^n\theta_0$ does not approach a zero of $\sin$ too closely (apart from a set of measure zero), so after division by $n$ this term tends to $0$. Hence $\lambda = \ln 2$.

∎

The value $\lambda = \ln 2 \approx 0.693$ means that at every step one binary digit of information about the initial error is lost. Let us feel this in concrete numbers.

Example 4.3（Sixteen digits of precision vanish in 54 steps）

A double-precision floating point number carries roughly $16$ digits of precision ($\delta_0 \approx 10^{-16}$). By Proposition 4.2 the error doubles at every step, so after $n$ steps it is $\delta_n \approx 2^n \times 10^{-16}$. This reaches $1$ — the size of the whole interval $[0,1]$, that is, the state of knowing nothing — when

$$2^n \times 10^{-16} \ge 1 \iff n \ge \frac{16 \ln 10}{\ln 2} = \frac{16 \times 2.3026}{0.6931} = 53.2 ,$$

that is, at step $n = 54$. In practice, two computations started from $x_0 = 0.4$ and $x_0 = 0.4 + 10^{-16}$ appear to agree to several decimal places for the first forty steps or so, but past step 54 they become completely unrelated sequences.

The important point is that this is not a defect of the computer. Doubling the precision to $32$ digits buys only $53$ more steps, a factor of two. The predictable time grows only like the logarithm of the precision.

# Checking sensitive dependence on initial conditions in the logistic map (standard library only)
def orbit(x, n):
    for _ in range(n):
        x = 4.0 * x * (1.0 - x)
    return x

x0 = 0.4
for n in (10, 30, 50, 54, 60):
    a, b = orbit(x0, n), orbit(x0 + 1e-16, n)
    print(f"n={n:3d}  A={a:.10f}  B={b:.10f}  |A-B|={abs(a-b):.3e}")

It was Edward Lorenz who discovered this phenomenon in meteorology. In a 1963 paper he showed that a system of ordinary differential equations obtained by truncating atmospheric convection to three variables behaves neither periodically nor convergently, and that tiny differences in the initial data are amplified. The famous question of whether the flap of a butterfly’s wings in Brazil sets off a tornado in Texas comes from this work. In present-day numerical weather prediction the Lyapunov time of the atmosphere — the time for an error to grow by a factor of $e$ — is estimated at a few days, which is the theoretical reason behind the statement that weather forecasts are limited to about two weeks ahead.

Example 4.4（The time at which the solar system itself becomes unpredictable）

Planetary motion is the stronghold of Newtonian mechanics, yet chaos lives there too. In 1989 Jacques Laskar reported, from a long-term numerical integration of the whole solar system, that the Lyapunov time of the orbital elements of the inner planets is about $5$ million years. That is, $\lambda = 1/(5\times 10^6\ \text{yr})$.

Suppose we know the position of a planet to within $15\ \mathrm{m}$. After $100$ million years the error is

$$\delta = 15\ \mathrm{m} \times \exp\left( \frac{10^8}{5 \times 10^6} \right) = 15 \times e^{20} = 15 \times 4.85\times 10^{8} \approx 7.3 \times 10^{9}\ \mathrm{m} .$$

Now $7.3\times 10^9\ \mathrm{m}$ is about $7.3$ million km, roughly $5\%$ of the Sun–Earth distance ($1.5\times10^8$ km). Since the diameter of the Earth is $1.3\times 10^4$ km, where the Earth will be along its orbit $100$ million years from now cannot be stated even in principle. We can still say which ellipse the planet is travelling on; we cannot say where on that ellipse it sits. The demon is not even allowed an error of 15 m.

Caution

Chaos is not a denial of determinism. Theorem 3.2 remains true: identical initial states give a single future. What chaos destroys is a different expectation, namely that nearby causes produce nearby effects. Being deterministic and being predictable are entirely different matters.

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5. The second counterattack: quantum mechanics — the initial condition does not exist

Chaos made the demon’s condition 1 (zero error) impossible in practice. Quantum mechanics strikes something more fundamental: a state in which position and momentum simultaneously have definite values does not exist in nature.

Theorem 5.1（Kennard's uncertainty relation）

In any state of a one-dimensional quantum system, the standard deviation $\Delta x$ of the measured position $\hat{x}$ and the standard deviation $\Delta p$ of the measured momentum $\hat{p}$ always satisfy

$$\Delta x \, \Delta p \ \ge\ \frac{\hbar}{2} ,$$

where $\hbar = h/2\pi \approx 1.055 \times 10^{-34}\ \mathrm{J\cdot s}$ is Planck’s constant divided by $2\pi$.

