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  • Letter
  • Open Access

Role of charge in thermodynamic uncertainty relations

David Christian Ohnmacht1,*,†, Wolfgang Belzig1, and Juan Carlos Cuevas2,3

  • *Contact author: D.C.Ohnmacht@tudelft.nl
  • †Present address: QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands.
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Phys. Rev. B 114, L171401 – Published 8 September, 2026

DOI: https://doi.org/10.1103/hcg9-14zg

Abstract

We demonstrate that the charge value of transport mechanisms heavily impacts the validity of thermodynamic uncertainty relations (TURs). Specifically, we show within the framework of full counting statistics, that the recently established quantum TUR can be violated by the presence of transport processes that carry more than one charge, like Andreev reflection processes in normal metal-superconductor junctions. We propose a modified quantum TUR, which incorporates the charge value and demonstrate that this charge-dependent quantum TUR can only be violated if the highest charge transport process exceeds this charge value. In particular, we establish that the breaking of the quantum TUR solely originates from the charge value of the highest charge transport process. Namely, our analytical considerations do not invoke the existence of superconductivity, and these considerations generally hold for noninteracting electronic transport which can be described by the scattering formalism.

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References (40)

  1. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  2. T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  3. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  4. P. Pietzonka, A. C. Barato, and U. Seifert, Universal bounds on current fluctuations, Phys. Rev. E 93, 052145 (2016).
  5. P. Pietzonka, A. C. Barato, and U. Seifert, Universal bound on the efficiency of molecular motors, J. Stat. Mech. (2016) 124004.
  6. P. Pietzonka and U. Seifert, Universal trade-off between power, efficiency, and constancy in steady-state heat engines, Phys. Rev. Lett. 120, 190602 (2018).
  7. A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from exchange fluctuation theorems, Phys. Rev. Lett. 123, 090604 (2019).
  8. B. K. Agarwalla and D. Segal, Assessing the validity of the thermodynamic uncertainty relation in quantum systems, Phys. Rev. B 98, 155438 (2018).
  9. G. Manzano and R. López, Quantum-enhanced performance in superconducting Andreev reflection engines, Phys. Rev. Res. 5, 043041 (2023).
  10. D. Andrieux and P. Gaspard, A fluctuation theorem for currents and non-linear response coefficients, J. Stat. Mech. (2007) P02006.
  11. K. Brandner, T. Hanazato, and K. Saito, Thermodynamic bounds on precision in ballistic multiterminal transport, Phys. Rev. Lett. 120, 090601 (2018).
  12. E. Potanina, C. Flindt, M. Moskalets, and K. Brandner, Thermodynamic bounds on coherent transport in periodically driven conductors, Phys. Rev. X 11, 021013 (2021).
  13. R. López, J. S. Lim, and K. W. Kim, Optimal superconducting hybrid machine, Phys. Rev. Res. 5, 013038 (2023).
  14. K. Prech, P. Johansson, E. Nyholm, G. T. Landi, C. Verdozzi, P. Samuelsson, and P. P. Potts, Entanglement and thermokinetic uncertainty relations in coherent mesoscopic transport, Phys. Rev. Res. 5, 023155 (2023).
  15. S. Katz, J. L. Lebowitz, and H. Spohn, Phase transitions in stationary nonequilibrium states of model lattice systems, Phys. Rev. B 28, 1655 (1983).
  16. S. Kheradsoud, N. Dashti, M. Misiorny, P. P. Potts, J. Splettstoesser, and P. Samuelsson, Power, efficiency and fluctuations in a quantum point contact as steady-state thermoelectric heat engine, Entropy 21, 777 (2019).
