Lecture 0 began with two clouds that passed through each other thousands of times and did not become one thermal hump. Lecture 17 gave the gas its equation of state, and Lecture 19 proved relaxation for a quench of the Ising chain; here the gas is assumed to be locally in a generalised Gibbs ensemble. So the question is

The answer: the density of each rapidity is carried at its dressed velocity ((20.5), (20.4)); for hard rods the dressed velocity is computed in one line (Proposition 20.1), and in the trap’s two free limits the flow is a rigid rotation that returns exactly (Figure 20.3). The worked example is hard rods.

Free fermions made the argument exact, and easy, because the charges were quadratic. The replacement for an interacting model is already built: a macrostate in the thermodynamic limit is a root density $\rho(\bethe)$, or the filling fraction $\vartheta=\rho/\rho_t$ of Definition 17.1, and every extensive charge is $$\frac{\langle Q_a\rangle}{L}=\int q_a(\bethe)\,\rho(\bethe)\,\dd\bethe ,$$(20.1) with $q_a$ the charge’s one-particle eigenvalue — $q=k^{2}$ for the energy of the gas, $q=k$ for its momentum. A family of charges fixes $\rho$ exactly when its $q_a$ separate root densities; given such a complete family, the generalised Gibbs ensemble of Definition 19.5 is a single function — for a one-species gas; one filling per quasiparticle species in general, as the chain’s strings (Lecture 17’s reading) already needed — the same $\vartheta$ that Lecture 17 minimised the free energy over, now fixed by the initial state instead of by a temperature. Maximising (17.4) at fixed (20.1) is the generalised thermodynamic Bethe ansatz, and it returns (17.7) in dimensionless form: with $\eta=\ln(\rho_h/\rho)$ and $w=\sum_a\beta_aq_a$, $$\eta(\bethe)=w(\bethe)-\frac1{2\pi}\int K(\bethe-\mu) \ln\big(1+e^{-\eta(\mu)}\big)\,\dd\mu , \qquad \vartheta=\frac1{1+e^{\eta}}$$(20.2) cited. The multipliers carry the temperature, so (17.7)’s explicit $T$’s go: at thermal equilibrium $\eta=\varepsilon/T$, and $w=(\varepsilon_0-\mu)/T$ returns Lecture 17’s equation exactly.

Now let the state vary in space: $\vartheta(\bethe,x,t)$, slowly on the scale of the interactions — a local generalised Gibbs ensemble. That is the assumption, and the only one; there is no gradient expansion beyond leading order, which is what Euler scale means.

Lecture 12 found that a collision permutes the momenta and leaves a displacement behind (Theorem 12.2), and Lecture 11 found its quantum counterpart (11.7). A quasiparticle crossing a gas therefore collects one displacement per particle it passes, and does not travel at its bare velocity.

Equation (20.3) is Equation (17.1) with the constant kernel $K=-a$ — not a local one: a delta-function kernel would dress every function by the same factor and cancel in the ratio. Dressing with it gives $2\pi\rho_t=(1)^{\mathrm{dr}}=1-an$, the free volume fraction, and $(\varepsilon_0')^{\mathrm{dr}}=\bethe-a\jmath$, whose ratio is (20.3). Lecture 17’s large-$c$ limit (the paragraph after Proposition 17.9) says this is no curiosity: to leading order in $1/c$, and in the rest frame, the Lieb–Liniger gas is free fermions in a box lengthened by $2/c$ per particle, that is, a hard-rod gas of length $a=-2/c$. In general $$v^{\mathrm{eff}}(\bethe,x,t) =\frac{(\varepsilon_0')^{\mathrm{dr}}}{(k_0')^{\mathrm{dr}}} ,$$(20.4) with the dressing of (17.1) at the local $\vartheta$ and $k_0$ the excitation’s own bare momentum (Remark 17.3). Write it this way and not as $\varepsilon'/2\pi\rho_t$, which is Equation (17.10). Lecture 17 got that by differentiating a thermal Yang–Yang equation driven by $\varepsilon_0-\mu$; differentiating (20.2) gives $\eta'=(w')^{\mathrm{dr}}$ instead, which dresses the combination $w$ and not the physical energy. (Free fermions with $w=\bethe^{4}$: there $2\pi\rho_t=1$ and $\eta'=4\bethe^{3}$, against $2\bethe$ from $\varepsilon_0=\bethe^{2}$.) The two coincide at thermal equilibrium, where $\eta=\varepsilon/T$ and the $T$ cancels from the ratio — which is where Lecture 17 used it, saying explicitly that calling the ratio a propagation speed was not proved there. This section is where it is used, and the rods are the reason to believe it.

