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Probability Distribution for Vacuum Energy Flux Fluctuations in Two Spacetime Dimensions

Christopher J. Fewster, L. H. Ford

TL;DR

This work derives exact probability distributions for vacuum fluctuations of the energy flux in two-dimensional conformal field theories by modeling time-averaged stress-tensor components as independent shifted-Gamma variables and convolving them to obtain the flux distribution. The flux distribution is shown to be symmetric and governed by a closed form involving a modified Bessel function, with tails that decay exponentially and a variance fixed by the sampling parameters. The authors also construct the joint distribution of energy flux and energy density, revealing that flux fluctuations are more centrally concentrated and providing conditional distributions when the energy density is negative. By analyzing specific sampling functions (Gaussian-like, Lorentzian-like, and compactly supported), they obtain explicit parameters and behaviors, offering insights applicable to four-dimensional models, numerical simulations, and analog condensed-matter systems where stress-tensor fluctuations play a role.

Abstract

The probability distribution for vacuum fluctuations of the energy flux in two dimensions will be constructed, along with the joint distribution of energy flux and energy density. Our approach will be based on previous work on probability distributions for the energy density in two dimensional conformal field theory. In both cases, the relevant stress tensor component must be averaged in time, and the results are sensitive to the form of the averaging function. Here we present results for two classes of such functions, which include the Gaussian and Lorentzian functions. The distribution for the energy flux is symmetric, unlike that for the energy density. In both cases, the distribution may possess an integrable singularity. The functional form of the flux distribution function involves a modified Bessel function, and is distinct from the shifted Gamma form for the energy density. By considering the joint distribution of energy flux and energy density, we show that the distribution of energy flux tends to be more centrally concentrated than that of the energy density. We also determine the distribution of energy fluxes, conditioned on the energy density being negative. Some applications of the results will be discussed.

Probability Distribution for Vacuum Energy Flux Fluctuations in Two Spacetime Dimensions

TL;DR

This work derives exact probability distributions for vacuum fluctuations of the energy flux in two-dimensional conformal field theories by modeling time-averaged stress-tensor components as independent shifted-Gamma variables and convolving them to obtain the flux distribution. The flux distribution is shown to be symmetric and governed by a closed form involving a modified Bessel function, with tails that decay exponentially and a variance fixed by the sampling parameters. The authors also construct the joint distribution of energy flux and energy density, revealing that flux fluctuations are more centrally concentrated and providing conditional distributions when the energy density is negative. By analyzing specific sampling functions (Gaussian-like, Lorentzian-like, and compactly supported), they obtain explicit parameters and behaviors, offering insights applicable to four-dimensional models, numerical simulations, and analog condensed-matter systems where stress-tensor fluctuations play a role.

Abstract

The probability distribution for vacuum fluctuations of the energy flux in two dimensions will be constructed, along with the joint distribution of energy flux and energy density. Our approach will be based on previous work on probability distributions for the energy density in two dimensional conformal field theory. In both cases, the relevant stress tensor component must be averaged in time, and the results are sensitive to the form of the averaging function. Here we present results for two classes of such functions, which include the Gaussian and Lorentzian functions. The distribution for the energy flux is symmetric, unlike that for the energy density. In both cases, the distribution may possess an integrable singularity. The functional form of the flux distribution function involves a modified Bessel function, and is distinct from the shifted Gamma form for the energy density. By considering the joint distribution of energy flux and energy density, we show that the distribution of energy flux tends to be more centrally concentrated than that of the energy density. We also determine the distribution of energy fluxes, conditioned on the energy density being negative. Some applications of the results will be discussed.

Paper Structure

This paper contains 10 sections, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Vacuum Energy Density
  3. Vacuum Energy Flux
  4. The joint distribution of averaged flux and energy density
  5. Specific Sampling Functions
  6. Gaussian-like Functions
  7. Lorentzian-like Functions
  8. Compactly Supported Functions
  9. Summary
  10. Factorization of the joint moment generating function

Figures (4)

  • Figure 1: The probability distribution, $P_F(F)$, with Gaussian averaging is plotted as a function of the dimensionless flux, $F$.
  • Figure 2: The cumulative probability distribution, $P_<(F)$, with Gaussian averaging is plotted as a function of the dimensionless flux, $F$, on two different scales. These plots may be obtained by numerical integration of Eq. \ref{['eq:PF']} or numerical evaluation of the exact expression, Eq. \ref{['eq:PCE']}.
  • Figure 3: The probability distribution of flux, conditioned on the energy density being negative.
  • Figure 4: The cumulative probability distribution, $P_<(F)$, with Lorentzian averaging is plotted as a function of the dimensionless flux, $F$.