Open problems in the rigorous theory of Hamiltonian friction for a tracer particle in a Bose-Einstein condensate: sharp decay rates, two-body bound states, and interacting condensates

Fröhlich, Gang and Soffer (Comm. Math. Phys. 315 (2012) 401-444) rigorously established that a heavy tracer particle moving through an ideal Bose-Einstein condensate is decelerated to rest by emission of Cherenkov radiation of gapless modes, in a classical mean-field limit: for small initial data the particle momentum obeys $|P_t| \le c\,t^{-1/2-\delta}$ for every $\delta$ in an explicit interval $I$ with $\sup I \approx 0.66$, and the condensate fluctuation field relaxes to a static splash. The E-model (Bogoliubov limit of an interacting condensate) has since been substantially analyzed (subsonic ballistic stability, supersonic deceleration to the speed of sound in dimension three, and arbitrary-velocity scattering in dimension five or more), while the G-model with fully nonlinear condensate self-interaction, the dynamics in the presence of bound states or zero-energy resonances of the two-body Schrödinger operator, the sharp long-time decay exponent of the momentum, and the removal of the small-data and spherical-symmetry hypotheses remain open.

Background

How irreversible, dissipative-looking motion emerges from fundamentally reversible Hamiltonian dynamics is a classical problem of mathematical physics. Fröhlich, Gang and Soffer introduced a concrete and mathematically tractable family of models: a very heavy quantum particle (a tracer) moves through a very dense, weakly interacting Bose gas exhibiting Bose-Einstein condensation (the state of a dilute boson gas in which a macroscopic fraction of atoms occupies a single quantum wave function). In a mean-field limit (particle mass and gas density growing with a parameter $N$ while couplings are scaled appropriately), the quantum dynamics approaches classical nonlinear Hamiltonian evolution equations for the tracer position $X_t$ and momentum $P_t$ coupled to a complex order-parameter (Ginzburg-Landau) field describing the condensate; the physical picture is that a tracer moving faster than the speed of sound in the condensate emits sound waves (the gapless Goldstone modes of the broken $U(1)$ symmetry) - Cherenkov radiation - and thereby experiences a friction force with memory that decelerates it.

The family is parametrized by the tracer-atom coupling $g$, the atom-atom coupling $\kappa$, and the condensate density scale $\rho_0$: the B-model (ideal gas, $\kappa = 0$, weak coupling $g \to 0$), the C-model ($\kappa = 0$, $g \neq 0$), the E-model (the Bogoliubov limit in which $\kappa\rho_0/g^2$ is held fixed while $g, \kappa \to 0$, so that the condensate dynamics becomes quadratic in the fluctuation field), and the G-model ($\kappa > 0$, $g \neq 0$, with the full nonlinear condensate self-interaction). A convenient normalization of the E-model equations, due to Egli, Fröhlich, Gang, Shao and Sigal, is

$$M\dot X_t = P_t, \quad \dot P_t = -\nabla V(X_t) + \int_{\mathbb R^3} \nabla W(\cdot - X_t)\, \mathrm{Re}\,\beta_t, \quad i\dot\beta_t = -\Delta\beta_t + \mu\, \mathrm{Re}\,\beta_t + W(\cdot - X_t),$$

where $W$ is the two-body tracer-atom potential, $\beta_t$ is the condensate fluctuation field (the deviation of the order parameter from its value at infinity), and $\sqrt{\mu}$ is the speed of sound. These equations have traveling-wave solutions (inertial motion of the particle accompanied by a deformation, or splash, of the condensate) precisely at subsonic speeds, and no finite-energy traveling waves at supersonic speeds.

For the B-model, Fröhlich, Gang and Soffer proved (Theorem 2.1 of the source paper) that for smooth, spherically symmetric, rapidly decaying $W$ and small initial data ($\|\langle x\rangle^4 \beta_0\|_2$ and $|P_0|$ below a threshold depending on $\delta$), the momentum satisfies $|P_t| \le c\, t^{-1/2-\delta}$ as $t \to \infty$ for every $\delta$ in an explicitly defined interval $I \subset (0,1)$ with $\sup I \approx 0.66$, so that $X_t$ converges to a limiting position and $\beta_t$ converges to a static splash $-2(-\Delta)^{-1}W^{X_\infty}$ which is not square-integrable. Egli and Gang proved the analogous decay-to-rest theorem for the C-model at small coupling $g$, under the spectral assumption that the two-body Schrödinger operator $-(1/2m)\Delta + gW$ has no bound states (eigenfunctions below the continuum) and no zero-energy resonance (a non-square-integrable solution at zero energy, the borderline case where dispersive estimates degenerate).

