{
  "schema_version": 1,
  "id": "PHYS-2026-08-10-01",
  "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-10-01/",
  "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-10-01.md",
  "metadata": {
    "schema_version": 1,
    "id": "PHYS-2026-08-10-01",
    "date": "2026-08-10",
    "updated_at": "2026-08-10",
    "title": "Interactions can dress motion without thermalizing rapidities",
    "summary": "Derive generalized hydrodynamics from Bethe root-density conservation, separate elastic factorized scattering from Boltzmann relaxation, and follow weak integrability breaking back to ordinary hydrodynamics.",
    "language": "en",
    "entry_kind": "daily",
    "status": "published",
    "level": "research",
    "user_difficulty": "unrated",
    "domains": [
      "condensed-matter",
      "statistical-mechanics",
      "quantum-theory",
      "mathematical-physics"
    ],
    "estimated_minutes": 100
  },
  "content_markdown": "\n## The claim under pressure\n\n**An Euler-scale kinetic equation with no Boltzmann collision integral need not describe a free gas.**\n\nIn an interacting integrable system, scattering can remain elastic and factorized. It shifts trajectories and dresses velocities while preserving the stable quasiparticle content.\n\nThe resulting equation looks like advection,\n\n$$\n\\partial_t\\vartheta(\\lambda,x,t)\n+v^{\\rm eff}[\\vartheta](\\lambda,x,t)\\,\n\\partial_x\\vartheta(\\lambda,x,t)=0,\n$$\n\nbut its velocity is a nonlinear functional of the local many-body state. The absence of a rapidity-redistributing collision term is not the absence of interactions.\n\nThis distinction is the spine of generalized hydrodynamics (GHD).\n\n## 1. Four processes that should not share one name\n\nThe word “collision” hides several different statements.\n\n### Microscopic encounter\n\nParticles or wave packets overlap and scatter. This can occur in both integrable and nonintegrable systems.\n\n### Elastic factorized scattering\n\nIn a stable integrable quasiparticle basis, an incoming set of rapidities reappears after scattering. The many-body $S$-matrix factorizes into two-body amplitudes, subject to the usual integrability assumptions.\n\n### Dressing\n\nAdding one excitation shifts $O(L)$ background Bethe roots by $O(L^{-1})$ in a system of size $L$. Their sum is finite. Energy, momentum, charge, and velocity therefore depend on the occupied state.\n\n### Thermalizing redistribution\n\nA Boltzmann collision operator transfers occupation between quasiparticle modes while preserving only its collision invariants, usually particle number, momentum, and energy.\n\nGHD contains the first three processes at Euler scale. The fourth appears after integrability is broken, or when one has chosen variables that are not stable quasiparticles.\n\n“Collisionless” is therefore risky shorthand. **Collision-integral-free at Euler scale** is more precise.\n\n## 2. Lieb–Liniger thermodynamics fixes the notation\n\nConsider the repulsive Lieb–Liniger gas in units $\\hbar=2m=1$:\n\n$$\nH=\\int dx\\,\n\\left[\n\\partial_x\\psi^\\dagger\\partial_x\\psi\n+c\\,\\psi^\\dagger\\psi^\\dagger\\psi\\psi\n\\right],\n\\qquad c>0.\n$$\n\nIts thermodynamic states are described by a particle root density $\\rho_{\\rm p}(\\lambda)$ and a total density of available Bethe states $\\rho_{\\rm s}(\\lambda)$. Define the filling\n\n$$\n\\vartheta(\\lambda)=\n\\frac{\\rho_{\\rm p}(\\lambda)}{\\rho_{\\rm s}(\\lambda)},\n\\qquad 0\\leq\\vartheta\\leq1.\n$$\n\nFor the kernel\n\n$$\nK(\\lambda)=\\frac{2c}{\\lambda^2+c^2},\n$$\n\nuse the convolution convention\n\n$$\n(\\mathcal K f)(\\lambda)\n=\\int\\frac{d\\mu}{2\\pi}\\,\nK(\\lambda-\\mu)f(\\mu).\n$$\n\nThe thermodynamic Bethe equation is\n\n$$\n\\rho_{\\rm s}(\\lambda)\n=\\frac{p'(\\lambda)}{2\\pi}\n+(\\mathcal K\\rho_{\\rm p})(\\lambda).\n$$\n\nFor Lieb–Liniger, $p(\\lambda)=\\lambda$ and $E(\\lambda)=\\lambda^2$ in these units. The formulas below are written so that the same logic applies more widely.\n\n## 3. Dressing is many-body backflow\n\nFor any one-particle function $h(\\lambda)$, define its dressing by\n\n$$\nh^{\\rm dr}(\\lambda)\n=h(\\lambda)\n+\\mathcal K\\!\\left(\\vartheta h^{\\rm dr}\\right)(\\lambda).\n$$\n\nSince $\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}$, the Bethe equation gives\n\n$$\n2\\pi\\rho_{\\rm s}(\\lambda)\n=\\bigl(p'(\\lambda)\\bigr)^{\\rm dr}.\n$$\n\nThe effective velocity is\n\n$$\nv^{\\rm eff}(\\lambda)\n=\\frac{\\bigl(E'(\\lambda)\\bigr)^{\\rm dr}}\n{\\bigl(p'(\\lambda)\\bigr)^{\\rm dr}}.\n$$\n\nThis ratio is not a thermodynamic correction pasted onto free motion. It is the propagation speed of an excitation after the background backflow has been included.\n\nFor a conserved charge with bare one-particle eigenvalue $h(\\lambda)$, the density and Euler current are\n\n$$\nq[h]=\\int d\\lambda\\,\\rho_{\\rm p}(\\lambda)h(\\lambda),\n$$\n\n$$\nj[h]=\\int d\\lambda\\,\n\\rho_{\\rm p}(\\lambda)v^{\\rm eff}(\\lambda)h(\\lambda).\n$$\n\nInteractions enter both the allowed state density and the current. Replacing $v^{\\rm eff}$ by the bare velocity generally destroys the interacting hydrodynamics.\n\n## 4. Root-density conservation produces filling advection\n\nThe Euler equation is most safely stated first for the quasiparticle density, as in the [foundational Lieb–Liniger construction](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.041065):\n\n$$\n\\partial_t\\rho_{\\rm p}(\\lambda)\n+\\partial_x\\!\\left(\nv^{\\rm eff}(\\lambda)\\rho_{\\rm p}(\\lambda)\n\\right)=0.\n$$\n\nThe density of available states obeys the corresponding equation\n\n$$\n\\partial_t\\rho_{\\rm s}(\\lambda)\n+\\partial_x\\!\\left(\nv^{\\rm eff}(\\lambda)\\rho_{\\rm s}(\\lambda)\n\\right)=0.