A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles

Luís Simão Ferreira

doi:10.3934/krm.2021045

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2022, Volume 15, Issue 1: 91-117. Doi: 10.3934/krm.2021045

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A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles

Luís Simão Ferreira^,

Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande 016, 1749-016 Lisbon

^*Corresponding author: Luís Simão Ferreira
^*Corresponding author: Luís Simão Ferreira

Received: July 31, 2021

Revised: October 31, 2021

Published: February 2022

Supported by CMAF-CIO grant UIDP/04561/2020

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Abstract

In this paper, we proceed as suggested in the final section of [2] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around $ 0.02 $, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.

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Mathematics Subject Classification: Primary: 82C40, 39B42; Secondary: 82M31.

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Figure 1. Sampled and implied entropy decay for $ \alpha = 2 $, and sampled entropy decay for $ \alpha = 0 $, respectively

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Figure 2. $ \alpha = 0 $, sampled velocity

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Figure 3. $ \alpha = 2 $, sampled velocity

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Figure 4. $ \alpha = 2 $, implied velocity

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Figure 5. First 300 values of $ \kappa_{n, \ell} $, for $ \ell = 5, 15 $, and overlap of the plots from $ \ell = 6 $ to $ 10 $

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