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For transportation system analysis in a new space dimension with respect to individual trips’ remaining distances, vehicle trips demand has two main components: the departure time and the trip distance. In particular, the trip distance distribution (TDD) is a direct input to the bathtub model in the new space dimension, and is a very important variable to consider in many applications, such as the development of distance-based congestion pricing strategies or mileage tax. For a good understanding of the demand pattern, both the distribution of trip initiation and trip distance should be calibrated from real data. In this paper, it is assumed that the demand pattern can be described by the joint distribution of trip distance and departure time. In other words, TDD is assumed to be time-dependent, and a calibration and validation methodology of the joint probability is proposed, based on log-likelihood maximization and the Kolmogorov–Smirnov test. The calibration method is applied to empirical for-hire vehicle trips in Chicago, and it is concluded that TDD varies more within a day than across weekdays. The hypothesis that TDD follows a negative exponential, log-normal, or Gamma distribution is rejected. However, the best fit is systematically observed for the time-dependent log-normal probability density function. In the future, other trip distributions should be considered and also non-parametric probability density estimation should be explored for a better understanding of the demand pattern.
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This content will become publicly available on April 9, 2026
Probability distribution for vacuum energy flux fluctuations in two spacetime dimensions
The probability distribution for vacuum fluctuations of the energy flux in two dimensions is constructed, along with the joint distribution of energy flux and energy density. Our approach is 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 are also discussed.
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- Award ID(s):
- 2207903
- PAR ID:
- 10616274
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review D
- Volume:
- 111
- Issue:
- 8
- ISSN:
- 2470-0010
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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