Dean-Kawasaki fluctuating hydrodynamics for backscattering hard rods

This paper employs a Dean-Kawasaki fluctuating hydrodynamic formulation to demonstrate that a system of one-dimensional backscattering hard rods, where particles flip velocities at a rate $\gamma$, exhibits a crossover from ballistic to diffusive spreading in density correlations as time progresses from $t \ll 1/\gamma$ to $t \gg 1/\gamma$.

The Gist

The Big Picture: A Crowd of Bouncing Balls

Imagine a long, narrow hallway filled with people (the "hard rods") who can only move forward or backward. In a normal, perfectly organized hallway, if two people bump into each other, they simply swap places and keep moving. This is a "perfectly integrable" system—it's predictable, and information travels in straight lines forever.

But in this paper, the author introduces a twist: The "Flip".

Imagine that every person in this hallway has a tiny, invisible coin. Every now and then (at a rate called $\gamma$), they flip the coin. If it lands on "Heads," they keep walking. If it lands on "Tails," they instantly turn around and walk the other way. This is the "backscattering" noise.

The Problem: Breaking the Perfect Order

In the world of physics, systems that are perfectly predictable (integrable) are rare and fragile. Adding this "flip" mechanism is like throwing a wrench into a clockwork machine. It breaks the perfect order.

The author wanted to know: How does this chaos change the way things move through the crowd?

• Before the flip: If you dropped a drop of dye in the hallway, it would travel in a straight, fast line (ballistic motion).
• After the flip: The dye gets scattered, slowed down, and eventually spreads out like a drop of ink in water (diffusive motion).

The paper asks: How long does it take for the "perfect straight line" behavior to turn into "spreading out" behavior?

The Tool: The "Dean-Kawasaki" Microscope

To solve this, the author uses a mathematical tool called Dean-Kawasaki fluctuating hydrodynamics.

Think of this tool as a special microscope that doesn't just look at the average position of the people, but also looks at the tiny, random jitters (fluctuations) of every single person.

• Standard Hydrodynamics: Like looking at a crowd from a helicopter and seeing the general flow.
• Dean-Kawasaki: Like being in the crowd, hearing the individual footsteps, the random turns, and the collisions, and using that noise to predict how the whole group behaves.

The Discovery: Two Different Worlds

The paper finds that the behavior of the crowd depends entirely on time.

1. The Short Time: The Sprint ($t \ll 1/\gamma$)

If you watch the crowd for a very short time (shorter than the average time it takes for someone to flip their direction), the "flip" hasn't happened yet.

• Analogy: Imagine a sprinter running down a track. They haven't tripped or turned around yet.
• Result: The density waves (like a ripple in the crowd) travel ballistically. They move fast and straight, just like the original, perfect system. The "flip" noise hasn't had time to mess things up.

2. The Long Time: The Shuffle ($t \gg 1/\gamma$)

If you watch for a long time, everyone has flipped directions many times. The perfect order is gone.

• Analogy: Imagine a crowded dance floor where everyone is randomly turning left and right. If you drop a ball, it doesn't go in a straight line; it bounces around, gets lost, and slowly spreads out.
• Result: The density waves travel diffusively. The "ripple" spreads out slowly, like heat spreading through a metal rod. The system has forgotten its initial perfect order and behaves like a normal, messy fluid.

The "Magic" Formula

The author derived a mathematical formula (Equation 23) that describes this transition. It's like a recipe that tells you exactly how the "ripple" looks at any given moment:

• It has a ballistic part (the fast, straight line) that fades away quickly.
• It has a diffusive part (the slow spread) that takes over as time goes on.

Why Does This Matter?

This isn't just about math; it helps us understand real-world physics.

1. Quantum Computers & Cold Atoms: Scientists are building systems with atoms that act like these "hard rods." Sometimes, these systems are supposed to be perfect, but tiny imperfections (noise) break the rules. This paper helps predict how those imperfections will ruin the system's performance over time.
2. Traffic and Crowds: It gives us a way to model how traffic jams or crowd surges behave when people suddenly change their minds or directions.

Summary

The paper is about a crowd of bouncing rods that occasionally turn around randomly.

• Short term: They act like a perfect, fast-moving train (Ballistic).
• Long term: They act like a slow, spreading gas (Diffusive).
• The Bridge: The author used a sophisticated mathematical microscope (Dean-Kawasaki) to show exactly how and when the system switches from being a "train" to being a "gas," proving that even a little bit of random noise can completely change how a system transports energy and matter.

Technical Summary

1. Problem Statement and Context

The paper addresses the challenge of understanding transport properties in integrable systems when subjected to integrability-breaking perturbations.

• Background: Generalized Hydrodynamics (GHD) has been successful in describing out-of-equilibrium dynamics in integrable systems (e.g., 1D cold atoms). However, most physical systems are not perfectly integrable; they contain perturbations that break integrability, leading to a transition from ballistic (integrable) to diffusive (non-integrable) transport.
• The Model: The author studies a 1D system of Backscattering Hard Rods (BHRs). These are hard rods of length $a$ and unit mass that interact via elastic collisions. Unlike standard hard rods, BHRs possess an additional stochastic mechanism: they randomly flip the sign of their velocity at a rate $\gamma$.
• Physical Consequence: This flipping mechanism breaks integrability. It causes the decay of odd moments of velocity (momentum) while preserving even moments (energy/mass). Consequently, the number of conserved quantities is halved.
• Objective: The goal is to derive a Dean-Kawasaki fluctuating hydrodynamic formulation for this system to compute unequal space-time correlations of density and understand the crossover from ballistic to diffusive transport regimes.

2. Methodology

The author employs a multi-step theoretical framework combining exact mappings, stochastic calculus, and linear response theory.