\n$$\n\nUse $\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}$ in the first equation:\n\n$$\n\\rho_{\\rm s}\n\\left(\\partial_t+v^{\\rm eff}\\partial_x\\right)\\vartheta\n+\\vartheta\n\\left[\n\\partial_t\\rho_{\\rm s}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm s})\n\\right]=0.\n$$\n\nThe bracket vanishes. Wherever $\\rho_{\\rm s}>0$,\n\n$$\n\\boxed{\n\\left(\\partial_t+v^{\\rm eff}[\\vartheta]\\partial_x\\right)\n\\vartheta=0\n}.\n$$\n\nThe equation is diagonal in rapidity, but not linear. Every velocity depends on the full local filling through the dressing integral equation.\n\nIn an inhomogeneous force field, the phase-space equation also contains rapidity acceleration. Schematically,\n\n$$\n\\partial_t\\rho_{\\rm p}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm p})\n+\\partial_\\lambda(a^{\\rm eff}\\rho_{\\rm p})=0.\n$$\n\nThe simple advection equation above assumes no such force term.\n\n## 5. Why phase shifts become velocities\n\nAn elastic two-body phase shift has a spatial interpretation. A wave packet exits a collision displaced relative to free propagation.\n\nLet $\\Delta x(\\lambda,\\mu)$ denote the trajectory shift of a $\\lambda$ quasiparticle caused by a $\\mu$ quasiparticle. During a time interval, the collision rate is set by their relative effective velocity.\n\nThis leads to a self-consistent kinetic form,\n\n$$\nv^{\\rm eff}(\\lambda)\n=v^{\\rm bare}(\\lambda)\n+\\int d\\mu\\,\\rho_{\\rm p}(\\mu)\n\\Delta x(\\lambda,\\mu)\n\\left[\nv^{\\rm eff}(\\lambda)-v^{\\rm eff}(\\mu)\n\\right].\n$$\n\nThe sign assigned to $\\Delta x$ depends on whether one defines a delay or an advance. The invariant content is the self-consistency: collision frequency depends on dressed relative motion, and accumulated shifts determine dressed motion.\n\nThis is the kinetic meaning of dressing. The quasiparticle keeps its rapidity in elastic factorized scattering, yet its ray $x/t$ changes because all other occupied rapidities alter its trajectory.\n\nThe statement must remain qualified. It concerns stable asymptotic quasiparticles and the integrable scattering description. It is not a literal account of distinguishable particles bouncing pairwise inside every integrable model.\n\n## 6. Hard rods expose the same mechanism\n\nA classical gas of rods of length $a$ offers a clean analogue. Let $\\rho(v)$ be the density of velocity quasiparticles,\n\n$$\nn=\\int dv\\,\\rho(v),\n\\qquad\nu=\\frac{1}{n}\\int dv\\,v\\rho(v),\n\\qquad an<1.\n$$\n\nEqual-mass rods exchange physical velocities at contact. If velocity labels are followed through the collision, they pass through one another with a spatial shift $a$.\n\nTheir effective velocity satisfies\n\n$$\nv^{\\rm eff}(v)\n=v+a\\int dw\\,\\rho(w)\n\\left[v^{\\rm eff}(v)-v^{\\rm eff}(w)\\right].\n$$\n\nThe solution is\n\n$$\nv^{\\rm eff}(v)=\\frac{v-an u}{1-an}.\n$$\n\nHence\n\n$$\nv^{\\rm eff}(v)-v^{\\rm eff}(w)\n=\\frac{v-w}{1-an}.\n$$\n\nExcluded volume accelerates relative quasiparticle motion in physical coordinates. No velocity is randomized, but the gas is not free.\n\nQuantum GHD replaces the fixed rod displacement by a rapidity-dependent scattering shift. The analogy is structural, not an identity between microscopic models.\n\n## 7. Local GGE is not manufactured by Boltzmann collisions\n\nOrdinary hydrodynamics assumes local Gibbs states characterized by a few conserved densities. An integrable model has an extensive family of conserved charges, so its local macrostate requires an entire filling function.\n\nGHD assumes local quasistationarity: after microscopic transients and coarse graining, a fluid cell is represented by a local generalized Gibbs ensemble, or an equivalent thermodynamic Bethe state.\n\nDephasing and scale separation support this description. A Boltzmann operator that destroys all but three moments is not the mechanism that defines it.\n\nFor Lieb–Liniger-type thermodynamic Bethe ansatz, the Yang–Yang entropy density is\n\n$$\ns_{\\rm YY}\n=\\int d\\lambda\\,\\rho_{\\rm s}(\\lambda)\n\\left[\n-\\vartheta\\log\\vartheta\n-(1-\\vartheta)\\log(1-\\vartheta)\n\\right].\n$$\n\nAt Euler scale it obeys an entropy continuity equation,\n\n$$\n\\partial_t s_{\\rm YY}+\\partial_x j_s=0.\n$$\n\nThere is no Euler-scale entropy production. This does not mean that fine-grained dynamics has become trivial or that all observables retain memory forever. It specifies what the Euler closure preserves.\n\n## 8. Beyond Euler: diffusion without thermalizing rapidities\n\nIntegrable transport is not synonymous with perfectly ballistic transport in every channel.\n\nFluctuations in the number and timing of collisions broaden quasiparticle trajectories by $O(t^{1/2})$ around their ballistic rays. This produces diffusive corrections to GHD even though the integrable charges remain conserved.\n\nNear a homogeneous state, the conventional Navier–Stokes-scale description contains a diffusion operator in rapidity space. It broadens profiles and produces coarse-grained entropy while respecting the full integrable conservation structure ([De Nardis, Bernard, and Doyon, 2018](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.160603)).\n\nThat mechanism is distinct from a nonintegrable Boltzmann operator. It does not generically collapse the filling to an ordinary thermal Gibbs distribution.\n\nThe scope of a local diffusion equation also matters. A [2025 analysis](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.134.187101) showed that, beyond linear response for linearly degenerate hydrodynamics, diffusive-order corrections can be controlled by nonlocal long-range correlations and may be reversible.\n\nThis result does not erase linear-response diffusive GHD. It warns against promoting its local gradient form to a universal nonlinear law far from homogeneous states.\n\nSymmetry can change the scaling further. At special points such as the isotropic Heisenberg chain, large quasiparticles and non-Abelian symmetry lead to superdiffusive rather than ordinary diffusive spin transport; the [2025 GHD perspective](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.15.010501) reviews this mechanism and its limits.