Uniform Partition-Function Estimates for Coulomb Modulated Energy at All Positive Temperatures
Abstract
We prove uniform-in-N partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel--Riesz kernels on ℝd in the locally square-integrable range 0≤ s<d/2. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal N−1 normalized relative-entropy rate for the three-dimensional Coulomb flow; and an N−1/2 Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman--Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.
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BibTeX
@misc{wang2026uniform,
title = {{Uniform Partition-Function Estimates for Coulomb Modulated Energy at All Positive Temperatures}},
author = {Zhenfu Wang and Xianliang Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00867},
eprint = {2609.00867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00867},
}
RIS
TY - UNPB
TI - Uniform Partition-Function Estimates for Coulomb Modulated Energy at All Positive Temperatures
AU - Zhenfu Wang
AU - Xianliang Zhao
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00867
UR - https://arxiv.org/abs/2609.00867
AB - We prove uniform-in-N partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel--Riesz kernels on ℝ^d in the locally square-integrable range 0≤ s<d/2. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal N^−1 normalized relative-entropy rate for the three-dimensional Coulomb flow; and an N^−1/2 Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman--Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.
ER -