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The partition function in the quantum-to-classical transition

Bingyu Cui

TL;DR

The paper develops a phase-space quantum partition function Z_u based on Bohmian trajectories, unifying quantum and classical statistical mechanics within a canonical ensemble. The stationary part of Z_u reproduces the conventional quantum partition function Z_q, while in the limit of sharply localized distributions it reduces to the classical partition function Z_cl, providing a natural quantum-to-classical crossover. A key result is a temperature- and localization-dependent criterion for classicalization, linked to the width σ of the wavepacket and the thermal de Broglie wavelength, with concrete analysis for a single harmonic oscillator and implications for Caldeira–Leggett baths. This framework offers a phase-space perspective on quantum statistics, ETH-related considerations, and potential extensions to open systems, while preserving standard thermodynamic relations through Z_u.

Abstract

In classical statistical mechanics, the partition function is defined in phase space. We extend this concept to quantum statistical mechanics using Bohmian trajectories. The quantum partition function in phase space captures the ensemble of positions and momenta, along with the probability distribution that accounts for the inherent uncertainty in measuring particle locations. Within this framework, the quantum-to-classical transition arises naturally, maintaining consistency between dynamics and statistical mechanics.

The partition function in the quantum-to-classical transition

TL;DR

The paper develops a phase-space quantum partition function Z_u based on Bohmian trajectories, unifying quantum and classical statistical mechanics within a canonical ensemble. The stationary part of Z_u reproduces the conventional quantum partition function Z_q, while in the limit of sharply localized distributions it reduces to the classical partition function Z_cl, providing a natural quantum-to-classical crossover. A key result is a temperature- and localization-dependent criterion for classicalization, linked to the width σ of the wavepacket and the thermal de Broglie wavelength, with concrete analysis for a single harmonic oscillator and implications for Caldeira–Leggett baths. This framework offers a phase-space perspective on quantum statistics, ETH-related considerations, and potential extensions to open systems, while preserving standard thermodynamic relations through Z_u.

Abstract

In classical statistical mechanics, the partition function is defined in phase space. We extend this concept to quantum statistical mechanics using Bohmian trajectories. The quantum partition function in phase space captures the ensemble of positions and momenta, along with the probability distribution that accounts for the inherent uncertainty in measuring particle locations. Within this framework, the quantum-to-classical transition arises naturally, maintaining consistency between dynamics and statistical mechanics.

Paper Structure

This paper contains 12 sections, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical and quantum mechanics
  3. Classical dynamics and statistical mechanics
  4. The Bohmian interpretation of quantum mechanics
  5. Partition functions in quantum statistical physics
  6. From dynamics to statistical mechanics
  7. Quantum to classical crossover
  8. Single harmonic oscillator
  9. Conclusion
  10. The time evolution of the energy
  11. Free particle
  12. The quantum-to-classical transition of a heat bath

Figures (1)

  • Figure 1: Time evolution of the marginal partition function $Z_{(x_0,p_0)}$ in Eq. \ref{['eq:margin']} for a quantum harmonic oscillator with mass $m$ and frequency $\omega$. The initial state is the Gaussian wavepacket in Eq. \ref{['eq:ho']} with specified $(x(0),p(0))\equiv(x_0,p_0)$. Curves show $\sigma=0.45,k_BT=2$ (red); $\sigma=0.45,k_BT=5$ (blue) and $\sigma=0.65,k_BT=2$ (green). All curves are normalized to $Z_{(x_0,p_0)}=1$ at $t=0$ (black dashed line). The displacement is measured in units of $x_0$; energy in units of $m\omega^2x_0^2$ and time in units of $\omega^{-1}$.