\n\nThe useful hierarchy is\n\n$$\n\\text{ballistic Euler rays}\n\\;\\to\\;\n\\text{integrability-preserving broadening}\n\\;\\to\\;\n\\text{rapidity redistribution if integrability breaks}.\n$$\n\nThese arrows label increasing levels of description, not a universal sequence in every observable.\n\n## 9. Weak integrability breaking restores a collision operator\n\nLet a perturbation of strength $g$ break the higher conserved charges. The kinetic equation becomes\n\n$$\n\\partial_t\\rho_{\\rm p}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm p})\n=\\mathcal I_{\\rm br}[\\rho_{\\rm p}].\n$$\n\nThe collision operator redistributes rapidities. It must still preserve every exact charge. If particle number, momentum, and energy survive, then\n\n$$\n\\int d\\lambda\\,\n\\begin{pmatrix}\n1\\\\\np(\\lambda)\\\\\nE(\\lambda)\n\\end{pmatrix}\n\\mathcal I_{\\rm br}(\\lambda)=0.\n$$\n\nAll other generalized charges can decay.\n\nA naive relaxation-time ansatz,\n\n$$\n\\mathcal I_{\\rm naive}\n=-\\frac{\\rho_{\\rm p}-\\rho_{\\rm p}^{\\star}}{\\tau},\n$$\n\nviolates conservation unless $\\rho_{\\rm p}^{\\star}$ has the same exact conserved moments as $\\rho_{\\rm p}$. A sound approximation projects out the collision invariants.\n\nThe crossover has at least two time regimes:\n\n$$\nt\\ll\\tau_{\\rm br}:\n\\quad \\text{prethermal GHD},\n$$\n\n$$\nt\\gg\\tau_{\\rm br}:\n\\quad \\text{hydrodynamics of the exact conserved modes}.\n$$\n\nOften $\\tau_{\\rm br}\\propto g^{-2}$ by a golden-rule estimate. Selection rules, kinematic restrictions, and low-dimensional phase space can change that scaling, so it is not a theorem.\n\nIf number, momentum, and energy remain exact in a Galilean fluid, a Chapman–Enskog expansion should recover viscous and thermal hydrodynamics. If momentum relaxes, the late regime may contain only particle and energy diffusion.\n\nIn one dimension, nonlinear fluctuating hydrodynamics can control still longer scales and produce KPZ sound broadening under the appropriate mode-coupling conditions. KPZ behavior is not guaranteed by weak integrability breaking alone.\n\n## 10. Smooth Euler flow is a model-dependent result\n\nNonlinear advection often suggests shock formation. GHD is unusual because its characteristic fields are linearly degenerate in the integrable setting.\n\nA [2025 result](https://journals.aps.org/prb/abstract/10.1103/5w4p-bxnt) proved absence of shocks for a broad class of Lieb–Liniger initial states under stated regularity and support conditions. It sharpens the reason Euler GHD can remain smooth in that problem.\n\nIt should not be recited as “integrable hydrodynamics never forms shocks.” Other models, singular states, forcing, boundaries, or integrability-breaking terms require separate analysis.\n\nThis is a useful methodological lesson: an exact structural property can be strong without being universal.\n\n## 11. What experiment and current theory can actually test\n\nThe [2021 quantum-gas experiment](https://www.science.org/doi/10.1126/science.abf0147) on one-dimensional Bose gases observed dynamics that conventional hydrodynamics could not capture and found quantitative GHD behavior across interacting regimes.\n\nThe discriminating object is not a fitted velocity alone. It is the state-dependent propagation encoded by a rapidity distribution and its dressed velocities.\n\nRecent work has pushed the framework in three useful directions.\n\n### Long-range correlations at diffusive order\n\nDiffusive corrections far from homogeneous states can depend on correlations that no local one-point constitutive law contains. Measuring two-point structure becomes part of testing hydrodynamics, not an optional refinement.\n\n### Wave-packet simulations of generalized fluids\n\nA [2026 construction](https://journals.aps.org/prb/abstract/10.1103/b587-8yyt) represents generalized fluids by interacting wave packets. It gives a concrete trajectory picture, incorporates integrability breaking, and provides access to fluctuating correlations beyond one-point Euler evolution.\n\nOne-point observables can look thermal while long-range correlations still record the generalized fluid. “Apparent thermalization” must therefore specify which observables have been tested.\n\n### Controlled crossover models\n\n[Relaxation-time models](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.L060302) are being used to connect GHD with conventional hydrodynamics. Their value is diagnostic: they reveal which modes survive and how transport crosses over.\n\nTheir conclusions depend on the chosen collision operator. A relaxation-time model is not a microscopic derivation unless its rates and conserved subspace follow from the perturbation.\n\n## The deliberately false inference\n\n**If the kinetic equation contains no collision integral, then the gas is free and interactions can affect only its equation of state.**\n\nThe interacting Lieb–Liniger gas is a counterexample. Its filling advects without an Euler collision integral, but\n\n$$\nv^{\\rm eff}(\\lambda)\n=\\frac{(E')^{\\rm dr}}{(p')^{\\rm dr}}\n$$\n\ndepends on the complete local filling through the scattering kernel.\n\nThe interactions change currents, propagation rays, expansion dynamics, and correlations. What they do not do at integrable Euler scale is erase rapidity occupations down to a few thermal moments.\n\n## Collision: four levels\n\n### Scattering data\n\nElastic factorized scattering preserves the stable quasiparticle rapidities and supplies phase shifts.\n\n### Thermodynamic state\n\nBackflow dresses momentum, energy, charges, and available states. This is an $O(1)$ collective effect.