Remark 5.2

This inequality was proved by Earle Kennard in 1927. The proof takes a few lines, starting from the canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar$ and the Cauchy–Schwarz inequality in Hilbert space. It lies beyond the level of this article, so we omit the details; a complete derivation appears in any standard textbook, for instance Chapter 1 of Sakurai–Napolitano, Modern Quantum Mechanics. What matters is that this is not a technical limitation arising from clumsy instruments but a theorem about the mathematical nature of states.

Example 5.3（The velocity uncertainty of an electron confined inside an atom）

Suppose the position of an electron (mass $m = 9.11\times 10^{-31}\ \mathrm{kg}$) is fixed to within $\Delta x = 1.0 \times 10^{-10}\ \mathrm{m}$, roughly the size of an atom. By Theorem 5.1,

$$\Delta p \ \ge\ \frac{\hbar}{2\Delta x} = \frac{1.055\times 10^{-34}}{2 \times 1.0\times 10^{-10}} = 5.28 \times 10^{-25}\ \mathrm{kg\cdot m/s} .$$

Converting to a velocity,

$$\Delta v = \frac{\Delta p}{m} \ \ge\ \frac{5.28\times 10^{-25}}{9.11\times 10^{-31}} = 5.8 \times 10^{5}\ \mathrm{m/s} ,$$

that is, $580\ \mathrm{km}$ per second. The moment we say “this electron is here inside the atom”, its velocity acquires a spread of $\pm 580$ km/s. The demon’s requirement that both position and momentum be known with zero error means $\Delta x \Delta p = 0 < \hbar/2$, in direct contradiction with Theorem 5.1. The demon’s condition 1 is not merely hard to carry out; it is inconsistent by definition.

Example 5.4（Why quantum effects are invisible in everyday life）

We repeat the same computation for a pachinko ball of mass $m = 1.0\times 10^{-3}\ \mathrm{kg}$ (1 g) localised to $\Delta x = 1.0\times 10^{-6}\ \mathrm{m}$ (one micrometre).

$$\Delta v \ \ge\ \frac{\hbar}{2 m \Delta x} = \frac{1.055\times 10^{-34}}{2 \times 10^{-3} \times 10^{-6}} = 5.3\times 10^{-26}\ \mathrm{m/s}$$

Moving at this speed for the age of the universe ($13.8$ billion years $= 4.35\times 10^{17}\ \mathrm{s}$), the distance covered is only

$$5.3\times 10^{-26} \times 4.35\times 10^{17} = 2.3\times 10^{-8}\ \mathrm{m} = 23\ \mathrm{nm} ,$$

about $100$ atomic diameters. This is why quantum uncertainty is entirely negligible at everyday scales. That $\hbar$ is the absurdly small number $10^{-34}$ is the reason our world looks “classical”.

At this point something must be added quickly: quantum mechanics has not killed determinism. The time evolution of a quantum state $|\psi(t)\rangle$ obeys the Schrödinger equation

$$i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle .$$

This is a first-order differential equation for $|\psi\rangle$, and hence a perfectly good deterministic system in the sense of Definition 2.2. Given the initial state $|\psi(0)\rangle$, the state $|\psi(t)\rangle$ at any later time is uniquely determined.

Where, then, does randomness enter? From measurement. Given the deterministically evolved $|\psi(t)\rangle$, the question “where will the particle be found if we measure its position?” has only a probabilistic answer (the probability is supplied by the Born rule（Definition 2.2）［Schrödinger's Cat］). The question of why a deterministic wave function yields probabilistic measurement outcomes is called the measurement problem (Remark 5.1［Schrödinger's Cat］), and a century after the birth of quantum mechanics interpretations still diverge. The entrance to that controversy is Schrödinger’s Cat.

So we may retrofit the demon like this: an intellect that knows the wave function of the universe completely and can solve the Schrödinger equation. This upgraded demon can know every future wave function. Ask it, however, what tomorrow’s experiment will yield, and all it returns is a probability. Laplace held that probability expresses ignorance; the probabilities of quantum mechanics, at least on the standard interpretation, express not ignorance but a property of nature itself. Laplace’s world view was inverted at precisely the place where he spoke of probability.

