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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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Deep learning the equilibrium of multi-stage Bayesian games
We are the first to introduce the energy function into game theory literature and prove that the zero point of the energy function is a Nash equilibrium. 18 examples are tested using the algorithm.
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- Award ID(s):
- 2238838
- PAR ID:
- 10505941
- Publisher / Repository:
- INFORMS Annual Meeting 2023
- Date Published:
- Format(s):
- Medium: X
- Location:
- Phoenix, AZ
- Sponsoring Org:
- National Science Foundation
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