We study the classical Toda lattice with domain wall initial conditions, for which left and right half lattice are in thermal equilibrium but with distinct parameters of pressure, mean velocity, and temperature. In the hydrodynamic regime the respective space-time profiles scale ballisticly. The particular case of interest is a jump from low to high pressure at uniform temperature and zero mean velocity. Thereby the scaling function for the average stretch (also free volume) is forced to change sign. By direct inspection, the hydrodynamic equations for the Toda lattice seem to be singular at zero stretch. In our contribution we report on numerical solutions and convincingly establish that nevertheless the self-similar solution exhibits smooth behavior.
1. Introduction, 2. Dynamics of the classical Toda lattice, 3. Static properties, 4. Mean-field Dyson Brownian motion, 5. Hydrodynamics for hard rods, 6. Generalized hydrodynamic equations, 7. Linearized hydrodynamics and GGE dynamical correlations, 8. Domain wall initial states, 9. Toda fluid, 10. Hydrodynamics for the Lieb-Liniger delta-Bose gas, 11. Quantum Toda lattice, 12. Beyond the Euler time scale.
For 1D Hamiltonian systems with periodic solutions, Helmholtz formalism provides a tantalizing interpretation of classical thermodynamics, based on time integrals of purely mechanical quantities and without need of statistical description. Here we extend this approach to include heat flux and pressure at the walls, thereby enabling it to describe actual thermodynamic transformations, such as isothermal compressions and expansions. The presence of hard walls, which gives rise to non zero pressure, is justified by means of the virial theorem, while the heat fluxes are introduced as quasi-static …
The well-known Wall theorem states a simple and precise relation among temperature, pressure and density of a fluid at contact with a confining hard wall in thermodynamic equilibrium. In this Letter we develop an extension of the Wall theorem to out-of-equilibrium conditions, providing an exact relation between pressure, density and temperaure at the wall, valid for strong non-equilibrium situations. We derive analytically this Non-equilibrium Wall theorem for stationary states and validate it with non-equilibrium event-driven molecular-dynamics simulations. We compare the analytical expressio…
The concept of the partial osmotic pressure of ions in an electrolyte solution is critically examined. In principle these can be defined by introducing a solvent-permeable wall and measuring the force per unit area which can certainly be attributed to individual ions. Here I demonstrate that although the total wall force balances the bulk osmotic pressure as required by mechanical equilibrium, the individual partial osmotic pressures are extra-thermodynamic quantities dependent on the electrical structure at the wall, and as such they resemble attempts to define individual ion activity coeffic…