\n\n### Euler dynamics\n\nRoot densities satisfy continuity equations. The filling is transported by a state-dependent effective velocity, with no rapidity-redistributing collision operator.\n\n### Loss of integrability\n\nA true collision operator relaxes the nonprotected generalized charges. Its null space determines the variables of late-time ordinary hydrodynamics.\n\n## Do this now\n\nWork with the classical hard-rod gas. Let\n\n$$\nn=\\int dv\\,\\rho(v),\n\\qquad\nu=\\frac{1}{n}\\int dv\\,v\\rho(v),\n\\qquad an<1,\n$$\n\nand suppose\n\n$$\nv^{\\rm eff}(v)\n=v+a\\int dw\\,\\rho(w)\n\\left[v^{\\rm eff}(v)-v^{\\rm eff}(w)\\right].\n$$\n\n### Target A: solve the self-consistency equation\n\nDefine\n\n$$\nj=\\int dv\\,\\rho(v)v^{\\rm eff}(v).\n$$\n\nShow that $j=nu$ and derive\n\n$$\nv^{\\rm eff}(v)=\\frac{v-an u}{1-an}.\n$$\n\n### Target B: interpret the result\n\nFind $v^{\\rm eff}(v)-v^{\\rm eff}(w)$. Explain the limits $a\\to0$ and $an\\to1^{-}$.\n\nWhy does the result describe an interacting gas even though the velocity distribution is not randomized?\n\n### Target C: reconstruct filling advection\n\nAssume\n\n$$\n\\partial_t\\rho_{\\rm p}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm p})=0,\n$$\n\n$$\n\\partial_t\\rho_{\\rm s}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm s})=0.\n$$\n\nSet $\\vartheta=\\rho_{\\rm p}/\\rho_{\\rm s}$ and derive its material-advection equation without assuming that $v^{\\rm eff}$ is constant.\n\n### Target D: audit a relaxation-time model\n\nTake\n\n$$\n\\mathcal I\n=-\\frac{\\rho_{\\rm p}-\\rho_{\\rm p}^{\\star}}{\\tau}.\n$$\n\nIf particle number, momentum, and energy are exactly conserved, what three matching conditions must $\\rho_{\\rm p}^{\\star}$ satisfy?\n\nExplain why choosing one fixed global Gibbs state generally fails.\n\n### Oral check 1\n\nWhat survives a collision in an integrable quasiparticle description, and what changes?\n\n### Oral check 2\n\nCan an integrable system have diffusion or superdiffusion without ordinary thermalization?\n\n<details>\n<summary>Hints</summary>\n\n- Integrate the effective-velocity equation against $\\rho(v)$.\n- The double integral cancels after exchanging $v$ and $w$.\n- Apply the product rule to $\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}$.\n- Collision invariants are left zero modes of $\\mathcal I$.\n\n</details>\n\n<details class=\"solution\">\n<summary>Solution outline</summary>\n\nIntegrating the hard-rod equation gives\n\n$$\nj=nu+a\\int dv\\,dw\\,\n\\rho(v)\\rho(w)\n\\left[v^{\\rm eff}(v)-v^{\\rm eff}(w)\\right].\n$$\n\nThe double integral is antisymmetric under $v\\leftrightarrow w$, so it vanishes and $j=nu$.\n\nThe original equation then becomes\n\n$$\nv^{\\rm eff}(v)\n=v+a\\left[nv^{\\rm eff}(v)-j\\right],\n$$\n\nwhich yields\n\n$$\nv^{\\rm eff}(v)=\\frac{v-an u}{1-an}.\n$$\n\nTherefore\n\n$$\nv^{\\rm eff}(v)-v^{\\rm eff}(w)\n=\\frac{v-w}{1-an}.\n$$\n\nAs $a\\to0$, bare motion is recovered. As $an\\to1^{-}$, the free volume vanishes and the quasiparticle-coordinate description becomes singular. The growing relative effective speed counts how excluded volume converts label crossings into larger physical displacements.\n\nFor the filling, substitute $\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}$:\n\n$$\n\\rho_{\\rm s}\n(\\partial_t+v^{\\rm eff}\\partial_x)\\vartheta\n+\\vartheta\n\\left[\n\\partial_t\\rho_{\\rm s}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm s})\n\\right]=0.\n$$\n\nThe second continuity equation removes the bracket, leaving\n\n$$\n(\\partial_t+v^{\\rm eff}\\partial_x)\\vartheta=0.\n$$\n\nNo derivative of $v^{\\rm eff}$ was discarded; its contribution is contained in the cancelled continuity term.\n\nFinally, conservation requires\n\n$$\n\\int d\\lambda\\,\n\\mathcal I(\\lambda)=0,\n$$\n\n$$\n\\int d\\lambda\\,\np(\\lambda)\\mathcal I(\\lambda)=0,\n$$\n\n$$\n\\int d\\lambda\\,\nE(\\lambda)\\mathcal I(\\lambda)=0.\n$$\n\nFor the relaxation-time ansatz, $\\rho_{\\rm p}^{\\star}$ must match the instantaneous particle, momentum, and energy densities of $\\rho_{\\rm p}$. A fixed global target cannot do so for a generic local state.\n\n</details>\n\n## Exit ticket\n\nA one-dimensional gas shows state-dependent ballistic fronts at intermediate times and ordinary diffusion at very late times.\n\nGive two explanations that fit the observation but imply different microscopic physics. Then name one measurement that would distinguish them.\n\nA strong answer should consider:\n\n1. weak breaking of an integrable quasiparticle hierarchy;\n2. coexistence of ballistic and diffusive sectors in an exactly integrable system;\n3. whether higher conserved charges decay;\n4. whether long-range correlations retain nonthermal information;\n5. how the crossover time scales with the perturbation strength.\n\n## Research checks worth doing next\n\n1. **Dressing reconstruction.** Infer $v^{\\rm eff}(\\lambda)$ at several local fillings and test one scattering kernel across all states.\n2. **Charge-decay hierarchy.** Measure several higher Bethe charges under a tunable integrability-breaking perturbation.\n3. **Null-space audit.** Verify that any kinetic collision model preserves exactly the microscopic unbroken charges.\n4. **Correlation test.** Compare one-point apparent thermalization with two-point and full-counting statistics.\n5. **Crossover scaling.** Determine whether $\\tau_{\\rm br}\\sim g^{-2}$ or whether selection rules produce another law.\n6. **Shock diagnostic.** Test smoothness assumptions before applying the Lieb–Liniger no-shock result to a new protocol.\n7. **Fluctuation class.** Separate ordinary diffusion from symmetry-driven superdiffusion using dynamical structure factors.\n\n## Further reading\n\n- [Generalized Hydrodynamics: A Perspective](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.15.010501) — a 2025 account of the framework, experiments, fluctuations, and open problems.