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Relativistic Stochastic Mechanics I: Langevin Equation from Observer’s Perspective

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Abstract

Two different versions of relativistic Langevin equation in curved spacetime background are constructed, both are manifestly general covariant. It is argued that, from the observer’s point of view, the version which takes the proper time of the Brownian particle as evolution parameter contains some conceptual issues, while the one which makes use of the proper time of the observer is more physically sound. The two versions of the relativistic Langevin equation are connected by a reparametrization scheme. In spite of the issues contained in the first version of the relativistic Langevin equation, it still permits to extract the physical probability distributions of the Brownian particles, as is shown by Monte Carlo simulation in the example case of Brownian motion in \((1+1)\)-dimensional Minkowski spacetime.

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Data Availability

All data used in this research was generated from Eqs. (28) and (29) by use of numeric programs written in C++ and python. The P-values given in Sect. 5 are calculated using Wolfram Language. All programs are available upon request.

Notes

  1. Since the space time is assumed to be endowed with a non-degenerate metric \(g_{\mu \nu }(x)\), we can identify the cotangent vector at any event with its dual tangent vector. Therefore, we are free to take the tangent space description instead of the cotangent space description in this work.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under the Grant No. 12275138.

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  1. School of Physics, Nankai University, Tianjin, 300071, China

    Yifan Cai, Tao Wang & Liu Zhao

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  1. Yifan Cai
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  2. Tao Wang
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  3. Liu Zhao
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Correspondence to Liu Zhao.

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Communicated by Jae Dong Noh.

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Cai, Y., Wang, T. & Zhao, L. Relativistic Stochastic Mechanics I: Langevin Equation from Observer’s Perspective. J Stat Phys 190, 193 (2023). https://doi.org/10.1007/s10955-023-03204-5

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