Generalized hydrodynamics of the classical Toda system

Abstract

We obtain the exact generalised hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorised scattering of Toda particles, we obtain the generalised free energy and exact average currents, and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta, and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and in particular the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorised scattering of Toda particles.

Kardar–Parisi–Zhang Physics in Integrable Rotationally Symmetric Dynamics on Discrete Space–Time Lattice

“…It would be interesting to see if the double-scaling ansatz µ → 0, t → ∞, which has been proposed for quantum XXX model [30] would be applicable here as well, but this would require much more refined simulations on this particular regime which are beyond the scope of the present work. A quantitative theoretical explanation of stationary correlation cross sections displayed in Figure 4 is within the scope of generalized hydrodynamics of classical integrable systems [29], since C(x, t) can be related to an inhomogeneous quench problem for a step initial state in the linear response limit [9], provided one could facilitate local conserved charges constructed in section 2.4. This is an interesting problem for future research.…”

Hydrodynamics of massless integrable RG flows and a non-equilibrium c-theorem

“…The appearance of non-trivial physics in this limit is in accordance with the ballistic spreading of quasi-particles in integrable models and in many relevant situations the GHD predictions become valid after a relatively short transient time interval [37][38][39][40]. The GHD approach has been applied to various systems including spin chains and the Hubbard model [43,[45][46][47][48][49][50][51][52], classical gases and fields [53][54][55][56] and quantum gases and fields [38,42,57,58]. Interesting view points on the GHD approach are given in [59][60][61].…”

Linearized regime of the generalized hydrodynamics with diffusion

“…At large times the density of quasi-particles gets spread uniformly over the space as (34) indicates. To quantify the process of equilibration we consider the evolution of modes δn k (t) defined in (25).…”

“…Such bi-partite quenches or similar inhomogeneous initial states were studied in different contexts ranging from the CFT [1][2][3][4][5][6][7] and more generally QFT [8][9][10] to lattice models including 1d spin chains [11][12][13][14][15][16][17][18][19][20][21][22] and 1d Hubbard-like models [23][24][25]. The hydrodynamics solution to bi-partite quench protocol was originally proposed in [26,27] and further developed [28][29][30][31][32][33][34]. The resulting theory, Generalized Hydrodynamics (GHD), was successfully applied to variants of bi-partite quench protocols [27,[35][36][37][38] and other inhomogeneous setups [28,39].…”