Does hydrodynamic theory match Newtonian dynamics in confined Riesz gases?

Two descriptions of the same confined gas

The paper asks whether the real-time evolution of a confined Riesz gas, a one-dimensional system of classical particles with power-law interactions, can be reproduced by conventional hydrodynamics. The Riesz gas is a unifying model: depending on the power-law exponent k, it interpolates between the one-component plasma, Dyson's log-gas, the integrable Calogero model, and the hard-rod gas. Its equilibrium properties are well studied, but the paper states that the connection between microscopic dynamics, macroscopic evolution, and thermalization has remained largely unexplored.

The comparison is between two descriptions of the same system. On one side are large-scale Newtonian simulations of the full Hamiltonian, with particles confined by a quadratic trap and interacting with up to d neighbors on either side. On the other side is a hydrodynamic framework built on three coarse-grained fields: density, velocity, and temperature. The paper reports a direct and quantitative correspondence between them, which is the central claim to evaluate.

Newtonian simulations with two initial profiles

The microscopic side solves the equations of motion for N classical particles in a harmonic trap with pairwise power-law interactions. The simulations use a symplectic velocity-Verlet integration scheme with a time step dt = 10^-6, and the paper fixes d = 1, k = 1.5, m = 1, and omega = 1. Positions are initialized from prescribed spatial density profiles, while velocities are first drawn from a Maxwell distribution and then rescaled so that the total energy has a fixed prescribed value.

Two representative initial conditions are used. The first is a dome-like profile drawn from an equilibrium distribution at a chosen temperature. The second is a Newton-cradle-type profile consisting of two blobs, each of finite width and separated by a finite distance. For both, the paper reports N = 64 and an average over 10^4 realizations. The initial temperature for the dome-like case is T_init approximately 50, and the system reaches a steady state around t approximately 240 for the dome-like profile and around t approximately 500 for the Newton-cradle-type profile.

The steady-state temperature is obtained from the generalized equipartition relation and is approximately 39 for both initial conditions, consistent with the expectation that the steady state should not depend on the initial condition when the total energy is held fixed. The steady-state density profile is also compared with an analytical saddle-point prediction, using the unscaled coordinate x = y N^{alpha_k}.

Density, momentum, and temperature as coarse-grained fields

The hydrodynamic description is formulated in terms of three fields constructed from the microscopic data. The density field rho(x,t) is the ensemble average of the sum of delta functions at particle positions. The momentum density p(x,t) is the ensemble average of the particle velocities weighted by the same delta functions. The temperature field T(x,t) is defined through the local variance of the velocity distribution, with particle velocities measured relative to the local mean velocity v(x,t) = p(x,t)/rho(x,t).

This subtraction matters for interpretation. It separates the kinetic energy carried by the coherent bulk motion of the fluid from the energy associated with random thermal motion about that mean flow. The paper notes that the local kinetic energy density splits into a flow contribution rho v^2 / 2 and a thermal contribution rho T / 2, and that in the steady state, where v(x,t) = 0 and T is uniform, integrating the temperature definition recovers the equipartition relation. Readers should not confuse the local temperature field, which is a variance about the local mean velocity, with a simple second moment of the velocity distribution.

Two transport coefficients close the hydrodynamic equations

The hydrodynamic equations conserve particle number and energy, while momentum is not conserved because of the external harmonic trap. The momentum equation therefore gains an extra term related to the external force. At the Euler level, the equations are known exactly for this system. At the Navier-Stokes level, the paper introduces two phenomenological transport coefficients: the bulk viscosity xi and the thermal conductivity kappa, which encode dissipative effects at the macroscopic level.

To solve the equations, the pressure must be expressed as a function of the density field at finite temperature. The paper derives this from the free energy of the finite-range Riesz gas, obtaining P(x,t) = J zeta_d(k) k rho(x,t)^{k+1} + T(x,t) rho(x,t). In the noninteracting limit J = 0, this reduces to the ideal gas form P = rho T. The internal energy per unit mass is similarly assembled from the harmonic confinement, the interaction contribution, the kinetic contribution T/2, and the mean-flow contribution v^2/2.

The transport coefficients are not derived from the microscopic dynamics in this work. They are phenomenological inputs, and the paper explicitly lists a microscopic derivation of the transport coefficients as a natural direction for future investigation. This is a key boundary: the agreement reported below tests whether a conventional three-field hydrodynamic theory with two adjustable dissipative parameters can reproduce the Newtonian evolution, not whether those parameters can be predicted from first principles.

