A landmark 2015 experiment (Langen et al., Science 348, 207) observed a generalized Gibbs ensemble (GGE) in a coherently split one-dimensional Bose gas and found that roughly ten Lagrange multipliers sufficed to describe the measured steady-state correlation functions, although integrability supplies far more conserved quantities; the authors asked, in general, how many multipliers are needed to describe the steady state of a realistic integrable quantum system and conjectured that generic experimentally preparable initial states are described, to reasonable precision, by far fewer multipliers than the number N of degrees of freedom. In the decade of subsequent literature, complete-GGE theory and generalized hydrodynamics have identified which (infinite) sets of charges are exact in specific models; truncated-GGE studies in XXZ chains and the Ising field theory have shown that finite sets of 10-100 well-chosen charges can be highly accurate; squeezed-ensemble failures and non-Gaussian-correlation experiments have begun to identify where the canonical description breaks down; prethermalization theory bounds how long GGEs survive weak integrability breaking. What does not exist is the general principle asked for by Langen et al. (2015): a quantitative theory of the number of relevant multipliers as a function of model, initial-state class, observable family, and precision. Two related questions raised in the later literature - a general identification of the observables in which deviations first appear, and the time window over which a GGE description remains valid in realistic, only approximately integrable systems - likewise remain open, though neither is part of the objective pursued here. The problem therefore remains open.

Background

When an isolated quantum many-body system is driven far from equilibrium by a sudden change of a parameter (a "quantum quench"), it cannot exchange heat or particles with a bath, yet local observables typically relax to stationary values. For generic (non-integrable) systems, the eigenstate thermalization hypothesis explains why these stationary values are described by an ordinary thermal Gibbs ensemble fixed only by the conserved energy and particle number. Integrable systems, such as the one-dimensional Bose gas with repulsive contact interactions (the Lieb-Liniger model), instead possess an extensive family of independent conserved quantities - in the thermodynamic limit, a continuum of them - which prevent ordinary thermalization. Rigol, Dunjko and Olshanii proposed that the stationary state of such a system is described by a generalized Gibbs ensemble (GGE): a maximum-entropy density matrix of the form ρ^GGE∝exp⁡(−∑mλmI^m)\hat\rho_{\mathrm{GGE}}\propto\exp(-\sum_m\lambda_m\hat{I}_m), in which every conserved quantity I^m\hat{I}_m enters with its own Lagrange multiplier λm\lambda_m fixed by the initial expectation value. Each multiplier behaves as a temperature-like parameter, one per charge.

The 2015 experiment realized this program with a degenerate one-dimensional Bose gas coherently split into two halves. The splitting quench populates the phononic (sound-wave) modes of the gas, described at low energies by the Luttinger-liquid theory (the universal effective description of one-dimensional fluids as noninteracting sound waves), and the occupation numbers of these modes are the conserved quantities of the corresponding integrable model. Matter-wave interferometry between the two halves measured two-point phase correlation functions of the steady state, and higher-order (up to tenth-order) correlation functions, and showed that they are reproduced by a GGE whose multipliers were fixed independently. A chi-square analysis showed that the occupations of the nine lowest modes plus one additional multiplier for all higher modes - about ten Lagrange multipliers in total - already described the measured observables to the precision of the data, although the integrable model provides far more conserved quantities (the supplementary material attributes part of this saturation to the decay of higher-mode contributions and the finite imaging resolution). The experiment also noted that a full tomography of all GGE parameters would require exponentially many measurements, and verified the ensemble against higher-order correlation functions, motivated by the expectation that deviations from a GGE description should manifest first in such higher-order observables.

The number of multipliers is not a technical fitting detail: it measures how much information about the initial state an integrable system retains at late times, i.e. whether the steady state is, like an equilibrium state of ordinary thermodynamics, characterized by a handful of parameters, or whether genuinely extensive information survives the relaxation. This is why the authors asked how many multipliers are needed in general for a realistic integrable quantum system, and conjectured that most experimentally obtainable initial states evolve into steady states describable to reasonable precision by far fewer than N multipliers, where N is the number of degrees of freedom.

Three later lines of theory transformed the landscape without answering the general question. First, "complete" GGEs were constructed: for the spin-1/2 Heisenberg chain, including quasilocal conserved quantities (charges that are sums of locally supported operators with tails decaying fast enough to give finite thermodynamic weight) makes the GGE exact for arbitrary quenches, and for interacting Bethe-ansatz gases (one-dimensional models exactly solvable by the scattering-based Bethe ansatz), generalized hydrodynamics (the exact thermodynamic-limit theory of transport in integrable systems) shows that the exact ensemble requires the full occupation function over rapidity, the quasiparticle momentum parameter of the Bethe ansatz: a continuum of charges rather than any finite number. Second, truncated GGEs, which retain only a finite number of best-localized charges, were shown in specific models (XXZ spin chains, the Ising field theory) to reproduce local steady-state observables with remarkable accuracy using on the order of 10-100 charges, with the convergence depending on which basis of charges is chosen. Third, prethermalization theory establishes that under weak integrability breaking (small perturbations that spoil exact conservation of the charges) a GGE-like state persists as a long-lived prethermal plateau before eventual thermalization, with lifetimes set by the perturbation strength. What is still missing is precisely what Langen et al. (2015) asked for: a general principle connecting the model, the initial state, the observable family, and the required precision to the number of multipliers needed. The identification of the observables in which such a description fails first, and the time window over which it remains valid in realistic - trapped, finite-temperature, only approximately integrable - systems, are related questions raised in the later literature; Langen et al. (2015) note only that approximately conserved quantities keep the GGE description valid for significant timescales.

Problem Statement

Establish the general principle that fixes how many Lagrange multipliers - and, beyond the count, which finite subset of the extensive family of conserved charges - a generalized Gibbs ensemble must contain in order to reproduce the long-time steady state of a realistic integrable quantum many-body system, such as the coherently split one-dimensional Bose gas of the 2015 experiment, to any specified precision, for a specified family of observables and class of initial states. The single scientific objective is a quantitative theory of the economy of the generalized-thermal description, and it comprises the following jointly supporting aspects: (i) how the truncation error of a finite-charge GGE decreases with the number and choice (basis) of included charges, and how the required number scales with the order and locality of the observables, with properties of the initial state, and with system size; and (ii) whether the conjecture of Langen et al. (2015) holds, that generic experimentally preparable initial states relax to steady states described, to reasonable precision, by far fewer than N Lagrange multipliers, where N is the number of degrees of freedom. Two related questions raised in the later literature frame the realistic setting of this objective without being part of it: identifying the observables in which deviations from a truncated (or canonical local-charge) GGE first become measurable - Langen et al. (2015) expect higher-order correlation functions to play this role - and determining the time window over which such a description remains valid in a realistic, only approximately integrable system, given that weak integrability breaking turns the GGE into a prethermal plateau of finite lifetime. A complete answer must address both aspects (i) and (ii), and may take any of several forms: a proven general framework; controlled model-spanning results (bounds or scaling laws covering interacting lattice models and continuum gases); a decisive refutation of the conjecture for experimentally preparable states; or a systematic experimental characterization establishing the scaling of the required multiplier count on a realistic platform.

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