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On some mathematical problems for open quantum systems with varying particle number

Benedikt M. Reible, Luigi Delle Site

Abstract

We derive the effective Hamiltonian $H - μN$ for open quantum systems with varying particle number from first principles within the framework of non-relativistic quantum statistical mechanics. We prove that under physically motivated assumptions regarding the size of the system and the range of the interaction, this form of the Hamiltonian is unique up to a constant. Our argument relies firstly on establishing a rigorous version of the surface-to-volume ratio approximation, which is routinely used in an empirical form in statistical mechanics, and secondly on showing that the Hilbert space for systems with varying particle number must be isomorphic to Fock space. Together, these findings provide a rigorous mathematical justification for the standard grand canonical formalism employed in statistical physics.

On some mathematical problems for open quantum systems with varying particle number

Abstract

We derive the effective Hamiltonian for open quantum systems with varying particle number from first principles within the framework of non-relativistic quantum statistical mechanics. We prove that under physically motivated assumptions regarding the size of the system and the range of the interaction, this form of the Hamiltonian is unique up to a constant. Our argument relies firstly on establishing a rigorous version of the surface-to-volume ratio approximation, which is routinely used in an empirical form in statistical mechanics, and secondly on showing that the Hilbert space for systems with varying particle number must be isomorphic to Fock space. Together, these findings provide a rigorous mathematical justification for the standard grand canonical formalism employed in statistical physics.
Paper Structure (26 sections, 14 theorems, 111 equations, 2 figures)

This paper contains 26 sections, 14 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prolegomenon
  3. Main Results
  4. Outline of the paper
  5. Notation
  6. Basic physical assumptions and mathematical setup
  7. Open quantum systems
  8. Interactions
  9. Density operators
  10. Surface-to-volume ratio approximation
  11. Short-range interactions
  12. Volume of the interaction corridor
  13. Formulation of the approximation
  14. Varying particle number and Fock space
  15. Particle number operator
  16. ...and 11 more sections

Key Result

Lemma 2.5

Let $T \in \mathscr{L}(\mathcal{H}_\mathsf{S})$ be self-adjoint and $\rho \in \mathscr{S}(\mathcal{H}_\mathsf{S} \otimes \mathcal{H}_\mathsf{R}, T \otimes \mathop{}\!\mathrm{Id}_\mathsf{R})$ be a $T \otimes \mathop{}\!\mathrm{Id}_\mathsf{R}$-compatible density operator. Then $\mathop{\mathrm{tr}}\no

Figures (2)

  • Figure 1: Division of the total system according to \ref{['asm:systemDivision']}.
  • Figure 2: Illustration of the set $\varOmega_\mathsf{S}$, circumscribed by the solid line, and the interaction corridor $(\partial \varOmega_\mathsf{S})_{\delta}^+ \cup (\partial \varOmega_\mathsf{S})_{\delta}^-$ of width $2 \delta$ around its boundary, highlighted as the gray region delimited by the dashed lines.

Theorems & Definitions (36)

  • Remark 2.2
  • Example 2.3
  • Definition 2.4: Compatible density operators
  • Lemma 2.5
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4: Surface-to-volume ratio approximation
  • proof
  • Example 3.6
  • ...and 26 more