Kinetic Equation for Stochastic Vector Bundles
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929315974742016 |
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| author | Zhong, De-yu Wang, Guang-qian |
| author_facet | Zhong, De-yu Wang, Guang-qian |
| contents | The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker-Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker-Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_16754 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Kinetic Equation for Stochastic Vector Bundles Zhong, De-yu Wang, Guang-qian Fluid Dynamics Mathematical Physics The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker-Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker-Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention. |
| title | Kinetic Equation for Stochastic Vector Bundles |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2211.16754 |