Typicality

Typicality says that a property holds on a set assigned overwhelmingly large measure. It is useful in deterministic and stochastic theories alike, but “most” has no mathematical meaning until a measure is specified. A typicality explanation must therefore justify both the measure and the link between its large-measure sets and empirical expectations.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Measure before majority

Let $\Omega$ describe allowed configurations or histories and let $\mu$ be a normalized probability measure, $\mu(\Omega)=1$. A property $T$ is typical at tolerance $0<\epsilon<1$ if

$$\mu(T)\ge 1-\epsilon.$$

Different measures can disagree on which sets are typical. Counting equal-sized coordinate bins is not invariant under a nonlinear coordinate change. A physical theory should supply structure that selects an appropriate measure or class of equivalent measures.

In pilot-wave theory, the squared wavefunction defines an equivariant configuration measure. In an Everettian analysis, squared amplitudes define a measure on decohered records. These are mathematically related constructions with different ontological roles.

A finite concentration argument

Suppose a product preparation induces independent binary variables with success probability $p$ under a stated measure. Let $F_N=N^{-1}\sum_jX_j$. Then

$$\mathbb E(F_N)=p,\qquad \operatorname{Var}(F_N)=\frac{p(1-p)}N.$$

Chebyshev’s inequality gives

$$\mu(|F_N-p|\ge\delta) \le \frac{p(1-p)}{N\delta^2}.$$

For $p=1/2$, $N=10^4$, and $\delta=0.05$, the bound is $0.01$. At least $99\%$ of the measure lies on records whose frequency differs from $p$ by less than $0.05$. The bound is conservative but explicit.

The argument establishes concentration under the chosen measure. It does not establish that no exceptional sequence exists or that every initial configuration relaxes to equilibrium.

Equivariance and equilibrium

A measure is equivariant if evolving configurations and evolving the wavefunction transport the measure consistently. For Bohmian motion, the same continuity equation governs a distribution initially equal to $|\psi|^2$ and the squared wavefunction itself.

Equivariance is not convergence. A nonequilibrium density need not become exactly $|\psi|^2$ under deterministic invertible evolution. Coarse-grained relaxation may occur under additional dynamical assumptions; it must not be substituted for the exact preservation result.

Typicality arguments can avoid imagining repeated preparations of the entire universe: they assess which subsystem records occur for most universal initial configurations under a nominated measure. Whether this adequately explains probability is a separate foundational question.

Limits of the explanation

A very small exceptional set can contain physically possible histories. Typicality reasoning does not identify the actual universal configuration. Nor does it establish the measure by declaring measured frequencies typical after seeing them; that would supply little predictive content.

Useful applications specify the initial macroconditions, dynamical law, measure, observed subsystem, and approximation error. Claims that rely on independence should state why the relevant correlations are negligible.

Exercises

Does conservation of the ratio $f=\rho/|\psi|^2$ along Bohmian trajectories prove relaxation to equilibrium?

Solution

No. Conservation carries any initial departure along trajectories. Fine-grained equality is not generated from arbitrary inequality. Apparent relaxation can arise after coarse-graining when the distribution develops fine structure, but it requires additional assumptions and does not contradict conservation of $f$.

References

• D. Dürr, S. Goldstein, and N. Zanghì, “Quantum equilibrium and the origin of absolute uncertainty,” Journal of Statistical Physics 67, 843–907 (1992), author manuscript.
• A. Valentini, “Signal-locality, uncertainty, and the subquantum H-theorem. I,” Physics Letters A 156, 5–11 (1991), doi:10.1016/0375-9601(91)90116-P.