Stochastic dynamics of quasiparticles in the hard rod gas

TL;DR

This paper reveals that in a one-dimensional hard-rod gas, tagged quasiparticles undergo drifting Brownian motion with a velocity-dependent drift and diffusion, elucidated through a microscopic hard-point mapping and Euler GHD. In a homogeneous background, two quasiparticles with the same velocity remain strongly correlated, effectively moving as a rigid body, while fluctuations originate from initial phase-space fluctuations carried by Euler flow. A Dean–Kawasaki–type fluctuating hydrodynamics is formulated to describe these fluctuations, and the authors extend the analysis to inhomogeneous backgrounds and quenched initial conditions, deriving explicit mean, variance, and covariance expressions. The results bridge microscopic dynamics with fluctuating hydrodynamics, highlight important correlations, and raise puzzles about the precise fluctuating descriptions in integrable systems, suggesting avenues for future work in other models like the Toda chain.

Abstract

We consider a one-dimensional gas of hard rods, one of the simplest examples of an interacting integrable model. It is well known that the hydrodynamics of such integrable models can be understood by viewing the system as a gas of quasiparticles. Here, we explore the dynamics of individual quasiparticles for a variety of initial conditions of the background gas. The mean, variance, and two-time correlations are computed exactly and lead to a picture of quasiparticles as drifting Brownian particles. For the case of a homogeneous background, we show that the motion of two tagged quasiparticles is strongly correlated, and they move like a rigid rod at late times. Apart from a microscopic derivation based on the mapping to point particles, we provide an alternate derivation which emphasizes that quasiparticle fluctuations are related to initial phase-space fluctuations, which are carried over in time by Euler scale dynamics. For the homogeneous state, we use the Brownian motion picture to develop a Dean-Kawasaki-type fluctuating hydrodynamic theory, formally having the same structure as that derived recently by Ferrari and Olla. We discuss differences with existing proposals on the hydrodynamics of hard rods and some puzzles.

Figures (6)

• Figure 1: Schematic trajectories of three quasiparticles in a one-dimensional integrable system.
• Figure 2: The schematic diagram (a) illustrates the stochastic trajectories of hard rods encountering collisions and exchanging velocities with each other. The trajectory marked in deep blue indicates the path of a marked rod. In (b), we show trajectories of two quasiparticles (black and red) tagged by their bare velocities. At each collision, a quasiparticle jumps by a distance $\pm a$ while retaining its velocity. Since the collisions occur at random times for random initial configurations, a quasiparticle follows a stochastic path around a mean ballistic motion with an effective velocity.
• Figure 4: Numerical simulation results (circles) for the variance of quasiparticle separation are compared with theoretical predictions given in Eq. \ref{['var-sep-gen']} (dashed lines) for two cases: (a) quasiparticles moving with identical velocities, $v_0=u_0=0.5$; (b) quasiparticles moving with different velocities $v_0=0.4$, $u_0=0.5$. The main plots correspond to the homogeneous density profile with $\varrho_0=0.4$, while the plots in the inset are for the domain-wall profile with $\varrho_l= 0.4, \varrho_m=0.6, \varrho_r=0.5$. Simulations were performed with $N=5000, a=0.5, T=1$, $Y_0=(\frac{N}{10}+1)a$, with $\bar{N}=\varrho_0 Y_0$ for the homogeneous case and $\bar{N}=\varrho_m Y_0$ for the domain-wall case. The average has been done over $10^4$ independent initial configurations. The red solid lines in Figure (b) here represent the small and large $t$ asymptotics given in Eq. \ref{['t_asymptotic']}.
• Figure 5: Scaling collapse of the variance of separation $\frac{\langle (Y-X)^2\rangle_c}{t}$ plotted as a function of time $\epsilon^{-1}=\varphi_0t/ \bar{N}$ in the log scale for different values of $\bar{N}=\varrho_0 ({N}/{2}+1)a$ under homogeneous initial condition. For each $\varrho_0$, the data for different $\bar{N}$ collapses to a scaling curve. The scaling curve for smaller $\varrho_0$ appears to converge to the theoretical scaling function in Eq. \ref{['var-sep-scaling-function']} (red dashed line), which corresponds to the limit $\bar{N}/N \to 0$. The parameters used are: $a=0.5$, $v_0=u_0=0.5$, $Y_0=(N/2+1)a$. Average has been done over $10^4$ independent initial configurations for each $N$.
• Figure 6: Numerical simulation (black squares) and theoretical results (black dashed lines) for (a) mean, (b) variance, and (c) covariance of two quasiparticles $X(t), Y(t)$ from quenched initial condition following Eqs. \ref{['eq:2quasiparticle_origin_var_QIC_uniform']}-\ref{['eq:2quasiparticle_origin_covar_QIC_uniform']}. Insets show the corresponding results for quasiparticle $X(t)$. Results from quenched initial conditions are compared with those for annealed initial conditions (red stars: simulations; red dashed lines: theory). The parameters are $a=0.5, v_0=0.5, ~ u_0=1.0, ~T=1$, $N=3000,\varrho_0 =0.6$ and $Y_0=a$. The average has been done over $10^4$ realizations.