A. Mapping to Run-and-Tumble Particles (RTPs)

The system of interacting hard rods is mapped to a system of non-interacting Run-and-Tumble Particles (RTPs) (or "hard point particles").

• Mapping: The position $x_i$ of a hard rod is related to the position $x_i^{PP}$ of a point particle by subtracting the excluded volume of other rods: $x_i = x_i^{PP} - a \sum_{j \neq i} \Theta(x_i^{PP} - x_j^{PP})$.
• Dynamics: In this mapped picture, the hard rods behave as non-interacting RTPs that move ballistically between collisions and flip direction (tumble) with rate $\gamma$.
• Phase Space Density (PSD): The mean PSD $\bar{f}_\sigma(x, w, t)$ for the hard rods is expressed in terms of the PSD of the RTPs using a combinatorial formula involving the cumulative density.

B. Dean-Kawasaki Fluctuating Hydrodynamics

Instead of relying solely on mean-field equations, the author derives a stochastic equation for the fluctuating Phase Space Density $f_\sigma(x, w, t)$.

• Derivation: Starting from the trajectory of individual rods and the inverse mapping, the author derives the equation of motion for the fluctuating PSD.
• Key Equation (Eq. 13):
  $$\partial_t f_\sigma + \partial_x (v_\sigma^{\text{eff}} f_\sigma) = \gamma(f_{-\sigma} - f_\sigma) + \sigma \sqrt{\gamma \bar{f}} \xi(x, w, t)$$
  • $v_\sigma^{\text{eff}}$: Effective velocity incorporating hard-core interactions (excluded volume effects).
  • $\gamma(f_{-\sigma} - f_\sigma)$: Deterministic relaxation term due to velocity flipping.
  • $\xi(x, w, t)$: Stochastic noise term arising from the Poissonian nature of the flipping events. The noise has zero mean and delta-correlated statistics.

C. Linearization and Correlation Calculation

To compute transport properties, the system is linearized around a homogeneous stationary state.

• Perturbation: The PSD is written as $f_\sigma = \bar{f}_0/2 + \delta f_\sigma$.
• Fourier Analysis: The linearized equations are solved in Fourier space. The solution consists of a decaying initial condition term and an integral over the noise.
• Correlation Function: The unequal space-time correlation of mass densities, $\langle \rho(x, t)\rho(x', t') \rangle_c$, is computed by integrating the noise correlations over the solution of the linearized equation.

3. Key Results

The primary result is the analytical expression for the density-density correlation function (Eq. 23), which reveals two distinct transport regimes depending on the time scale relative to the flipping rate $\gamma$.

The Correlation Function

The correlation function is composed of two terms:
$$\langle \delta \rho(x, t) \delta \rho(x', t') \rangle_c = \underbrace{\text{Ballistic Term}}_{\propto e^{-\gamma \tau} \exp(-\frac{x^2}{\tau^2})} + \underbrace{\text{Diffusive Term}}_{\propto e^{-\gamma \tau} K_0(\frac{|x|}{\sqrt{\tau}})}$$
(Where $\tau = t - t'$ and $K_0$ is the modified Bessel function of the second kind).

Transport Regimes

1. Short Time ($t \ll 1/\gamma$):
   • The system exhibits ballistic transport.
   • The correlation spreads with a scaling of $x \sim t$.
   • The odd moments of velocity have not yet decayed significantly; the system retains memory of its integrable, ballistic nature.
2. Long Time ($t \gg 1/\gamma$):
   • The system exhibits diffusive transport.
   • The correlation spreads with a scaling of $x \sim \sqrt{t}$.
   • The flipping noise has randomized the direction of motion, destroying the ballistic conservation laws and leading to standard diffusion.
   • The diffusive term is dominated by the Bessel function $K_0$, characteristic of 1D diffusion with a specific diffusion coefficient derived from the model parameters.

Numerical Verification

The analytical results were verified against Molecular Dynamics (MD) simulations of $N=400$ backscattering hard rods.

• Fig. 2 demonstrates excellent agreement between the Dean-Kawasaki theory and MD simulations.
• For small $\gamma$ (long ballistic regime), the correlation follows the ballistic scaling.
• For large $\gamma$ (short ballistic regime), the correlation quickly transitions to diffusive scaling.

4. Significance and Contributions

• Validation of Dean-Kawasaki Formalism: The paper successfully applies the Dean-Kawasaki fluctuating hydrodynamics framework to an integrable system perturbed by stochastic noise. This demonstrates the method's utility in capturing both the mean dynamics and the fluctuation-induced transport properties.
• Mechanism of Crossover: It provides a precise analytical description of how integrability breaking (via velocity flipping) drives the crossover from ballistic to diffusive transport. It quantifies the time scale ($1/\gamma$) required for this transition.
• Conserved Quantities: It explicitly shows how the reduction in conserved quantities (from infinite in integrable systems to just mass/energy in this perturbed system) alters the hydrodynamic modes.
• General Applicability: The authors note that this approach is not limited to backscattering noise but can be extended to other perturbations, such as Brownian hard rods or systems confined in harmonic traps, offering a robust tool for studying non-equilibrium dynamics in interacting particle systems.

Conclusion

Mrinal Jyoti Powdel's work bridges the gap between integrable hydrodynamics (GHD) and conventional hydrodynamics by using a Dean-Kawasaki formulation. By deriving the fluctuating hydrodynamic equations for backscattering hard rods, the paper analytically proves that the introduction of a velocity-flipping rate $\gamma$ induces a crossover from ballistic to diffusive transport, a result rigorously confirmed by numerical simulations. This provides a deeper understanding of how integrability breaking perturbations manifest in macroscopic transport properties.