$$\boxed{\ \partial_t\rho(\bethe,x,t) +\partial_x\big[v^{\mathrm{eff}}(\bethe,x,t)\,\rho(\bethe,x,t)\big]=0\ } \qquad\text{for every }\bethe .$$(20.5) One continuity equation per rapidity sketched: Castro-Alvaredo, Doyon and Yoshimura 2016; Bertini, Collura, De Nardis and Fagotti 2016: the number of quasiparticles of each rapidity is separately conserved — what integrability buys — and they are transported at the dressed velocity, which is what interaction costs. Ordinary hydrodynamics has five such equations in three dimensions; this has one per point of a continuum.

Figure 20.2 is the theory in one picture, and the one place in this course where an Euler-scale claim is tested against the microscopic model it describes: the circles are rods, evolved exactly, not a second solution of (20.5). The agreement is a law of large numbers and needs many particles per hydrodynamic cell — at one rod per ten units of length the same comparison fails outright.

Equation (20.5) has an equivalent form: the filling fraction is not conserved but advected, $\partial_t\vartheta+v^{\mathrm{eff}}\partial_x\vartheta=0$ in the homogeneous case — not obvious, since $v^{\mathrm{eff}}$ depends on $\vartheta$ cited: Castro-Alvaredo, Doyon and Yoshimura 2016. There a trapping potential $V(x)$ enters as a force on the rapidity, and the content is that quasiparticles follow characteristics: $$\partial_t\vartheta+v^{\mathrm{eff}}\partial_x\vartheta -\partial_xV\,\partial_{\bethe}\vartheta=0 , \qquad\text{i.e.}\qquad \dot x=v^{\mathrm{eff}},\quad \dot{\bethe}=-\partial_xV ,$$(20.6) with $\vartheta$ constant along them cited: Doyon and Yoshimura 2017. In Lecture 11’s units, $\hbar=2m=1$, the rapidity is the momentum: $\bethe$ obeys Newton’s second law and $x$ a dressed version of the first.

One caution first: (20.6) does not say the trapped model is integrable. A position-dependent potential breaks the homogeneous charges, no Bethe ansatz solves the trapped Lieb–Liniger Hamiltonian, and (20.6) is an Euler-scale approximation, good while $V$ varies slowly on the scale of the interactions cited: Doyon and Yoshimura 2017; Caux et al. 2019, App. H. It says two things.

First, at the two ends of the coupling range the trap costs nothing at all. There $v^{\mathrm{eff}}=2\bethe$ — the ideal gas, and the impenetrable limit, where Girardeau’s map makes the gas free fermions in every state (Lecture 18’s reading) — and (20.6) is a linear flow of period $2\pi/\omega$: the distribution turns rigidly on ellipses of fixed $x^{2}+(2\bethe/\omega)^{2}$ and returns exactly to itself (Figure 20.3). That is the cradle. In between, $v^{\mathrm{eff}}$ depends on the local state, the return is only approximate and the shape drifts — which is what the experiments see. Second, $\int\rho\,\dd x$ is in general not conserved in a trap, since (20.6) moves $\bethe$: “the momentum distribution does not relax” means a shape returning, not a constant.

The theory, and then the data. Caux, Doyon, Dubail, Konik and Yoshimura applied GHD to the cradle geometry in 2019, at a coupling close to the original experiment’s. The first experimental test is Schemmer, Bouchoule, Doyon and Dubail’s atom chip: one cloud after a longitudinal trap quench, where GHD works and conventional hydrodynamics, which assumes local thermal equilibrium, does not. The quantitative test is Malvania, Zhang, Le, Dubail, Rigol and Weiss, in the same laboratory’s gases, measuring rapidity distributions directly through dozens of trap oscillations at $\gamma=c/n$ from $0.3$ to $9.3$ cited. There is, as of writing, no published comparison of generalised hydrodynamics with the 2006 data itself; theory and experiment meet in the later apparatus, not in the original plot.

