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bibkey: novaes2023absorption
authors: M. Novaes
year: 2023
title: "Time delay statistics for chaotic cavities with absorption"
doi: 10.1007/s10955-023-03183-7
url: https://arxiv.org/abs/2305.18235v1
claim: "Section 5: Eqs. (54) and (55), strong-absorption moment expansions for every positive channel count and order; Eq. (58), the first two cumulant coefficients for every M ≥ 1 and n > 1."
strata_touched:
  - D5/S3/Quantum/Transport/StrongAbsorption/Foundations
  - D5/S3/Quantum/Transport/StrongAbsorption/LaguerreMoments
  - D5/S3/Quantum/Transport/StrongAbsorption/Density
  - D5/S3/Quantum/Transport/StrongAbsorptionCumulants
  - D5/S3/Quantum/Transport/StrongAbsorption/SecondOrderDensity
  - D5/S3/Quantum/Transport/StrongAbsorptionMoments
license: citation-only
triage: anchor

Verified locator

DOI: https://doi.org/10.1007/s10955-023-03183-7

Source: https://arxiv.org/abs/2305.18235v1

M. Novaes, Journal of Statistical Physics 190, 182 (2023). The following page numbers and equation numbers refer to arXiv v1.

Model and conjecture

Section 1, Eqs. (2)–(3), uses dwell-time units and defines $Q(\gamma )=(1_M-S{S}^{†})/\gamma$ and $\gamma ={\tau }_d/{\tau }_a$. The source says: “τW = (1/M) Tr(Q) is called the Wigner time delay”.

Section 5, p. 7:

we conjecture that, for n > 1,

$$(-1{)}^n{k}_n(\gamma )=\frac{(n-1)!}{M^{2n-2}{\gamma }^{2n}}-\frac{(2n-1)n(n-1)!}{M^{2n-2}{\gamma }^{2n+1}}+O({\gamma }^{-2n-2}).\text{\hspace{1em}(58)}$$

The channel number is any positive integer. The cumulant order is any integer at least two. The constants in the remainder may depend on both. The author-hosted journal text numbers the same conjecture Eq. (65).

The joint law is supplied by A. Grabsch, Distribution of the Wigner–Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases, Journal of Physics A 53, 025202 (2020), DOI: 10.1088/1751-8121/ab5c87, https://arxiv.org/abs/1909.01002v2, Eqs. (3)–(4). At β = 2 and N = M, write the inverse delay rates as Γᵢ = γ + xᵢ. The weight on xᵢ ≥ 0 and 0 ≤ yⱼ ≤ γ is $e^{-M(\sum x_i+\sum y_j)}\Delta(x)^2\Delta(y)^2 \prod_{i,j}(\gamma+x_i-y_j)$. Its total mass normalizes the measure, and ${\tau }_W=M^{-1}{\sum }_i(\gamma +x_i{)}^{-1}$.

The formal cumulant is the n-th derivative at zero of the cumulant generating function of this observable under this normalized measure. The conclusion gives an explicit eventual absolute bound with denominator ${\gamma }^{2n+2}$. It determines two coefficients; the next coefficient, uniform bounds in M, and the β = 1, 4 classes remain open here.

Strong-absorption moments

Section 5, arXiv v1, p. 7:

We conjecture that (1/M)⟨Tr(Qⁿ)⟩ is given by 1/γⁿ − n/γⁿ⁺¹ + n²/γⁿ⁺² − (n + 1)[n(5n − 2)M² + n(n + 2)]/(6M²γⁿ⁺³) + O(γ^{−n−4}) (54) and that (1/Mⁿ)⟨[Tr(Q)]ⁿ⟩ is given by 1/γⁿ − n/γⁿ⁺¹ + n[(n + 1)M² + (n − 1)]/(2M²γⁿ⁺²) + O(γ^{−n−3}), (55) for every positive integer n.

The proper delay times are qᵢ = 1/(γ + xᵢ). The normalized spectral observable is M⁻¹ Σᵢ qᵢⁿ, and the Wigner observable is (M⁻¹ Σᵢ qᵢ)ⁿ. The eventual absolute remainder constants may depend on both M and n. Uniformity in either parameter and higher asymptotic orders are open here.