An independent, fully classical derivation of the effective velocity and dressing equations of generalized hydrodynamics for classical integrable field theories with radiative modes (sinh-Gordon prototype)

Generalized hydrodynamics (GHD) describes the Euler-scale transport of integrable systems in terms of quasiparticle densities, a dressing operation, and a state-dependent effective velocity. For classical integrable field theories with purely radiative (continuous-spectrum) modes, exemplified by the classical sinh-Gordon model, these equations were first written down by Bastianello, Doyon, Watts and Yoshimura (SciPost Phys. 4, 045 (2018)), who obtained them by taking the semiclassical limit of the quantum theory; they explicitly left open the task of deriving the effective velocity (and the accompanying dressing and current-average formulae) fully within classical field theory. The problem as stated here is to construct that derivation: starting only from the classical field theory, its generalized Gibbs ensembles, and its classical inverse-scattering description, recover the phase-shift kernel, the dressing operation, the current averages, and the effective velocity without any input from the quantum Bethe ansatz. No derivation of these equations fully within classical field theory is currently known; constructing one is the problem.

Background

The sinh-Gordon model is a classical relativistic field theory in one spatial dimension for a scalar field $\Phi$ with Lagrangian density $\mathcal{L}=(1/2)\partial_\mu\Phi\,\partial^\mu\Phi-(m^2/g^2)(\cosh(g\Phi)-1)$, where $m$ sets the mass scale and $g$ the interaction strength. It is integrable: it possesses infinitely many local conserved charges, and it can be solved by the inverse scattering transform (IST), a method that linearizes the nonlinear evolution by mapping each field configuration to spectral data of an associated linear auxiliary problem. The IST spectrum decomposes into discrete parts (solitons — localized particle-like waves) and a continuous part (radiation — dispersive wave trains). The sinh-Gordon model is the simplest interacting relativistic example whose spectrum is purely radiative: it has no solitons, only dispersive radiation.

When such a system is prepared far from equilibrium with weak, large-scale inhomogeneities, its local stationary states are described by generalized Gibbs ensembles (GGEs): statistical ensembles with weight $\exp(-\sum_i \beta_i Q_i)$ over all (infinitely many) local conserved charges $Q_i$. The thermodynamics of these ensembles is captured by a thermodynamic Bethe ansatz (TBA): the entropy of the quasiparticle modes is expressed through a pseudo-energy $\varepsilon(\theta)$, a function of the rapidity $\theta$ (the parametrization of energy and momentum as $E(\theta)=m\cosh\theta$, $p(\theta)=m\sinh\theta$), which solves a nonlinear integral equation whose kernel is the two-body differential scattering phase $\varphi(\theta,\gamma)$ (the derivative of the scattering phase shift accumulated when two excitations cross). For the classical sinh-Gordon model this kernel is $\varphi(\theta,\gamma)=(g^2/4)\,\cosh(\theta-\gamma)/\sinh^2(\theta-\gamma)$. Since $\varphi$ has a double pole at $\theta=\gamma$, the TBA is written equivalently, after an integration by parts, with the simple-pole kernel $\tilde\varphi(\theta,\gamma)=(g^2/4)/\sinh(\theta-\gamma)$ (so that $\varphi=-\partial_\theta\tilde\varphi$), and the singular integrals are understood as Cauchy principal-value integrals of this equivalent form at $\theta=\gamma$. Notably, this TBA thermodynamics has a genuinely classical pedigree: it was derived by Bullough, Pilling and Timonen in 1986 and by De Luca and Mussardo in 2016 through the inverse-scattering description of the classical field, as well as through semiclassical limits of the quantum theory. A peculiarity of classical radiative fields is an ultraviolet catastrophe: because the density of radiative modes grows without bound at high rapidity, the GGE is divergent unless the higher conserved charges are included with appropriately growing chemical potentials, so any classical statistical treatment of radiative ensembles must control this regularization.

The hydrodynamics is newer territory. In 2016, Castro-Alvaredo, Doyon and Yoshimura, and independently Bertini, Collura, De Nardis and Fagotti, founded generalized hydrodynamics (GHD) for quantum integrable many-body systems: on the Euler scale — the longest hydrodynamic scale, on which the dynamics is purely convective and diffusive corrections are neglected — the dynamics is carried by quasiparticle densities $n(x,t;\theta)$ obeying $\partial_t n+v^{\mathrm{eff}}\partial_x n=0$, where the effective velocity $v^{\mathrm{eff}}$ is a state-dependent renormalization of the bare group velocity produced by elastic two-body scattering. Its central structure is the dressing operation

$$h^{\mathrm{dr}}(\theta)=h(\theta)+\int \frac{d\gamma}{2\pi}\,\varphi(\theta,\gamma)\,n(\gamma)\,h^{\mathrm{dr}}(\gamma),$$

from which follow the quasiparticle density $2\pi\rho_p=(p')^{\mathrm{dr}}n$, the exact current averages of the conserved charges $\langle j_i\rangle=\int \frac{d\theta}{2\pi}(E')^{\mathrm{dr}}(\theta)\,n(\theta)\,h_i(\theta)$, and the effective velocity

$$v^{\mathrm{eff}}(\theta)=\frac{(E')^{\mathrm{dr}}(\theta)}{(p')^{\mathrm{dr}}(\theta)}.$$

Here a prime denotes differentiation with respect to rapidity, $E'=dE/d\theta$ and $p'=dp/d\theta$, and $h_i(\theta)$ is the single-rapidity eigenvalue of the conserved charge $Q_i$ (its one-particle dispersion, e.g. $h_1=E$ for the energy).