\n- [Hydrodynamics of the interacting Bose gas](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.041065) — one of the foundational 2016 derivations of GHD.\n- [Hydrodynamic Diffusion in Integrable Systems](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.160603) — quasiparticle scattering fluctuations and the diffusion operator near homogeneous states.\n- [Soliton Gas Kinetics and Generalized Hydrodynamics](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.045301) — the bridge between scattering shifts, classical soliton gases, and GHD.\n- [Quantum Generalized Hydrodynamics with the Atom Chip](https://www.science.org/doi/10.1126/science.abf0147) — experimental test of GHD in interacting one-dimensional Bose gases.\n- [Weak integrability breaking and relaxation](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.101.180302) — kinetic theory for the decay of generalized charges under weak perturbations.\n- [Generalized hydrodynamics with a relaxation-time approximation](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.L060302) — a controlled phenomenological route from generalized to conventional hydrodynamics.\n- [Diffusive hydrodynamics from long-range correlations](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.134.187101) — why nonlinear diffusive-order dynamics need not reduce to a local gradient expansion.\n- [Absence of shocks in the Lieb–Liniger gas](https://journals.aps.org/prb/abstract/10.1103/5w4p-bxnt) — a 2025 theorem for a broad specified class of initial states.\n- [Simulating generalized fluids via wave packets](https://journals.aps.org/prb/abstract/10.1103/b587-8yyt) — a 2026 trajectory and fluctuation framework, including integrability-breaking extensions.\n- [Relaxation-time GHD crossover model](https://arxiv.org/abs/2603.02158) — a 2026 preprint; useful as a model of crossover, not a universal microscopic derivation.\n\n## What to retain\n\n- Elastic factorized scattering can be strongly interacting without redistributing stable quasiparticle rapidities.\n- Dressing is the finite collective backflow of the thermodynamic Bethe state.\n- Euler GHD is a nonlinear advection theory because $v^{\\rm eff}$ depends on the full local filling.\n- The filling equation follows from continuity of both occupied roots and available Bethe states.\n- A local GGE is a quasistationary closure, not the product of ordinary Boltzmann thermalization.\n- Integrable systems can display diffusive or superdiffusive sectors without losing all higher conservation laws.\n- Weak breaking introduces a collision operator whose null space determines late-time hydrodynamics.\n- Relaxation-time models must project out exact conserved charges.\n- No-shock results and local diffusive corrections have stated domains of validity.\n- One-point thermal appearance does not prove erasure of generalized-fluid correlations.\n\nNext: derive the diffusion kernel from fluctuations of dressed two-body trajectory shifts, then compare it with a projected integrability-breaking collision operator.\n",
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      "display": false,
      "tex": "\\rho_{\\rm p}(\\lambda)",
      "line": 76,
      "sha256": "174e75a407782e3b52198f463bc9708569bafbad26953f4777b19ead86d40ceb"
    },
    {
      "index": 9,
      "display": false,
      "tex": "\\rho_{\\rm s}(\\lambda)",
      "line": 76,
      "sha256": "a9e851d8157d4e9afeee0e4717ae958580e926ce7c41a013ad0b3c35a6a66334"
    },
    {
      "index": 10,
      "display": true,
      "tex": "\\vartheta(\\lambda)=\n\\frac{\\rho_{\\rm p}(\\lambda)}{\\rho_{\\rm s}(\\lambda)},\n\\qquad 0\\leq\\vartheta\\leq1.",
      "line": 78,
      "sha256": "9536b1fc29db42968e0f4d97d4618e56d61fd675379ac344ee8e807ba61e3f74"
    },
    {
      "index": 11,
      "display": true,
      "tex": "K(\\lambda)=\\frac{2c}{\\lambda^2+c^2},",
      "line": 86,
      "sha256": "22a6c292dea0fc9744d6833fa3e8c2bba33facd6a0169ac0924c22ef4fa44d05"
    },
    {
      "index": 12,
      "display": true,
      "tex": "(\\mathcal K f)(\\lambda)\n=\\int\\frac{d\\mu}{2\\pi}\\,\nK(\\lambda-\\mu)f(\\mu).",
      "line": 92,
      "sha256": "7e00a3881cdecda9b3a0c91dd4c00cc82867e52d7e542d0aef6c3a0198932ba9"
    },
    {
      "index": 13,
      "display": true,
      "tex": "\\rho_{\\rm s}(\\lambda)\n=\\frac{p'(\\lambda)}{2\\pi}\n+(\\mathcal K\\rho_{\\rm p})(\\lambda).",
      "line": 100,
      "sha256": "56cb472ae371712c595cc3a0e2e71b053679c00d9301c2e0c3657ecb1f392a85"
    },
    {
      "index": 14,
      "display": false,
      "tex": "p(\\lambda)=\\lambda",
      "line": 106,
      "sha256": "b2313a172034b69333d708c1ce4ff5402711580f6d0aac50866b852c48531d90"
    },
    {
      "index": 15,
      "display": false,
      "tex": "E(\\lambda)=\\lambda^2",
      "line": 106,
      "sha256": "77511ad0389487de21ccfe83c4d21eddd4d1b65dc3584c2cc1717057c76feaae"
    },
    {
      "index": 16,
      "display": false,
      "tex": "h(\\lambda)",
      "line": 110,
      "sha256": "d4dd5d72b9aeda42207f3a254169009ba7915284a8403de2ba82ff23bff535ee"
    },
    {
      "index": 17,
      "display": true,
      "tex": "h^{\\rm dr}(\\lambda)\n=h(\\lambda)\n+\\mathcal K\\!\\left(\\vartheta h^{\\rm dr}\\right)(\\lambda).",
      "line": 112,
      "sha256": "3fc31d9e0b0103749deef8fe6718bcd2ae4c1af6f083b888a5d5595e94f11e6d"
    },
    {
      "index": 18,
      "display": false,
      "tex": "\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}",
      "line": 118,
      "sha256": "32401ba890fea2d021e3fad681cd070569878d353276c139f9f540c0095247d8"
    },
    {
      "index": 19,
      "display": true,
      "tex": "2\\pi\\rho_{\\rm s}(\\lambda)\n=\\bigl(p'(\\lambda)\\bigr)^{\\rm dr}.",
      "line": 120,
      "sha256": "91ead6df3baeb18ad688ea479631cbf428a8b32d820277f0cef75e2de58f9e73"
    },
    {
      "index": 20,
      "display": true,
