Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multitime fields and hard rod scaling limits

Pablo A. Ferrari, Stefano Olla

TL;DR

The paper develops a unified probabilistic framework that maps marked Poisson lines in space–velocity into a random surface H_N, whose diffusive scaling converges to the multitime Brownian (Lévy-Chentsov) field. It then connects this surface representation to hard-rod dynamics via dilations and surface-driven length transfers, establishing Euler- and diffusive-scale hydrodynamic limits and fluctuations through law of large numbers and central limit theorems for Poisson processes. Palm theory and invariant-measure analysis link ideal-gas equilibria to Poisson processes and enable a rigorous relationship between the hard-rod and ideal-gas invariant measures. The main contributions include a concrete surface-based encoding of hard-rod evolution, a detailed LLN for the Euler regime, and comprehensive CLT-type fluctuation results in both Euler and diffusive scalings, extending the generalized hydrodynamics framework to non-homogeneous settings with explicit covariance structures. These results provide a precise probabilistic foundation for multitime fields, their scaling limits, and their relevance to completely integrable systems and generalized hydrodynamics.

Abstract

A Poisson line process is a random set of straight lines contained in the plane, as the image of the map $(x,v)\mapsto (x+vt)_{t\in\mathbb{R}}$, for each point $(x,v)$ of a Poisson process in the space-velocity plane. By associating a step with each line of the process, a random surface called multitime walk field is obtained. The diffusive rescaling of the surface converges to the multitime Brownian motion, a classical Gaussian field also called Lévy-Chentsov field. A cut of the multitime fields with a perpendicular plane, reveals a one dimensional continuous time random walk and a Brownian motion, respectively. A hard rod is an interval contained in $\mathbb{R}$ that travels ballistically until it collides with another hard rod, at which point they interchange positions. By associating each line with the ballistic displacement of a hard rod and associating surface steps with hard rod jumps, we obtain the hydrodynamic limits of the hard rods in the Euler and diffusive scalings. The main tools are law of large numbers and central limit theorems for Poisson processes. When rod sizes are zero we have an ideal gas dynamics. We describe the relation between ideal gas and hard-rod invariant measures.

Multitime fields and hard rod scaling limits

TL;DR

The paper develops a unified probabilistic framework that maps marked Poisson lines in space–velocity into a random surface H_N, whose diffusive scaling converges to the multitime Brownian (Lévy-Chentsov) field. It then connects this surface representation to hard-rod dynamics via dilations and surface-driven length transfers, establishing Euler- and diffusive-scale hydrodynamic limits and fluctuations through law of large numbers and central limit theorems for Poisson processes. Palm theory and invariant-measure analysis link ideal-gas equilibria to Poisson processes and enable a rigorous relationship between the hard-rod and ideal-gas invariant measures. The main contributions include a concrete surface-based encoding of hard-rod evolution, a detailed LLN for the Euler regime, and comprehensive CLT-type fluctuation results in both Euler and diffusive scalings, extending the generalized hydrodynamics framework to non-homogeneous settings with explicit covariance structures. These results provide a precise probabilistic foundation for multitime fields, their scaling limits, and their relevance to completely integrable systems and generalized hydrodynamics.

Abstract

A Poisson line process is a random set of straight lines contained in the plane, as the image of the map , for each point of a Poisson process in the space-velocity plane. By associating a step with each line of the process, a random surface called multitime walk field is obtained. The diffusive rescaling of the surface converges to the multitime Brownian motion, a classical Gaussian field also called Lévy-Chentsov field. A cut of the multitime fields with a perpendicular plane, reveals a one dimensional continuous time random walk and a Brownian motion, respectively. A hard rod is an interval contained in that travels ballistically until it collides with another hard rod, at which point they interchange positions. By associating each line with the ballistic displacement of a hard rod and associating surface steps with hard rod jumps, we obtain the hydrodynamic limits of the hard rods in the Euler and diffusive scalings. The main tools are law of large numbers and central limit theorems for Poisson processes. When rod sizes are zero we have an ideal gas dynamics. We describe the relation between ideal gas and hard-rod invariant measures.

Paper Structure

This paper contains 17 sections, 17 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Poisson processes and white noise
  3. Surfaces generated by marked lines
  4. Multitime walk field
  5. Multitime Brownian field
  6. Surface fluctuations
  7. Euler scaling
  8. Diffusive scaling
  9. Hard rods
  10. Hard rods and multi time walk field
  11. Hard rod evolution as seen from a tagged particle
  12. Invariant measures for the ideal gas
  13. Palm measures and hard rod invariant measures
  14. Hard rod hydrodynamics
  15. Law of large numbers
  16. ...and 2 more sections

Key Result

Theorem 1

Assume $\mu\in{\mathcal{M}}$, then, where $B$ and ${\widehat{B }}$ are the Lévy-Chentsov fields associated to the distance $\mathrm{d}(a,b)= \mu_2(ab)$ and ${\widehat{\mathrm{d}}}(a,b)={\widehat{\mu}}_2(ab)$, respectively, where $d{\widehat{\mu}}_2(x,v,r) = dx\,r^2 d\mu(v,r|0)$. Furthermore, the processes $B$ and ${\widehat{B }}$ are

Figures (5)

  • Figure 1: The line $\ell(x,v)$ in the left figure belongs to $ab-$ while the line $\ell({\tilde{x}},{\tilde{v}})$ in the right figure belongs to $ab+$.
  • Figure 2: Mapping space-velocity points to lines contained in the space-time plane.
  • Figure 3: Surfaces generated by $(x,v,r)$ and $({\tilde{x}},{\tilde{v}},{\tilde{r}})$, respectively. We have $h_{(x,v,r)}(a)=-r$ because $o\in\mathop{\mathrm{\text{\rm right}}}\nolimits(x,v)$ and $a\in\mathop{\mathrm{\text{\rm left}}}\nolimits(x,v)$, and $h_{({\tilde{x}},{\tilde{v}},{\tilde{r}})}(a)=r$ because $o\in\mathop{\mathrm{\text{\rm left}}}\nolimits(x,v)$ and $a\in\mathop{\mathrm{\text{\rm right}}}\nolimits(x,v)$.
  • Figure 4: A point configuration $X$ with 3 lines with marks produces a surface $h_N$ with up to 6 possible heights. The left picture shows 3 lines with marks (invisible), and the right picture is a perspective of the surface, as seen from an observer located at $45^o$ to the left.
  • Figure 5: Quasi-particle evolution. At collision particles exchange positions.

Theorems & Definitions (29)

  • Theorem 1: Scaling limits for multitime processes
  • Theorem 2: Surface representation of the hard rod evolution ffgs22
  • Lemma 3: Law of large numbers for the empirical measure
  • proof
  • Lemma 4: Convergence to white noise
  • proof
  • proof : Proof of \ref{['e10']}
  • Remark 5: Comparing $H_N$ with Mandelbrot's $M_N$
  • Theorem 6: Joint convergence to multi time Brownian fields
  • Lemma 7: Flows and tagged quasiparticles in terms of surfaces
  • ...and 19 more