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6. The third counterattack: the demon lives inside the universe

Chaos and quantum mechanics were counterattacks from physics. There is one more, from logic. Reading condition 3 of Definition 3.1, “carries the computation to completion in finite time”, at face value, the following difficulties emerge.

Start with the sheer amount of information. As in Definition 2.1, writing down the state of $N$ particles requires $6N$ real numbers. For one mole of an ideal gas, $N = 6.02\times10^{23}$, so $6N = 3.6\times 10^{24}$ numbers. Recording each number in $8$ bytes gives $2.9\times 10^{25}$ bytes. The total volume of digital data in existence worldwide in the mid-2020s is estimated at around $10^{23}$ bytes, so merely noting down the state of a single mole of gas — a glassful of air — would take several hundred times all the data humanity holds. And “a real number in 8 bytes” is an approximation to about 16 digits, a precision that, as Example 4.3 showed, melts away in 54 steps of the logistic map.

More serious is self-reference. If the demon is a physical being inside the universe, then its own internal state is part of the state $\boldsymbol{x}$ of the universe. The demon must therefore compute the future including the results of its own computation. In a 1950 paper Karl Popper argued that even within classical physics, a predicting device inside a system cannot predict in full the future of the system that contains it, because it cannot know its own next prediction before carrying it out. If the demon stands outside the universe the situation is different, but that is no longer physics; it is theology.

Storing and erasing information also carries a physical price. By Landauer's principle（Theorem 6.4）［Maxwell's Demon］, erasing $1$ bit of information requires dumping at least $k_B T \ln 2$ of heat into an environment at temperature $T$. If the demon keeps recording and updating the state of the universe, entropy is necessarily generated in the process, and the state of the universe is thereby changed. The relation between an observing demon and thermodynamics is the theme of Maxwell’s Demon; that the demon’s books balance once the cost of recording and erasing is counted is recorded in Corollary 6.5［Maxwell's Demon］.

flowchart TD
A["Premise 1: the state of the world is a finite list of numbers"] --> D["Laplace's demon: future and past as plain as the present"]
B["Premise 2: change of state is fixed by the equations of motion"] --> D
C["Premise 3: given initial data the solution is unique"] --> D
D --> E["Counterattack 1: chaos. Initial error grows exponentially; prediction becomes meaningless in finite time"]
D --> F["Counterattack 2: quantum mechanics. No state has definite position and momentum at once"]
D --> G["Counterattack 3: self-reference and the physics of information. The demon is part of the universe, and recording has a cost"]
E --> H["Conclusion: determinism survives, predictability is dead"]
F --> H
G --> H

The three premises on which Laplace's demon stands, and the three counterattacks of the twentieth century

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7. Summary: is the demon dead?

Let us sort things out. The three propositions bound up in Laplace’s demon must be kept apart.

Proposition	Modern verdict
Determinism: the same initial state gives rise to only one future	A theorem for classical systems under the Lipschitz condition (Theorem 3.2). It holds in quantum systems too, at the level of the wave function
Predictability: the future can be stated with finite precision and finite computational resources	Refuted by chaos (Definition 4.1, Example 4.4)
Existence of a complete initial condition: position and momentum have definite values simultaneously	Refuted by quantum mechanics (Theorem 5.1)

Nineteenth-century thinkers took these three to be one continuous thing. Twentieth-century physics cut them into three and denied the middle one and the bottom one. What remains, the top one — determinism itself — is still alive inside the equations of physics.

You may want to ask next whether human beings have free will. To be honest, the tools of this article cannot answer that question. Neither chaos nor quantum randomness explains what it is for “me to have decided” something: however unpredictable a die may be, no one ordinarily says the die has free will. Still, Laplace’s demon left us a surprisingly useful distinction — that being predictable and being determined are entirely different things. Other large open questions in physics are collected in Unsolved Problems in Physics (for instance Definition 3.1［Unsolved Problems in Physics］ and Definition 4.1［Unsolved Problems in Physics］).

One last aside. When Napoleon remarked to Laplace that his book on celestial mechanics made no mention of God, Laplace is said to have replied, “Sire, I had no need of that hypothesis.” The details of the anecdote differ between the sources that transmit it, and how much of it was an actual conversation is unclear. It does, however, convey rather well the confidence his demon carried.