The sharp decay rate is not known. In Appendix E.2 of the source paper the authors analyze the linearized integro-differential equation for the momentum: they prove rigorously that its solution cannot decay like $t^{-1/2-\delta}$ for any $\delta \in (0,1/2)$, and that if it decays like $C\,t^{-1-\delta}$ then $\delta$ must solve a transcendental equation whose unique solution (located numerically) lies in $(0.15, 0.17)$; note that this linearized prediction $t^{-1.16\ldots}$ matches the fastest proven upper bound $t^{-1/2-\sup I} \approx t^{-1.16}$, but a matching lower bound, the exact asymptotics, and a non-numerical identification of the exponent for the full nonlinear problem are missing. Related particle-field systems (a Schrödinger equation coupled to a classical particle, or Dirac fields coupled to a particle) have been studied by Komech, Kopylova, Spohn and others with soliton-scattering methods, and the quantum many-body side of the impurity-in-condensate problem has an active rigorous literature (mean-field limits, the Bogoliubov-Fröhlich Hamiltonian, radiative corrections), but these do not address the classical friction questions below.

Problem Statement

The common objective is to complete the rigorous theory of Hamiltonian friction - the emergence, mechanism, and quantitative long-time behavior of the deceleration of a heavy tracer particle - for the Fröhlich-Gang-Soffer tracer-in-condensate mean-field models, by resolving the four obstructions that the source paper and its companion work explicitly identify. Each component addresses a different obstruction to the same physical conclusion (friction and relaxation derived from reversible dynamics), and together they delimit the boundary of the known theory. The four components are independent; each is to be settled in full.

(1) Sharp decay rate (B-model, and the C-model at small $g$). Determine the optimal long-time decay of the tracer momentum. In particular, prove or disprove that, for generic small data (the notion of genericity is fixed here, independently of the asserted decay: it means that only a meager (first-category) subset of the small-data class $\|\langle x\rangle^4\beta_0\|_2, |P_0| \le \epsilon_0$ of Theorem 2.1 may be exceptional, and that exceptional set may not be defined in terms of the decay statement being proved), $|P_t|$ has an asymptotic equivalent $C\,t^{-1-\delta_*}$, where $\delta_*$ is the solution (located numerically in $(0.15, 0.17)$, and unique on the numerical evidence reported in the source) of the transcendental equation, Eq. (E.7) of the source, for the linearized integro-differential equation; this includes a rigorous, non-numerical characterization of $\delta_*$ and establishing the matching lower bound showing that the proven family $|P_t| \le c\,t^{-1/2-\delta}$, $\delta \in I$, is sharp at $\sup I \approx 0.66$ (or exhibiting the correct rate if it is not).

(2) Two-body bound states and zero-energy resonances (C-model). Analyze the dynamics when the Schrödinger operator $-(1/2m)\Delta + gW$ has bound states or a zero-energy resonance, a regime excluded by all existing theorems: establish or rule out the expected instabilities and run-away solutions for the coupled particle-field system, and characterize the qualitatively different long-time behavior (including the possibility that the tracer forms a bound complex with condensate atoms, changing its effective mass).

(3) Interacting condensates. For the G-model ($\kappa > 0$, $g \neq 0$, full nonlinear self-interaction of the condensate, no Bogoliubov limiting): prove global well-posedness and determine the friction mechanism and long-time behavior of the tracer (deceleration laws, convergence to ballistic or resting motion, relaxation of the field). Within the E-model, complete the remaining three-dimensional regimes: initial speed at or arbitrarily near the speed of sound (the trans-sonic regime, where the linearized operator has a threshold eigenvalue), inter-atomic potentials beyond the contact case, and data beyond the smallness restrictions of the known proofs.

(4) Hypotheses. Determine whether the B- and C-model decay-to-rest theorems extend beyond smooth, spherically symmetric, rapidly decaying potentials $W$ and small initial data: global existence and momentum decay for non-spherically-symmetric $W$, for weaker smoothness or decay of $W$, and for large weighted norms of $\beta_0$ or large $|P_0|$ (where the known contraction-mapping arguments break down).

References

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