      "tex": "v^{\\rm eff}(\\lambda)\n=\\frac{\\bigl(E'(\\lambda)\\bigr)^{\\rm dr}}\n{\\bigl(p'(\\lambda)\\bigr)^{\\rm dr}}.",
      "line": 127,
      "sha256": "00fa08d3e0080f37ae9d53d89857dbfa62555b8ac0c218d1e540068a1757925a"
    },
    {
      "index": 21,
      "display": false,
      "tex": "h(\\lambda)",
      "line": 135,
      "sha256": "d4dd5d72b9aeda42207f3a254169009ba7915284a8403de2ba82ff23bff535ee"
    },
    {
      "index": 22,
      "display": true,
      "tex": "q[h]=\\int d\\lambda\\,\\rho_{\\rm p}(\\lambda)h(\\lambda),",
      "line": 137,
      "sha256": "95da91982654999ad03b64fb558121d8b71de29dc6c5af98cf2159aeb284737c"
    },
    {
      "index": 23,
      "display": true,
      "tex": "j[h]=\\int d\\lambda\\,\n\\rho_{\\rm p}(\\lambda)v^{\\rm eff}(\\lambda)h(\\lambda).",
      "line": 141,
      "sha256": "5880153bbfb88e2f8d450c84ebf86cf73a2e8d8b5203f891eb46256c639ba5e5"
    },
    {
      "index": 24,
      "display": false,
      "tex": "v^{\\rm eff}",
      "line": 146,
      "sha256": "103b687261390ab386309a7ce4f37226d21803492fc323cdc3d20965b26b78a0"
    },
    {
      "index": 25,
      "display": true,
      "tex": "\\partial_t\\rho_{\\rm p}(\\lambda)\n+\\partial_x\\!\\left(\nv^{\\rm eff}(\\lambda)\\rho_{\\rm p}(\\lambda)\n\\right)=0.",
      "line": 152,
      "sha256": "46814a52f9ad5262bf63cae31f703901737741576f3cbda41a7392273005fb68"
    },
    {
      "index": 26,
      "display": true,
      "tex": "\\partial_t\\rho_{\\rm s}(\\lambda)\n+\\partial_x\\!\\left(\nv^{\\rm eff}(\\lambda)\\rho_{\\rm s}(\\lambda)\n\\right)=0.",
      "line": 161,
      "sha256": "2dfd433bf853043884e8ed97bffcbeb0cb87a3c081ee42536da242050f07f20b"
    },
    {
      "index": 27,
      "display": false,
      "tex": "\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}",
      "line": 168,
      "sha256": "32401ba890fea2d021e3fad681cd070569878d353276c139f9f540c0095247d8"
    },
    {
      "index": 28,
      "display": true,
      "tex": "\\rho_{\\rm s}\n\\left(\\partial_t+v^{\\rm eff}\\partial_x\\right)\\vartheta\n+\\vartheta\n\\left[\n\\partial_t\\rho_{\\rm s}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm s})\n\\right]=0.",
      "line": 170,
      "sha256": "df75253afc28aa8b2c2a2568c8cf82881fec3b423c499414c65f6dced54f2155"
    },
    {
      "index": 29,
      "display": false,
      "tex": "\\rho_{\\rm s}>0",
      "line": 180,
      "sha256": "915148dcfa8e6b59ade388187b3e669e7d059fef40919d82c4d7c0cce4babfce"
    },
    {
      "index": 30,
      "display": true,
      "tex": "\\boxed{\n\\left(\\partial_t+v^{\\rm eff}[\\vartheta]\\partial_x\\right)\n\\vartheta=0\n}.",
      "line": 182,
      "sha256": "7023f399b45690fe0636ddc19c1bc909886075424a84157d1fb4ffda542ac692"
    },
    {
      "index": 31,
      "display": true,
      "tex": "\\partial_t\\rho_{\\rm p}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm p})\n+\\partial_\\lambda(a^{\\rm eff}\\rho_{\\rm p})=0.",
      "line": 193,
      "sha256": "80b6b14bb6b7ad57799bd6b17c67a8053940a816039ab2232ff02a6e3006806c"
    },
    {
      "index": 32,
      "display": false,
      "tex": "\\Delta x(\\lambda,\\mu)",
      "line": 205,
      "sha256": "252e37080f7be99f2184a464387457cd374e1288fbbbacf9d24d83b8df6f8ef8"
    },
    {
      "index": 33,
      "display": false,
      "tex": "\\lambda",
      "line": 205,
      "sha256": "583cf6290785d72cc1b71808962dd4bd72ca369f004fb5cb587694fc1b26149f"
    },
    {
      "index": 34,
      "display": false,
      "tex": "\\mu",
      "line": 205,
      "sha256": "492247e1bab15272d1e53b8015c042bf6fa0cb8f23ec52cad93d6ba80dafb804"
    },
    {
      "index": 35,
      "display": true,
      "tex": "v^{\\rm eff}(\\lambda)\n=v^{\\rm bare}(\\lambda)\n+\\int d\\mu\\,\\rho_{\\rm p}(\\mu)\n\\Delta x(\\lambda,\\mu)\n\\left[\nv^{\\rm eff}(\\lambda)-v^{\\rm eff}(\\mu)\n\\right].",
      "line": 209,
      "sha256": "af800fd7066091c47666d2c27294fb3740e1a348ab3bb63769510380582fe4b8"
    },
    {
      "index": 36,
      "display": false,
      "tex": "\\Delta x",
      "line": 219,
      "sha256": "b0defe9cacb385901ac7a290dccd8b2a923b2c33930eca3037efc5851f05705c"
    },
    {
      "index": 37,
      "display": false,
      "tex": "x/t",
      "line": 221,
      "sha256": "5262360baaca1ddc061b47a5ec47aa572c72ccbee37f23c10bb4095df580517f"
    },
    {
      "index": 38,
      "display": false,
      "tex": "a",
      "line": 227,
      "sha256": "ca978112ca1bbdcafac231b39a23dc4da786eff8147c4e72b9807785afee48bb"
    },
    {
      "index": 39,
      "display": false,
      "tex": "\\rho(v)",
      "line": 227,
      "sha256": "46ee596fccef3309bb099087bf1e365d46c5f0bdb4c778deabfb53042a6df72b"
    },
    {
      "index": 40,
      "display": true,
      "tex": "n=\\int dv\\,\\rho(v),\n\\qquad\nu=\\frac{1}{n}\\int dv\\,v\\rho(v),\n\\qquad an<1.",
      "line": 229,
      "sha256": "dc9df83d3c59dea9ac78979e17e4201eada999d20949832bd998009aff2f35b9"
    },
    {
      "index": 41,
      "display": false,
      "tex": "a",
      "line": 236,
      "sha256": "ca978112ca1bbdcafac231b39a23dc4da786eff8147c4e72b9807785afee48bb"
    },
    {
      "index": 42,
      "display": true,
      "tex": "v^{\\rm eff}(v)\n=v+a\\int dw\\,\\rho(w)\n\\left[v^{\\rm eff}(v)-v^{\\rm eff}(w)\\right].",
      "line": 240,
      "sha256": "8608b37079c678f2c9fac064284728b44c227d527ec6f010d214c80e3c3d0737"
    },
    {
      "index": 43,
      "display": true,
      "tex": "v^{\\rm eff}(v)=\\frac{v-an u}{1-an}.",
      "line": 248,
      "sha256": "d79b53a92a419a46a577c93561982652eb9e3e8f052c190c928e7949d0204344"
    },
    {
      "index": 44,
      "display": true,
      "tex": "v^{\\rm eff}(v)-v^{\\rm eff}(w)\n=\\frac{v-w}{1-an}.",
      "line": 254,
      "sha256": "3bc22a07be8034aafb12b7862608e6168570f51f024c5bd3c7656c92ddf08fd9"
    },
    {
      "index": 45,
      "display": true,
