Max Born's 1958 paper 'Vorhersagbarkeit in der klassischen Mechanik' (Zeitschrift für Physik 153, 372–388) analyzed the evolution of Gaussian ensembles with finite initial spread in integrable ('multiply periodic') Hamiltonian systems using action-angle variables. He showed that anharmonicity shears the ensemble so that, after a critical time, the angle variables become uniformly distributed — a complete loss of angle predictability — while the action variables remain sharply determined, an asymptotic state with the structure of Bohr's stationary states; uncoupled harmonic systems, and hence all free (uncoupled) fields, are the only exception. Born then conjectured that introducing nonlinear coupling terms between fields of different kinds must likewise produce the smearing of angle variables and sharpening of action variables, explicitly leaving this extension for further investigation. Whether, and under which conditions, Born's predictability-loss phenomenon extends to nonlinearly coupled classical fields is open: no proof, refutation, or reformulation of the conjecture is known in the literature.

Background

Classical mechanics is deterministic in the Laplacean sense only if initial conditions are known exactly. Max Born recast determinism as an operational question about predictability: since initial conditions are always known with finite precision, the physically meaningful object is not a single trajectory but an ensemble, i.e. a probability density on phase space transported by the Liouville equation (the linear first-order equation governing the evolution of a phase-space probability density under Hamiltonian flow). A system counts as predictable only if the evolved distribution becomes arbitrarily sharp when the initial spread is shrunk to zero after the evolution — the 'double limit': first evolve a distribution of finite width, then let the initial width go to zero.

In his 1958 paper, Born carried this program out for conservative systems that admit action-angle variables. Action-angle variables (φk,Ik)(\varphi_k, I_k) are special canonical coordinates available for integrable ('multiply periodic') systems: the actions IkI_k are constants of the motion (functions of the conserved momenta), and the angles φk\varphi_k are their conjugate cyclic variables, defined modulo a period, in which the motion is quasi-periodic,

I˙k=0,φ˙k=νk(I),νk=∂E/∂Ik,\dot I_k = 0, \qquad \dot\varphi_k = \nu_k(I), \quad \nu_k = \partial E/\partial I_k,

where E(I)E(I) is the Hamiltonian expressed through the actions and νk\nu_k are the frequencies. For an initial Gaussian ensemble in (φ,I)(\varphi, I), the dependence of the frequencies on the actions (the anharmonicity matrix μkl=∂2E/∂Ik∂Il\mu_{kl} = \partial^2 E/\partial I_k \partial I_l) shears the ensemble, exactly as velocity spread shears a free-packet: a localized blob is stretched along the angle directions and squeezed in the transverse direction. After a critical time tct_c (inversely proportional to the initial action spread and to the anharmonicity, tc=1/(mσ)t_c = 1/(m\sigma) with mm the smallest eigenvalue of the shear matrix and σ\sigma a measure of the initial action spread), the angle distribution becomes uniform over its period, so no prediction of the angle variables is possible, while the action variables remain sharply determined — arbitrarily sharp as the initial ensemble is narrowed in Born's double limit. Born emphasized that this asymptotic stationary ensemble, with sharp actions and completely undetermined angles, has exactly the structure of Bohr's stationary states in the old quantum theory (sharp, eventually quantized action variables with physically irrelevant angles), offering a classical-statistical rationale for why old quantum theory singled out actions for quantization.

The exceptional case is μkl=0\mu_{kl} = 0 identically, i.e. uncoupled harmonic oscillators, whose frequencies are amplitude-independent: nothing spreads, and the initial Gaussian persists forever. Since all free fields (fields without coupling terms) are equivalent to collections of independent harmonic normal modes, they share this exceptional, perfectly predictable status.

Born closed by conjecturing that the situation must change as soon as one introduces nonlinear coupling terms between fields of different kinds: the smearing of the angle variables and the sharpening of the action variables, he argued, must also occur there — and stated that this should be investigated further ('Dies sollte näher untersucht werden'). This conjecture is the open problem. It is not a routine extension, because generic nonlinearly coupled fields are not integrable: global action-angle variables no longer exist, and resonant interactions can transfer action between modes. Modern theory describes both possible fates of coupled Hamiltonian systems: strong sensitivity to initial conditions (chaos, with exponential divergence of nearby trajectories), and — in the opposite direction — Kolmogorov–Arnold–Moser (KAM) theory, which proves that sufficiently weak perturbations of integrable systems retain quasi-periodic motion on a large measure of invariant tori, and the Fermi–Pasta–Ulam–Tsingou (FPUT) problem of nonlinear oscillator chains, whose approach to thermalization (redistribution of mode energies) is famously subtle and, as current work states, still not fully understood. The rigorous wave-turbulence program similarly derives kinetic equations for the statistics of mode actions and phases of weakly nonlinear fields, in specific scaling limits. What plays the role of 'angles' and 'actions' in a coupled field, and whether Born's asymptotic picture of predictability loss survives the coupling, is precisely what has to be established.

Problem Statement

Born's 1958 analysis established, for conservative integrable (multiply periodic) systems evolving Gaussian ensembles of finite initial spread in action-angle variables, that the asymptotic state has sharply determined action variables and completely smeared — uniformly distributed — angle variables, i.e. a total loss of predictability of the angles after a critical time, with uncoupled harmonic systems (and hence all free, uncoupled fields) the only exception. The open problem is the extension Born himself conjectured but left unproven: introduce nonlinear coupling terms between fields of different kinds and determine whether the same phenomenon — smearing of the angle variables and sharpening of the action variables, with the associated loss of predictability — also occurs. The conjectured extension is the conjunction of two independently refutable components: (A) the angle-like variables become equidistributed, and (B) the action-like variables remain sharply determined. Because it is a conjunction, a counterexample to either component in a specified coupled classical field refutes the conjecture. The question is to make the conjecture precise for coupled classical fields — what plays the role of angle and action variables in the (generically non-integrable) coupled system, the initial ensemble, and the order of the long-time and vanishing-initial-spread limits — and to establish whether and under which conditions the conjectured asymptotics hold or fail. No restriction to a particular field theory, coupling strength, spatial dimension, or initial distribution is part of the question; delimiting the regimes in which the phenomenon does or does not occur is not a narrowing of it.

References

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