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 \(\hbar \rightarrow 0\).
This is a preview of subscription content, log in via an institution to check access.
References
W Schottky, Ann. d. Phys. 57, 541 (1918)
J B Johnson, Nature 119, 50 (1927)
J B Johnson, Phys. Rev. 32, 97 (1928)
H Nyquist, Phys. Rev. 32, 110 (1928)
J F Qu, S P Benz, H Rogalla, W L Tew, D R White and K L Zhou, Meas. Sci. Technol. 30, 112001 (2019)
P J Baxandall, Wireless World 74, 1397 (November 1968), 388–392 (December 1968), 454–459
E Milotti, 1/f noise: a pedagogical review, https://doi.org/10.48550/arXiv.physics/0204033
R H Dicke, Rev. Sci. Instrum. 17, 268 (1946)
A Einstein, Ann. d. Phys. 17, 549 (1905)
P Langevin, C R Acad. Sci. (Paris) 146, 530 (1908)
G E Uhlenbeck and L S Ornstein, Phys. Rev. 36, 823 (1930), Reprinted in Selected papers on noise and stochastic processes edited by N Wax (Dover Publications, New York, 2003)
D T Gillespie, Am. J. Phys. 64, 225 (1996)
N Wiener, Time series (MIT Press, Cambridge, Massachusetts, 1964) p. 42
C Chatfield, The analysis of time Series: An introduction, Fourth Edn (Chapman and Hall, London, 1989) pp. 94–95
R Kubo, Rep. Prog. Phys. 29, 255 (1966)
H B Callen and T A Welton, Phys. Rev. 83, 34 (1951)
J J Waterston and J W Strutt, Proc. R. Soc. London 5, 604 (1851)
P Bialas and J Łuczka, Entropy 20, 123 (2018)
J Spiechowicz, P Bialas and J Łuczka, Phys. Rev. A 98, 052107 (2018)
P Bialas, J Spiechowicz and J Łuczka, Sci. Rep. 8, 16080 (2018)
P Bialas, J Spiechowicz and J Łuczka, J. Phys. A: Math. Theor. 52, 15 (2019)
J Spiechowicz and J Łuczka, J. Stat. Mech. 2019, 064002 (2019)
J Łuczka, J. Stat. Phys. 179, 839 (2020)
J Spiechowicz and J Łuczka, Sci. Rep. 11, 4088 (2021)
J Kaur, A Ghosh and M Bandyopadhyay, Phys. Rev. E 104, 064112 (2021)
J Kaur, A Ghosh and M Bandyopadhyay, Physica A 599, 127466 (2022)
C Beck and E G D Cohen, Physica A 322, 267 (2003)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghosh, A. Generalised energy equipartition in electrical circuits. Pramana - J Phys 97, 82 (2023). https://doi.org/10.1007/s12043-023-02553-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-023-02553-w