      "tex": "s_{\\rm YY}\n=\\int d\\lambda\\,\\rho_{\\rm s}(\\lambda)\n\\left[\n-\\vartheta\\log\\vartheta\n-(1-\\vartheta)\\log(1-\\vartheta)\n\\right].",
      "line": 273,
      "sha256": "9c81e924e35a1f4a13c23c52f9f54aed7b026a31f8bb9a2e9014220620c2e4bf"
    },
    {
      "index": 46,
      "display": true,
      "tex": "\\partial_t s_{\\rm YY}+\\partial_x j_s=0.",
      "line": 284,
      "sha256": "79a07fecf8dd6b420fa23b1566a97a4819ad2ce8da59a61569c66b027fabd4fd"
    },
    {
      "index": 47,
      "display": false,
      "tex": "O(t^{1/2})",
      "line": 294,
      "sha256": "f392f376e23b9bf6fb33c997f39534a9fbfabeb6169b89439b29d1f5354af7d9"
    },
    {
      "index": 48,
      "display": true,
      "tex": "\\text{ballistic Euler rays}\n\\;\\to\\;\n\\text{integrability-preserving broadening}\n\\;\\to\\;\n\\text{rapidity redistribution if integrability breaks}.",
      "line": 308,
      "sha256": "abde2598dd69aab4fdb3285f951f122f0978522089f021feecd90ca605124e19"
    },
    {
      "index": 49,
      "display": false,
      "tex": "g",
      "line": 320,
      "sha256": "cd0aa9856147b6c5b4ff2b7dfee5da20aa38253099ef1b4a64aced233c9afe29"
    },
    {
      "index": 50,
      "display": true,
      "tex": "\\partial_t\\rho_{\\rm p}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm p})\n=\\mathcal I_{\\rm br}[\\rho_{\\rm p}].",
      "line": 322,
      "sha256": "f6c52d7372b3620910fc8b28e8d421602284020f5e364ac5a8b53cc983ea08cd"
    },
    {
      "index": 51,
      "display": true,
      "tex": "\\int d\\lambda\\,\n\\begin{pmatrix}\n1\\\\\np(\\lambda)\\\\\nE(\\lambda)\n\\end{pmatrix}\n\\mathcal I_{\\rm br}(\\lambda)=0.",
      "line": 330,
      "sha256": "3873814ade6c2a713fcfd1fd93ef0e0a2468e02559d15e54d9e1a80822a73003"
    },
    {
      "index": 52,
      "display": true,
      "tex": "\\mathcal I_{\\rm naive}\n=-\\frac{\\rho_{\\rm p}-\\rho_{\\rm p}^{\\star}}{\\tau},",
      "line": 344,
      "sha256": "b980e42ba36e596f2351aed1d8d0d60d8888de328cea70dcf15386f4580970ec"
    },
    {
      "index": 53,
      "display": false,
      "tex": "\\rho_{\\rm p}^{\\star}",
      "line": 349,
      "sha256": "2b722b7c977ec81cb168ad9b283c31f00428806e583a33e7e9044ed0fa7256ff"
    },
    {
      "index": 54,
      "display": false,
      "tex": "\\rho_{\\rm p}",
      "line": 349,
      "sha256": "0eba24b91c6a9a4a598ed327c030a7e6988b15cc7c99986f32814a100adfbdb7"
    },
    {
      "index": 55,
      "display": true,
      "tex": "t\\ll\\tau_{\\rm br}:\n\\quad \\text{prethermal GHD},",
      "line": 353,
      "sha256": "5567b9551e848098781d303de8756b68432218c31c74181699972072c1b58d83"
    },
    {
      "index": 56,
      "display": true,
      "tex": "t\\gg\\tau_{\\rm br}:\n\\quad \\text{hydrodynamics of the exact conserved modes}.",
      "line": 358,
      "sha256": "f016abf124bbd7d77e153bbfb2bcda30ff6249a173bda7ffca280f665ac4bbe3"
    },
    {
      "index": 57,
      "display": false,
      "tex": "\\tau_{\\rm br}\\propto g^{-2}",
      "line": 363,
      "sha256": "4bae7c22d9f1a5ea234c77c4b433e208cfa9a950a3b0723975829d6bd3e4424f"
    },
    {
      "index": 58,
      "display": true,
      "tex": "v^{\\rm eff}(\\lambda)\n=\\frac{(E')^{\\rm dr}}{(p')^{\\rm dr}}",
      "line": 409,
      "sha256": "0281cba1ee6a0d0d0e7906dad28564f7f4a851d00ddc4a363ab325a8585801af"
    },
    {
      "index": 59,
      "display": false,
      "tex": "O(1)",
      "line": 426,
      "sha256": "4a137b861e8d266f3c1bfeb6608fbf87f862ed7863127d6802f545b38d4ec25f"
    },
    {
      "index": 60,
      "display": true,
      "tex": "n=\\int dv\\,\\rho(v),\n\\qquad\nu=\\frac{1}{n}\\int dv\\,v\\rho(v),\n\\qquad an<1,",
      "line": 440,
      "sha256": "b7a6ad88e9af82bed907b37520c424fe8a2d32e5b8cef6c11050165838d2ab73"
    },
    {
      "index": 61,
      "display": true,
      "tex": "v^{\\rm eff}(v)\n=v+a\\int dw\\,\\rho(w)\n\\left[v^{\\rm eff}(v)-v^{\\rm eff}(w)\\right].",
      "line": 449,
      "sha256": "8608b37079c678f2c9fac064284728b44c227d527ec6f010d214c80e3c3d0737"
    },
    {
      "index": 62,
      "display": true,
      "tex": "j=\\int dv\\,\\rho(v)v^{\\rm eff}(v).",
      "line": 459,
      "sha256": "eec018b90e36687bb20dac07939c744b35a5fea96bc0d3828cff312f4bc2eea2"
    },
    {
      "index": 63,
      "display": false,
      "tex": "j=nu",
      "line": 463,
      "sha256": "9b060d130f40f5e327201f55b1da04c9263be8fe4ed8c1357aefb460e6ac71aa"
    },
    {
      "index": 64,
      "display": true,
      "tex": "v^{\\rm eff}(v)=\\frac{v-an u}{1-an}.",
      "line": 465,
      "sha256": "d79b53a92a419a46a577c93561982652eb9e3e8f052c190c928e7949d0204344"
    },
    {
      "index": 65,
      "display": false,
      "tex": "v^{\\rm eff}(v)-v^{\\rm eff}(w)",
      "line": 471,
      "sha256": "28b39165d35932dbe3fa361dbef291fa9cd62fd28e6b850e21f5c88a3185c348"
    },
    {
      "index": 66,
      "display": false,
      "tex": "a\\to0",
      "line": 471,
      "sha256": "039a73ba2bae14d90f17d42cdd909ff473e5ba5d9f93a4a4dc69988469fe222b"
    },
    {
      "index": 67,
      "display": false,
      "tex": "an\\to1^{-}",
      "line": 471,
      "sha256": "3e6fd5371dc492303a0c04f565b0b491f7107e45f64a6cdf64ccf1e280abd775"
    },
    {
      "index": 68,
      "display": true,
      "tex": "\\partial_t\\rho_{\\rm p}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm p})=0,",
      "line": 479,
      "sha256": "01a478219de2632d724adcafb1fa2a2caa60693900c8e52e9416cd35548767b8"
    },
    {
      "index": 69,
      "display": true,
      "tex": "\\partial_t\\rho_{\\rm s}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm s})=0.",
      "line": 484,
      "sha256": "eb53d12eab032e889629520405efbe8fefa1ca5c89b7596b040bb602b1d564b5"
    },
    {
      "index": 70,
      "display": false,
      "tex": "\\vartheta=\\rho_{\\rm p}/\\rho_{\\rm s}",
      "line": 489,
      "sha256": "191af0c8d4b60908ac6b2a9b9821f5896d51755c29bbd88717097a757a9b6215"
