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New convergence bound for the cluster expansion in canonical ensemble

Giuseppe Scola

Abstract

We perform a cluster expansion in the canonical ensemble with periodic boundary conditions, introducing a new choice of polymer activities that differs from the standard ones. This choice leads to an improved bound for the convergence of the cluster expansion, which we compare with the known one. We also recover the irreducible Mayer coefficients for the thermodynamic free energy. The results presented here can also be applied to the case of zero boundary conditions and to the convergence of correlation expansions.

New convergence bound for the cluster expansion in canonical ensemble

Abstract

We perform a cluster expansion in the canonical ensemble with periodic boundary conditions, introducing a new choice of polymer activities that differs from the standard ones. This choice leads to an improved bound for the convergence of the cluster expansion, which we compare with the known one. We also recover the irreducible Mayer coefficients for the thermodynamic free energy. The results presented here can also be applied to the case of zero boundary conditions and to the convergence of correlation expansions.

Paper Structure

This paper contains 7 sections, 1 theorem, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Model and main result
  4. Polymer model representation and convergence of cluster expansion
  5. From cluster coefficients to Mayer's coefficients (proof of \ref{['eq:TFE-final']})
  6. Free case
  7. Interacting case

Key Result

Theorem 2.1

Given inverse temperature $\beta>0$ and stability constant $B\ge 0$, there exists $K\ge1$, such that for all complex $\rho$ with $|\rho|\le \rho^*$, where the limit in eq:f-thermo exists and the thermodynamic free energy is given by where the coefficients $\beta_m$ are the irreducible Mayer coefficients, defined by and $\mathcal{B}_{m+1}$ denotes the family of $2$-connected (irreducible) graphs

Figures (2)

  • Figure 1: First graph: red line $g(a^*_{\beta,B,K})$, blue line $K=1.3$ with $e^{-\beta B}K\in(0,1.3]$. Second graph: particular with $e^{-\beta B}K=1.3\cdot e^{-\beta B}\in(0,0.2]$.
  • Figure 2: Red line $g(a^*_{\beta,B,K})$, blue line $K=1.1462$ with $e^{-\beta B}K\in(0,1.1462]$.

Theorems & Definitions (5)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 4.1