  17. K. Ptaszyński, Coherence-enhanced constancy of a quantum thermoelectric generator, Phys. Rev. B 98, 085425 (2018).
  18. J. Liu and D. Segal, Thermodynamic uncertainty relation in quantum thermoelectric junctions, Phys. Rev. E 99, 062141 (2019).
  19. T. Ehrlich and G. Schaller, Broadband frequency filters with quantum dot chains, Phys. Rev. B 104, 045424 (2021).
  20. F. Taddei and R. Fazio, Thermodynamic uncertainty relations for systems with broken time reversal symmetry: The case of superconducting hybrid systems, Phys. Rev. B 108, 115422 (2023).
  21. P. Menczel, E. Loisa, K. Brandner, and C. Flindt, Thermodynamic uncertainty relations for coherently driven open quantum systems, J. Phys. A: Math. Theor. 54, 314002 (2021).
  22. H. M. Friedman, B. K. Agarwalla, O. Shein-Lumbroso, O. Tal, and D. Segal, Thermodynamic uncertainty relation in atomic-scale quantum conductors, Phys. Rev. B 101, 195423 (2020).
  23. D. C. Ohnmacht, J. C. Cuevas, W. Belzig, R. López, J. S. Lim, and K. W. Kim, Thermodynamic uncertainty relations in superconducting junctions, Phys. Rev. Res. 7, L012075 (2025).
  24. K. Brandner and K. Saito, Thermodynamic uncertainty relations for coherent transport, Phys. Rev. Lett. 135, 046302 (2025).
  25. M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett. 57, 1761 (1986).
  26. F. Mayo, N. Sobrino, R. Fazio, F. Taddei, and M. Governale, Quantum thermodynamic uncertainty relation and macroscopic superconducting coherence, Phys. Rev. Res. 8, 013242 (2026).
  27. L. S. Levitov and G. B. Lesovik, Charge distribution in quantum shot noise, Sov. Phys. JETP Lett. 58, 230 (1993).
  28. Y. V. Nazarov, Universalities of weak localization, arXiv:cond-mat/9908143.
  29. J. C. Cuevas and W. Belzig, Full counting statistics of multiple Andreev reflections, Phys. Rev. Lett. 91, 187001 (2003).
  30. J. C. Cuevas and W. Belzig, Dc transport in superconducting point contacts:   A full-counting-statistics view, Phys. Rev. B 70, 214512 (2004).
  31. D. C. Ohnmacht, W. Belzig, and J. C. Cuevas, Full counting statistics of Yu-Shiba-Rusinov bound states, Phys. Rev. Res. 5, 033176 (2023).
  32. A. Komnik and A. O. Gogolin, Full counting statistics for the Kondo dot, Phys. Rev. Lett. 94, 216601 (2005).
  33. T. L. Schmidt, A. O. Gogolin, and A. Komnik, Full counting statistics of spin transfer through a Kondo dot, Phys. Rev. B 75, 235105 (2007).
  34. N. Sobrino, F. Taddei, R. Fazio, and M. Governale, Current precision in interacting hybrid normal-superconducting systems, Phys. Rev. B 113, 195407 (2026).
  35. L. S. Levitov and M. Reznikov, Counting statistics of tunneling current, Phys. Rev. B 70, 115305 (2004).
  36. W. Belzig, Full counting statistics of super-Poissonian shot noise in multilevel quantum dots, Phys. Rev. B 71, 161301 (2005).
  37. J. Koch, F. von Oppen, and A. V. Andreev, Theory of the Franck-Condon blockade regime, Phys. Rev. B 74, 205438 (2006).
  38. M. Vanević, Y. V. Nazarov, and W. Belzig, Elementary events of electron transfer in a voltage-driven quantum point contact, Phys. Rev. Lett. 99, 076601 (2007).
  39. F. Xu, C. Holmqvist, and W. Belzig, Overbias light emission due to higher-order quantum noise in a tunnel junction, Phys. Rev. Lett. 113, 066801 (2014).
  40. P.-J. Peters, F. Xu, K. Kaasbjerg, G. Rastelli, W. Belzig, and R. Berndt, Quantum coherent multielectron processes in an atomic scale contact, Phys. Rev. Lett. 119, 066803 (2017).

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Phys. Rev. B 114, L171401– Published 8 September, 2026
  • Received 23 December 2025
  • Revised 11 March 2026
  • Accepted 13 July 2026
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