    },
    {
      "index": 71,
      "display": false,
      "tex": "v^{\\rm eff}",
      "line": 489,
      "sha256": "103b687261390ab386309a7ce4f37226d21803492fc323cdc3d20965b26b78a0"
    },
    {
      "index": 72,
      "display": true,
      "tex": "\\mathcal I\n=-\\frac{\\rho_{\\rm p}-\\rho_{\\rm p}^{\\star}}{\\tau}.",
      "line": 495,
      "sha256": "3e97f71a995e0eb05904d0d43e2f2a18d638ceeef09174bb78bf2957557e91e8"
    },
    {
      "index": 73,
      "display": false,
      "tex": "\\rho_{\\rm p}^{\\star}",
      "line": 500,
      "sha256": "2b722b7c977ec81cb168ad9b283c31f00428806e583a33e7e9044ed0fa7256ff"
    },
    {
      "index": 74,
      "display": false,
      "tex": "\\rho(v)",
      "line": 515,
      "sha256": "46ee596fccef3309bb099087bf1e365d46c5f0bdb4c778deabfb53042a6df72b"
    },
    {
      "index": 75,
      "display": false,
      "tex": "v",
      "line": 516,
      "sha256": "4c94485e0c21ae6c41ce1dfe7b6bfaceea5ab68e40a2476f50208e526f506080"
    },
    {
      "index": 76,
      "display": false,
      "tex": "w",
      "line": 516,
      "sha256": "50e721e49c013f00c62cf59f2163542a9d8df02464efeb615d31051b0fddc326"
    },
    {
      "index": 77,
      "display": false,
      "tex": "\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}",
      "line": 517,
      "sha256": "32401ba890fea2d021e3fad681cd070569878d353276c139f9f540c0095247d8"
    },
    {
      "index": 78,
      "display": false,
      "tex": "\\mathcal I",
      "line": 518,
      "sha256": "e6884a0c40dcac51f9c57f14cc38e1e3298387944d4771e88dccf22dfe45c9bb"
    },
    {
      "index": 79,
      "display": true,
      "tex": "j=nu+a\\int dv\\,dw\\,\n\\rho(v)\\rho(w)\n\\left[v^{\\rm eff}(v)-v^{\\rm eff}(w)\\right].",
      "line": 527,
      "sha256": "5f017b5de8b6f51eaea79f689467bd142f01fb87d67c07768255e9d11077308a"
    },
    {
      "index": 80,
      "display": false,
      "tex": "v\\leftrightarrow w",
      "line": 533,
      "sha256": "741ff2583841ceed6f4ce55d1b521a99e0f120c6ee1df96bc8c359c65c8e18b1"
    },
    {
      "index": 81,
      "display": false,
      "tex": "j=nu",
      "line": 533,
      "sha256": "9b060d130f40f5e327201f55b1da04c9263be8fe4ed8c1357aefb460e6ac71aa"
    },
    {
      "index": 82,
      "display": true,
      "tex": "v^{\\rm eff}(v)\n=v+a\\left[nv^{\\rm eff}(v)-j\\right],",
      "line": 537,
      "sha256": "35949ec98c7ffc287406ad68793cd760cf7d42ab7a1d0494436753f3fed688d0"
    },
    {
      "index": 83,
      "display": true,
      "tex": "v^{\\rm eff}(v)=\\frac{v-an u}{1-an}.",
      "line": 544,
      "sha256": "d79b53a92a419a46a577c93561982652eb9e3e8f052c190c928e7949d0204344"
    },
    {
      "index": 84,
      "display": true,
      "tex": "v^{\\rm eff}(v)-v^{\\rm eff}(w)\n=\\frac{v-w}{1-an}.",
      "line": 550,
      "sha256": "3bc22a07be8034aafb12b7862608e6168570f51f024c5bd3c7656c92ddf08fd9"
    },
    {
      "index": 85,
      "display": false,
      "tex": "a\\to0",
      "line": 555,
      "sha256": "039a73ba2bae14d90f17d42cdd909ff473e5ba5d9f93a4a4dc69988469fe222b"
    },
    {
      "index": 86,
      "display": false,
      "tex": "an\\to1^{-}",
      "line": 555,
      "sha256": "3e6fd5371dc492303a0c04f565b0b491f7107e45f64a6cdf64ccf1e280abd775"
    },
    {
      "index": 87,
      "display": false,
      "tex": "\\rho_{\\rm p}=\\vartheta\\rho_{\\rm s}",
      "line": 557,
      "sha256": "32401ba890fea2d021e3fad681cd070569878d353276c139f9f540c0095247d8"
    },
    {
      "index": 88,
      "display": true,
      "tex": "\\rho_{\\rm s}\n(\\partial_t+v^{\\rm eff}\\partial_x)\\vartheta\n+\\vartheta\n\\left[\n\\partial_t\\rho_{\\rm s}\n+\\partial_x(v^{\\rm eff}\\rho_{\\rm s})\n\\right]=0.",
      "line": 559,
      "sha256": "e18889274a9f0809d3e62c3ef69eaaba96e678b7e57ee2ff03e5ea8f2c53c3c7"
    },
    {
      "index": 89,
      "display": true,
      "tex": "(\\partial_t+v^{\\rm eff}\\partial_x)\\vartheta=0.",
      "line": 571,
      "sha256": "aed73c3a03e3f640772e711d2172ed89a0b2f6e736bd2d49d5f106eadd5a5a41"
    },
    {
      "index": 90,
      "display": false,
      "tex": "v^{\\rm eff}",
      "line": 575,
      "sha256": "103b687261390ab386309a7ce4f37226d21803492fc323cdc3d20965b26b78a0"
    },
    {
      "index": 91,
      "display": true,
      "tex": "\\int d\\lambda\\,\n\\mathcal I(\\lambda)=0,",
      "line": 579,
      "sha256": "c29843a2c4ed2db7ac7e7d5ec2971fc339ff8d3c7d78d66ce2c3fd5e0503614e"
    },
    {
      "index": 92,
      "display": true,
      "tex": "\\int d\\lambda\\,\np(\\lambda)\\mathcal I(\\lambda)=0,",
      "line": 584,
      "sha256": "e58ce970b0d5fb661cfb9200c556df2f08e6948199f69daf3ed6b45b77a7339a"
    },
    {
      "index": 93,
      "display": true,
      "tex": "\\int d\\lambda\\,\nE(\\lambda)\\mathcal I(\\lambda)=0.",
      "line": 589,
      "sha256": "50a1e42b0275ba027ccaa0e183c983936cad370b3dc252db2553161911a51bd5"
    },
    {
      "index": 94,
      "display": false,
      "tex": "\\rho_{\\rm p}^{\\star}",
      "line": 594,
      "sha256": "2b722b7c977ec81cb168ad9b283c31f00428806e583a33e7e9044ed0fa7256ff"
    },
    {
      "index": 95,
      "display": false,
      "tex": "\\rho_{\\rm p}",
      "line": 594,
      "sha256": "0eba24b91c6a9a4a598ed327c030a7e6988b15cc7c99986f32814a100adfbdb7"
    },
    {
      "index": 96,
      "display": false,
      "tex": "v^{\\rm eff}(\\lambda)",
      "line": 614,
      "sha256": "52abca9c2b3e4b205a2dfefffb6f02af8bf696ecbe19aa4ba5e6b89b09e523db"
    },
    {
      "index": 97,
      "display": false,
      "tex": "\\tau_{\\rm br}\\sim g^{-2}",
      "line": 618,
      "sha256": "940e4381008fb71f616aa6a63e767f405454c3648a8cf3be3619963daff5fe12"
    },
    {
      "index": 98,
      "display": false,
      "tex": "v^{\\rm eff}",
      "line": 640,
      "sha256": "103b687261390ab386309a7ce4f37226d21803492fc323cdc3d20965b26b78a0"
    }
  ]
}