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8. Exercises

Exercise 8.1Easy

The solution of the differential equation $\dfrac{dx}{dt} = x$ is $x(t) = x(0)e^{t}$. For the initial values $x(0) = 1$ and $x(0) = 1.001$, find the difference between the two solutions at $t = 10$. Then find the Lyapunov exponent of this system (in the continuous-time version, $\lambda = \lim_{t\to\infty}\frac1t \ln|\delta(t)/\delta(0)|$).

Solution

By linearity the difference is computed directly:

$$\delta(10) = 1.001\,e^{10} - 1\cdot e^{10} = 0.001 \times e^{10} = 0.001 \times 22026.47 = 22.03 .$$

The initial gap of $0.001$ has grown to $22.03$, an amplification of about $22{,}000$ times.

For the Lyapunov exponent, since $\delta(t) = \delta(0)e^{t}$,

$$\lambda = \lim_{t\to\infty}\frac{1}{t}\ln\left|\frac{\delta(0)e^{t}}{\delta(0)}\right| = \lim_{t\to\infty}\frac{1}{t}\cdot t = 1 .$$

Note, however, that $\lambda > 0$ does not make this system chaotic. That is exactly why Definition 4.1 includes “motion confined to a bounded region” in the definition of chaos. The solutions of this system simply fly off to infinity; the orbits are never folded back and mixed, which is the behaviour characteristic of chaos. Chaos requires both stretching and folding.

Exercise 8.2Standard

Track the logistic map $x_{n+1} = 4x_n(1-x_n)$ of Proposition 4.2 with initial precision $\delta_0 = 10^{-30}$ (high-precision arithmetic far beyond double precision). After how many steps does the error reach $1$? Also, by what factor did raising the precision from $\delta_0 = 10^{-16}$ to $10^{-30}$ increase the number of predictable steps?

Solution

By Proposition 4.2 the error doubles at every step, so after $n$ steps it is $\delta_n = 2^n \delta_0$. The condition $\delta_n \ge 1$ reads

$$2^n \times 10^{-30} \ge 1 \iff n \ge \frac{30 \ln 10}{\ln 2} = \frac{30 \times 2.302585}{0.693147} = 99.66 ,$$

so it takes $n = 100$ steps.

As computed in Example 4.3, the case $\delta_0 = 10^{-16}$ gave $54$ steps, so the factor is merely $100/54 = 1.85$. Improving the precision by a factor of $10^{14}$ (a hundred trillion) fails even to double the predictable time.

In general the number of predictable steps is $n \approx \dfrac{1}{\lambda}\ln\dfrac{1}{\delta_0}$, which depends only on the logarithm of $\delta_0$. This logarithm is why the strategy of “measuring more precisely so as to predict” is hopeless in chaotic systems.

Exercise 8.3Standard

Consider an electron in a hydrogen atom. Taking the position uncertainty to be of the order of the atomic radius, $\Delta x = 5.3\times 10^{-11}\ \mathrm{m}$ (the Bohr radius), use Theorem 5.1 to find the lower bound on the momentum uncertainty $\Delta p$, and use it to estimate the rough size of the kinetic energy $E \sim (\Delta p)^2/(2m)$ in electronvolts ($1\ \mathrm{eV} = 1.602\times 10^{-19}\ \mathrm{J}$). Take the electron mass to be $m = 9.11\times10^{-31}\ \mathrm{kg}$.

Solution

First the lower bound on the momentum:

$$\Delta p \ \ge\ \frac{\hbar}{2\Delta x} = \frac{1.055\times 10^{-34}}{2 \times 5.3\times 10^{-11}} = 9.95\times 10^{-25}\ \mathrm{kg\cdot m/s} .$$

Converting to a kinetic energy,

$$E \sim \frac{(\Delta p)^2}{2m} = \frac{(9.95\times 10^{-25})^2}{2 \times 9.11\times 10^{-31}} = \frac{9.90\times 10^{-49}}{1.822\times 10^{-30}} = 5.43\times 10^{-19}\ \mathrm{J} ,$$

and in electronvolts

$$E \sim \frac{5.43\times 10^{-19}}{1.602\times 10^{-19}} = 3.4\ \mathrm{eV} .$$

The actual ionisation energy of hydrogen is $13.6\ \mathrm{eV}$, so the order of magnitude is right (the discrepancy of a factor of $4$ comes from replacing $\Delta p$ by its lower bound and ignoring the potential energy in this crude estimate).

