Exercise Sheet 20

(i) From Lecture 20’s rate equation $v(\lambda)=\lambda+a\int\rho(\alpha)\big[v(\lambda)-v(\alpha)\big]\dd\alpha$, show that $\int\rho v=\jmath$, and deduce $v^{\mathrm{eff}}=(\lambda-a\jmath)/(1-an)$. (ii) Show that dressing with the constant kernel $K=-a$ gives $(1)^{\mathrm{dr}}=1-an$ and $(\lambda)^{\mathrm{dr}}=\lambda-a\jmath$, and so the same $v^{\mathrm{eff}}$. (iii) Boost every rod by $u$. Show that $v^{\mathrm{eff}}$ is boosted by $u$ too.

(i) Show that $K(k)=2/c+O(k^{2}/c^{3})$. (ii) Dress the Lieb–Liniger gas with the constant kernel $K_0=2/c$ and show that $v^{\mathrm{eff}}=2(k-a\jmath)/(1-an)$ with $a=-2/c$: hard rods of negative length, as Lecture 17’s large-$c$ limit says. (iii) Show that a local kernel $K=\kappa\,\delta$ would multiply every $f$ by the same factor wherever $\kappa\vartheta\neq2\pi$, so that $v^{\mathrm{eff}}$ would not be dressed at all.

Lecture 20’s generalised TBA is $\eta(\lambda)=w(\lambda)-\frac1{2\pi}\int K(\lambda-\mu)\ln(1+e^{-\eta(\mu)})\dd\mu$, $\vartheta=1/(1+e^{\eta})$. (i) Show that $w=(\varepsilon_0-\mu)/T$ gives Lecture 17’s Yang–Yang equation, with $\eta=\varepsilon/T$. (ii) Show that $\eta'=(w')^{\mathrm{dr}}$, assuming $K\to0$ at infinity and $\ln(1+e^{-\eta})$ bounded. (iii) For free fermions ($K=0$) with $\varepsilon_0=\lambda^{2}$ and $w=\beta_2\lambda^{2}+\beta_4\lambda^{4}$, compare $v^{\mathrm{eff}}$ with $\eta'/(2\pi\rho_t)$.

In the ideal and the impenetrable limits $v^{\mathrm{eff}}=2\lambda$. In a harmonic trap Lecture 20’s characteristics are $\dot x=2\lambda$, $\dot\lambda=-\partial_xV$. (i) Why is $V=\omega^{2}x^{2}/4$ the harmonic trap of frequency $\omega$ in these units? Solve the characteristics, and show that they rotate the plane $(x,2\lambda/\omega)$ rigidly with period $2\pi/\omega$. (ii) Deduce that the filling returns exactly after each period. Show that at a quarter period positions and rescaled rapidities exchange roles, so that the momentum distribution $M_t(\lambda)=\int\rho\,\dd x$ is then $M_{\pi/2\omega}(\lambda)=\frac2\omega n_0(-2\lambda/\omega)$, with $n_0=\int\rho\,\dd\lambda$ at $t=0$. Is $M_t$ constant for every initial state?

(Computational.) Take hard rods with $a=1$ and the initial state $\rho_0(\lambda,x)=n_0(x)\,p(\lambda)$, $n_0(x)=0.1\big(1+3e^{-(x/6)^{2}}\big)$, with $p$ the normalised sum of two Gaussians of width $0.45$ centred at $\lambda=\pm2$: the state of Lecture 20’s figure. The background of density $0.1$ fills the whole line; the lump is the excess over it. (i) Solve generalised hydrodynamics by the exact method of Exercise 7*. Plot $n(x,t)$ at $t=0,4,8,12$, and the free ($a=0$) $n(x,12)$, and locate the right-moving peak of $n(x,12)$ for $a=1$ and for $a=0$. (ii) Solve it again by an upwind scheme in flux form on a periodic grid. Check that the scheme conserves the rods of each rapidity slice, and compare the two solutions at $t=12$. (iii) Why does the interacting lump run ahead?

The course’s closing outcome. For each Part I entry below, give its Part II counterpart, the lecture where it lives, and the type of the correspondence: T a theorem of this course (an explicit proved map or $\hbar\to0$ limit), A a structural analogy, C cited. Poisson bracket; $n$ integrals in involution; Poincaré sections; the linear action–angle system; $\oint p\cdot\dd q=2\pi\hbar n$; permuted momenta; the classical displacement; the Lax matrix; the Sklyanin bracket; $\rmat(\lambda)$; $\pb{\transfer(\lambda)}{\transfer(\mu)}=0$; $\tr L^{k}$; the classical Yang–Baxter equation; the spectral curve; bands and gaps; the frequency $\partial K/\partial I$; the measure $\dd I\dd\varphi$; Calogero–Moser by reduction, and isochrony; the roots of $\sllie_N$; elastic non-diffractive scattering; invariant tori; KAM.

Hard rods are contracted free particles. Let point particles stream freely in a coordinate $y$: $\partial_t\hat\rho+\lambda\partial_y\hat\rho=0$, with $\hat\rho\ge0$ smooth, and $\hat n=\int\hat\rho\,\dd\lambda$, $\hat\jmath=\int\lambda\hat\rho\,\dd\lambda$ finite and differentiable under the integral. No decay as $|y|\to\infty$ is assumed. (i) Show that $\hat N(y,t)=\int_0^y\hat n(y',t)\,\dd y'-\int_0^t\hat\jmath(0,t')\,\dd t'$ has $\partial_y\hat N=\hat n$ and $\partial_t\hat N=-\hat\jmath$.

Give each particle a length $a>0$ by the map $x=y+a\hat N(y,t)$, and let $\rho(\lambda,x,t)\,\dd x=\hat\rho(\lambda,y,t)\,\dd y$. (ii) Show that at each $t$ the map is increasing and onto the line; that $\rho=\hat\rho/(1+a\hat n)$, $n=\hat n/(1+a\hat n)$, $\jmath=\hat\jmath/(1+a\hat n)$ and $1-an=1/(1+a\hat n)$; and that the point $y$ at a fixed $x$ moves as $\partial_ty\big|_x=a\hat\jmath/(1+a\hat n)$. (iii) With $\hat C(\lambda,y,t)=\int_0^y\hat\rho(\lambda,y',t)\,\dd y'-\lambda\int_0^t\hat\rho(\lambda,0,t')\,\dd t'$ and $C(\lambda,x,t)=\hat C(\lambda,y,t)$, show that $\partial_xC=\rho$ and $\partial_tC\big|_x=-\rho\,v^{\mathrm{eff}}$ with $v^{\mathrm{eff}}=(\lambda-a\jmath)/(1-an)$, and deduce that $\rho$ solves generalised hydrodynamics for hard rods.