What else spoils it. The atoms of Lecture 0’s tube are not exactly one-dimensional: virtually exciting a transverse mode and coming back leaves an effective three-body collision along the tube, which no Bethe ansatz solves, and whose rate sets a thermalisation time cited: Mazets, Schumm and Schmiedmayer 2008. Integrability is therefore approximate on two counts, and the question is one of time scales.

Lecture 0’s map named five ideas. Here is what each half of the course proved about them, and what it cited.

• P1, integrability as a count. Part I proved it, with the statuses its lectures give: the Liouville–Arnold theorem (Theorem 5.7), action–angle variables (Theorem 6.5), and the Toda ring integrable (Theorem 9.10). In Part II the count is not a definition (Lecture 0’s caveat); a commuting family from one matrix takes its place.

• P2, one family from one local identity. Proved on both sides: the quadratic bracket makes the Toda ring’s transfer matrices commute (Corollary 9.8), $RTT$ the Heisenberg chain’s (Theorem 14.3), and the quantum Toda chain proves that the second becomes the first as $\hbar\to0$ (Corollary 14.6).

• P3, scattering that only rearranges momenta. The classical side cites Moser (Theorem 12.2); the quantum side derives the Yang–Baxter equation from three-body consistency (Theorem 13.3) and the Bethe equations from a ring: for the gas with a full proof that the Bethe wavefunctions are eigenstates (Lecture 11), and for the chain sketched in coordinates (Lecture 10) and proved algebraically (Lecture 15).

• P4, densities, dressing and flow. Quantum only. The density equation is proved from an assumption (Lecture 16), the Yang–Yang equation from a count (Lecture 17), and the hydrodynamics of this lecture is sketched, with hard rods as its check.

• P5, relaxation to a generalised ensemble. Proved for one quench and one observable of the Ising chain (Theorem 19.7); for the interacting gas it is the premise of this lecture’s hydrodynamics, and the cradle is only nearly integrable.

The dictionary between the two halves, typed: T a theorem of this course, an explicit proved map or $\hbar\to0$ limit between the columns; A a structural analogy, one role played by two separately established facts; C a citation. The column L is the lecture, or the optional lecture, where the Part II side lives.

• One continuity equation per rapidity (20.5), with the dressed velocity (20.4) as the speed at which each rapidity is carried; hard rods show why (Proposition 20.1).

• The cradle: in the two free limits the trapped distribution turns rigidly and returns exactly; between them it returns approximately.

• The course closed: what each idea of Lecture 0’s map rests on, proved, sketched or cited.

• Status. Proposition 20.1 and (20.5) are sketched; the generalised thermodynamic Bethe ansatz, the advection form and the trap equation (20.6) are cited.

• Euler scale. Generalised hydrodynamics is leading order in the gradients; the trap breaks the homogeneous charges, and the tube is not exactly one-dimensional, so the cradle is integrable only approximately.

• No comparison with the 2006 data has been published; theory and experiment meet in later apparatus.

• A complete family of charges is needed to fix the state, and for the XXZ chain the ultralocal ones are not enough (the reading).

• Only the rows marked T are theorems, and all of them are Lecture 14’s.

The ultralocal charges are not enough. For the XXZ chain the charges generated by $\transfer(\bethe)$ as in Lecture 15 do not fix $\vartheta$: two macrostates with different $\vartheta$ can share all of them, and quasi-local charges — the ones whose densities decay exponentially instead of stopping, exactly as $Q[\cos mk]$ did in Lecture 19 — are needed to close the ensemble cited: Ilievski, De Nardis, Wouters, Caux, Essler and Prosen 2015. This is where the free-fermion picture misleads if pushed: there the strictly local charges sufficed, because $\varepsilon$ never vanished.