What this computation shows is why atoms do not collapse. Pushing the electron closer to the nucleus means shrinking $\Delta x$, whereupon $\Delta p \ge \hbar/(2\Delta x)$ makes the momentum large and the kinetic energy grow like $1/(\Delta x)^2$. The Coulomb potential energy falls only like $-1/\Delta x$, so for sufficiently small $\Delta x$ the rise in kinetic energy wins. The uncertainty relation did not merely strip the demon of information; it also gives a size to the atoms our bodies are made of.

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References

• Pierre-Simon Laplace, Essai philosophique sur les probabilités, 1814 (Japanese translation: Kakuritsu no Tetsugakuteki Shiron, tr. Sōshichi Uchii, Iwanami Bunko, 1997, in Japanese) — the famous “demon” passage appears in the opening chapter on probability.
• E. N. Lorenz, “Deterministic Nonperiodic Flow”, Journal of the Atmospheric Sciences 20 (1963), 130–141. The landmark paper demonstrating sensitive dependence on initial conditions by numerical experiment.
• J. Laskar, “A numerical experiment on the chaotic behaviour of the Solar System”, Nature 338 (1989), 237–238. Evaluates the Lyapunov time of the solar system by numerical integration.
• E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen”, Zeitschrift für Physik 44 (1927), 326–352. The first rigorous proof of the uncertainty relation in the form $\Delta x \Delta p \ge \hbar/2$.
• S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview Press, 2015 — Chapter 9 (the Lorenz equations) and Chapter 10 (the logistic map). The material of Proposition 4.2 is treated carefully at the level of this book.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — Chapter 1 contains a complete proof of Theorem 5.1.

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Appendix: Why the Lipschitz condition is essential to the uniqueness proof

Why continuity alone is not enough. Looking back at the proof of Theorem 3.2, the Lipschitz condition was used in exactly one place, the inequality $\lVert \boldsymbol{F}(\boldsymbol{x}(s)) - \boldsymbol{F}(\boldsymbol{y}(s)) \rVert \le L\,u(s)$. This says that the rate at which the gap between two orbits grows is proportional to the current gap, and never faster. If the growth rate is proportional to the gap, then a gap that starts at $0$ can never grow: anything multiplied by $0$ is $0$.

What goes wrong when the Lipschitz condition fails. For $F(x) = 3x^{2/3}$ in Example 3.3, however, the “slope” of $F$ blows up near $x = 0$. Then even when the current gap is $0$, the growth rate takes the indeterminate form $0 \times \infty$, and the guarantee that the gap cannot grow is lost. Indeed $x_c(t) = (t-c)^3$ lifts off from the state $x = 0$ to a non-zero state in finite time.

Is differentiability enough? A convenient sufficient condition is the following fact: if $\boldsymbol{F}$ is $C^1$ (its components are continuously differentiable), then the Lipschitz condition holds on any bounded closed region. The reason is the mean value theorem. The operator norm of the Jacobian matrix $D\boldsymbol{F}$ is a continuous function, hence attains a maximum $L$ on a bounded closed set. With this $L$, for any two points $\boldsymbol{u}, \boldsymbol{v}$ joined by a segment,

$$\lVert \boldsymbol{F}(\boldsymbol{u}) - \boldsymbol{F}(\boldsymbol{v}) \rVert = \left\lVert \int_0^1 D\boldsymbol{F}\bigl(\boldsymbol{v} + \tau(\boldsymbol{u}-\boldsymbol{v})\bigr)(\boldsymbol{u}-\boldsymbol{v})\, d\tau \right\rVert \le L \lVert \boldsymbol{u} - \boldsymbol{v}\rVert .$$

The function $F(x) = 3x^{2/3}$ served as a counterexample precisely because it is not differentiable at $x=0$.

Implications for physics. Newtonian gravity $\boldsymbol{F} \propto \boldsymbol{r}/\lVert\boldsymbol{r}\rVert^3$ and the Coulomb force are both $C^\infty$ for $\boldsymbol{r} \ne \boldsymbol{0}$, so as long as the particles do not collide the Lipschitz condition holds and Theorem 3.2 applies. Conversely, at the instant of a collision ($\boldsymbol{r} = \boldsymbol{0}$) the force diverges and the guarantee of determinism is cut off. For gravitational many-body problems with three or more bodies it is known that there exist solutions escaping to infinity in finite time (collision and non-collision singularities), and there classical determinism itself ceases to be definable. Laplace’s demon had already stumbled on the singularities of the equations, long before it met chaos or quantum mechanics.