For classical integrable field theories with purely radiative spectra, these hydrodynamic equations were first written down by Bastianello, Doyon, Watts and Yoshimura for the sinh-Gordon model (SciPost Phys. 4, 045 (2018)), where they were also verified numerically against Metropolis simulations of the classical field. However, the effective velocity and current averages were obtained there only via the semiclassical limit of the quantum sinh-Gordon thermodynamic Bethe ansatz and GHD (their Appendix A); the authors state verbatim that 'it would be interesting to have an independent derivation of the effective velocity fully within classical field theory, which we leave for later works'. Providing that derivation is the problem recorded here.

Problem Statement

Provide an independent, fully classical derivation of the Euler-scale hydrodynamic equations of generalized hydrodynamics for classical integrable field theories with radiative modes, taking the classical sinh-Gordon model as the concrete prototype: a derivation that covers the radiative generalized Gibbs ensembles of the sinh-Gordon model settles the problem; the identical question extends verbatim to other classical integrable field theories whose spectra are purely or partly radiative, but this extension is descriptive and not an additional deliverable, and the solitonic (discrete-spectrum) sector — for which classical derivations already exist (see references 6–12 and 14) — is not part of the target. Starting solely from the classical field theory — its Hamiltonian, its generalized Gibbs ensembles, and the classical inverse-scattering/spectral description of its solutions — and without passing through the semiclassical ($\hbar\to 0$) limit of the quantum theory or otherwise taking quantum Bethe-ansatz results (quantum S-matrix, quantum TBA, quantum GHD) as inputs, the derivation must obtain, for the radiative modes parametrized by the rapidity $\theta$:

1. the two-body differential scattering phase kernel of the classical field theory,

$$\varphi(\theta,\gamma)=\frac{g^2}{4}\,\frac{\cosh(\theta-\gamma)}{\sinh^2(\theta-\gamma)},$$

2. the dressing operation on a test function $h$,

$$h^{\mathrm{dr}}(\theta)=h(\theta)+\int\frac{d\gamma}{2\pi}\,\varphi(\theta,\gamma)\,n(\gamma)\,h^{\mathrm{dr}}(\gamma),$$

3. the exact current averages of the conserved charges $Q_i$ in a generalized Gibbs ensemble (with $h_i(\theta)$ the single-rapidity eigenvalue of $Q_i$),

$$\langle j_i\rangle=\int\frac{d\theta}{2\pi}\,(E')^{\mathrm{dr}}(\theta)\,n(\theta)\,h_i(\theta),$$

4. the effective velocity

$$v^{\mathrm{eff}}(\theta)=\frac{(E')^{\mathrm{dr}}(\theta)}{(p')^{\mathrm{dr}}(\theta)},$$

and thereby establish the hydrodynamic equation

$$\partial_t n(x,t;\theta)+v^{\mathrm{eff}}(x,t;\theta)\,\partial_x n(x,t;\theta)=0$$

on purely classical grounds. Because the classical inverse-scattering/spectral description of the solutions is among the permitted starting points, and because the kernel of item 1 already has classical inverse-scattering derivations (Bullough–Pilling–Timonen 1986; De Luca–Mussardo 2016; see Background and references 4, 5), item 1 may be established either by re-deriving that kernel from the classical dynamics or by invoking those classical results; the open content of the problem is carried by items 2–4 and the hydrodynamic equation, which must be obtained without any quantum input.

The derivation should explain how a velocity renormalization arises from the classical dynamics and statistics of radiative modes (for example from the classical scattering of wave packets or from the statistical mechanics of the continuous inverse-scattering spectrum), and must recover the known free-field limit $g\to 0$ and the dilute (low-occupation, $n\to 0$) limit, in both of which the dressing trivializes and $v^{\mathrm{eff}}(\theta)$ reduces to the bare velocity $\tanh\theta$. No artificial restriction to special states or observables is admitted beyond the ultraviolet finiteness that classical radiative ensembles themselves require: as noted in the Background, radiative GGEs are ultraviolet divergent unless higher conserved charges are included, and — as in the source treatment — averages of individual conserved densities or currents are finite only for ensembles containing sufficiently high-spin charges, or for ultraviolet-finite combinations of observables; the current-average formula of item 3 is to be established on that domain of finite ensembles and observables (equivalently, as an identity formal in the GGE data, valid whenever both sides converge). The higher-charge ultraviolet regularization is a known feature of the classical setting which the derivation must respect, not an additional deliverable.

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