Abstract
In this brief note, we demonstrate a generalised energy equipartition theorem for a generic electrical circuit with Johnson–Nyquist (thermal) noise. From quantum mechanical considerations, the thermal modes have an energy distribution dictated by Planck’s law. For a resistive circuit with some inductance, it is shown that the real part of the admittance is proportional to a probability distribution function which modulates the contributions to the system’s mean energy from various frequencies of the Fourier spectrum. Further, we analyse the case with a capacitor connected in series with an inductor and a resistor. The results resemble superstatistics, i.e. a superposition of two statistics and can be reformulated in the energy representation. The correct classical limit is obtained as \(\hbar \rightarrow 0\).
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Acknowledgements
The author would like to thank Jasleen Kaur for several discussions, comments on the manuscript and for help in preparing the plots. The author also acknowledges Malay Bandyopadhyay for several discussions. This work is supported by Ministry of Education (MoE), Government of India in the form of Prime Minister’s Research Fellowship (ID: 1200454).
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Ghosh, A. Generalised energy equipartition in electrical circuits. Pramana - J Phys 97, 82 (2023). https://doi.org/10.1007/s12043-023-02553-w
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DOI: https://doi.org/10.1007/s12043-023-02553-w