Agreement from early times to steady state

For the dome-like initial condition, the paper compares density, momentum, and temperature from Newtonian dynamics and hydrodynamics over a broad time window spanning early-time dynamics to the steady state, using the representative dissipative parameters xi = 0.25 and kappa = 0.15. It reports good agreement between the two descriptions for this case.

For the Newton-cradle-type initial condition, the behavior depends on the size of the dissipative coefficients. For very small values of xi and kappa, the hydrodynamic and Newtonian profiles agree at short times but deviate at intermediate times, and the weak dissipation slows relaxation, delaying the approach to the steady state relative to the microscopic dynamics. For moderately large values of xi and kappa, the hydrodynamic description fails to capture the very early-time dynamics because of excessive damping, but beyond an early time, in this case t greater than or approximately 40, the hydrodynamic profiles agree quantitatively with the Newtonian results up to the steady state. The paper reports that this behavior is consistently observed across the density, momentum, and temperature fields.

The distinction between the two initial conditions is important. The dome-like case shows agreement across the reported window with the reference parameters. The Newton-cradle case shows a crossover: early-time disagreement under strong damping, followed by quantitative agreement after t approximately 40. The paper does not claim that hydrodynamics captures every instant of the Newton-cradle evolution.

A substantial window of transport coefficients

To test robustness, the paper solves the hydrodynamic equations for additional parameter sets close to the reference choice: (xi, kappa) = (0.2, 0.1), (0.3, 0.2), and (0.2, 0.3). It also tests values farther from the reference: (0.1, 0.05) and (0.55, 0.45). The comparison uses a Newton-cradle-type initial condition and examines density, momentum, and temperature profiles at t = 40, 80, 160, and 500.

Across the close parameter sets, the profiles remain in close agreement with each other and with the reference (0.25, 0.15), indicating that the hydrodynamic evolution is robust within this reasonably wide window and that agreement with the corresponding Newtonian dynamics is maintained. For the far parameter choices, the relaxation timescales are modified by weaker and stronger dissipation, and the profiles deviate from the reference. With (0.1, 0.05), the smaller dissipation increases the hydrodynamic timescale and the profiles do not agree with the reference; with (0.55, 0.45), the larger dissipation decreases the hydrodynamic timescale.

The practical reading is a window, not a point. The paper's robustness claim applies to a substantial but finite range of xi and kappa around the reference values. Outside that range, the hydrodynamic description still exists, but its relaxation timescale no longer matches the reference behavior.

Euler dynamics dominate the approach to steady state

The paper interprets the persistence of agreement across a broad parameter window as evidence that the nonequilibrium evolution to the steady state is predominantly controlled by the Euler dynamics. In other words, the gross approach to the Gibbs-Boltzmann state is captured by the conservative part of the hydrodynamic equations, while the two dissipative coefficients encode the residual effects needed to match the microscopic evolution quantitatively.

This interpretation is the authors' reading of their own comparison, not an independent measurement. The evidence supporting it is the robustness of the profiles across the tested xi and kappa values and the quantitative agreement with Newtonian dynamics over the reported timescales. The paper also states that the results provide a systematic validation of hydrodynamics in the Riesz gas and establish a concrete bridge between Newtonian dynamics and continuum theories in systems with power-law interactions, starting from first principles.

Finite range, one dimension, and open questions

The simulations use a finite-range model with d = 1, meaning each particle interacts with one neighbor on either side, and fix k = 1.5. The paper notes that the findings are valid for all-to-all coupling as long as the system is in the relatively short-ranged regime, which for the all-to-all case corresponds to k greater than 1. The reported agreement is therefore established for a specific one-dimensional, finite-range, nonintegrable setup, not for every Riesz gas parameter regime.

The paper also identifies regimes where hydrodynamics may break down, such as extreme initial states and finite-size effects, and points to special integrable points such as the Calogero-Moser case k = 2 with all-to-all coupling, where conventional hydrodynamics needs to be replaced by generalized hydrodynamics. Understanding how weak perturbations away from integrability lead to a crossover from generalized to conventional hydrodynamics is listed as an open direction. The transport coefficients themselves remain phenomenological inputs awaiting a microscopic derivation.

The most decision-relevant result is the parameter window. If a reader wants to know when a three-field hydrodynamic description can be trusted for this confined Riesz gas, the paper's answer is: with the reference parameters (0.25, 0.15) and across the nearby sets (0.2, 0.1), (0.3, 0.2), and (0.2, 0.3), the profiles agree with Newtonian dynamics and with each other, while the far sets (0.1, 0.05) and (0.55, 0.45) shift the relaxation timescale and break the agreement. The priority for follow-up is therefore not whether hydrodynamics works at all, but how far the transport-coefficient window extends and whether xi and kappa can be derived from the particle-level dynamics rather than fitted.

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