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2024 | OriginalPaper | Chapter

Mesoscale Mode Coupling Theory for the Weakly Asymmetric Simple Exclusion Process

Author : Gunter M. Schütz

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

The asymmetric simple exclusion process and its analysis by mode coupling theory (MCT) is reviewed. To treat the weakly asymmetric case at large space scale \(x\varepsilon ^{-1}\), large time scale \(t \varepsilon ^{-\chi }\) and weak hopping bias \(b \varepsilon ^{\kappa }\) in the limit \(\varepsilon \rightarrow 0\) we develop a mesoscale MCT that allows for studying the crossover at \(\kappa =1/2\) and \(\chi =2\) from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality. The dynamical structure function is shown to satisfy for all \(\kappa \) an integral equation that is independent of the microscopic model parameters and has a solution that yields a scale-invariant function with the KPZ dynamical exponent \(z=3/2\) at scale \(\chi =3/2+\kappa \) for \(0\le \kappa <1/2\) and for \(\chi =2\) the exact Gaussian EW solution with \(z=2\) for \(\kappa >1/2\). At the crossover point it is a function of both scaling variables which converges at macroscopic scale to the conventional MCT approximation of KPZ universality for \(\kappa <1/2\). This fluctuation pattern confirms long-standing conjectures for \(\kappa \le 1/2\) and is in agreement with mathematically rigorous results for \(\kappa >1/2\) despite the numerous uncontrolled approximations on which MCT is based.

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previous chapter The Extended Phase Space Method in Kinetic Theory
next chapter The Two Dimensional Lorentz Gas in the Kinetic Limit: Theoretical and Numerical Results
Footnotes
1
One may infer \(\chi ^*= 3/2+\kappa \) also from an old result from coupling theory for the one-dimensional ASEP (\(\kappa =0\)) [101] but this does not directly predict the range \(0 < \kappa <1/2\) nor the behaviour at \(\kappa = 1/2\) unlike the mesoscale mode coupling theory developed below.
 
2
There is actually one particular model where this hypothesis can be proved rigorously with vanishing error in the mode coupling approximation, see the discussion in [76]. However, there is no general understanding of when MCT is a good approximation or for what type of cases it may break down.
 
3
This can be exemplified by taking as microscopic scale the atomistic scale of the molecules of which the beads and the channel are composed (with van-der-Waals or similar forces acting between them and Newton’s equation governing the dynamics), as mescoscopic scale the description in terms of hard-core beads and the channel as a confining geometry (with Brownian motion and hard-repulsion determining the noisy dynamics, i.e., the microscopic description in the picture above), and as macroscopic scale description with mild zooming in (with the continuous diffusion equation as dynamics, i.e., the mesoscopic description in the picture above).
 
4
With a view on studying later the limit \(\varepsilon \rightarrow 0\), only the \(\varepsilon \)-dependence of the parameters \(\Lambda _{i}(\varepsilon )\) is indicated.
 
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Metadata
Title
Mesoscale Mode Coupling Theory for the Weakly Asymmetric Simple Exclusion Process
Author
Gunter M. Schütz
Copyright Year
2024
Book
From Particle Systems to Partial Differential Equations

Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-65195-3_16

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