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Using precision coefficients of recurrence times and integrated currents to construct a lower bound for the average dissipation rate

Alberto Garilli* and Diego Frezzato

  • *Contact author: alberto.garilli@unipd.it
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Phys. Rev. E 113, 054127 – Published 18 May, 2026

DOI: https://doi.org/10.1103/gycp-9xg4

Abstract

For continuous-time Markov jump processes on irreducible networks with time-independent rate constants, we employ a transition-based formalism to express the long-time precision of a single integrated current over an observable channel in terms of precisions of the recurrence times of the forward and backward jumps, and of an effective affinity that captures the thermodynamic driving on that channel. This leads to a general inequality that, similarly to the well-known thermodynamic uncertainty relation (TUR), links the stationary entropy production rate with the fluctuations of an integrated current, but also incorporates the statistics of the forward and backward recurrence times. Such inequality can be saturated in less-restrictive conditions than the TUR, and potentially offers new opportunities for the optimization and design of biological and chemical out-of-equilibrium systems at the nanoscale.

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References (46)

  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. P. Pietzonka, F. Ritort, and U. Seifert, Finite-time generalization of the thermodynamic uncertainty relation, Phys. Rev. E 96, 012101 (2017).
  4. J. M. Horowitz and T. R. Gingrich, Proof of the finite-time thermodynamic uncertainty relation for steady-state currents, Phys. Rev. E 96, 020103(R) (2017).
  5. K. Liu and J. Gu, Dynamical activity universally bounds precision of response in Markovian nonequilibrium systems, Commun. Phys. 8, 62 (2025).
  6. Y. Song and C. Hyeon, Thermodynamic uncertainty relation to assess biological processes, J. Chem. Phys. 154, 130901 (2021).
  7. J. D. Mallory, O. A. Igoshin, and A. B. Kolomeisky, Do we understand the mechanisms used by biological systems to correct their errors? J. Phys. Chem. B 124, 9289 (2020).
  8. P. Pietzonka, A. C. Barato, and U. Seifert, Universal bound on the efficiency of molecular motors, J. Stat. Mech. (2016) 124004.
  9. W. Hwang and C. Hyeon, Energetic costs, precision, and transport efficiency of molecular motors, J. Phys. Chem. Lett. 9, 513 (2018).
  10. A. Dechant, Multidimensional thermodynamic uncertainty relations, J. Phys. A: Math. Theor. 52, 035001 (2019).
  11. G. Falasco and M. Esposito, Dissipation-time uncertainty relation, Phys. Rev. Lett. 125, 120604 (2020).
  12. A. Pal, S. Reuveni, and S. Rahav, Thermodynamic uncertainty relation for first-passage times on Markov chains, Phys. Rev. Res. 3, L032034 (2021).
  13. L. Oberreiter, U. Seifert, and A. C. Barato, Universal minimal cost of coherent biochemical oscillations, Phys. Rev. E 106, 014106 (2022).
  14. R. Marsland III, W. Cui, and J. M. Horowitz, The thermodynamic uncertainty relation in biochemical oscillations, J. R. Soc. Interface 16, 20190098 (2019).
  15. T. Kamijima, K. Funo, and T. Sagawa, Finite-time thermodynamic bounds and trade-off relations for information processing, Phys. Rev. Res. 7, 013329 (2025).
  16. J. van der Meer, B. Ertel, and U. Seifert, Thermodynamic inference in partially accessible Markov networks: A unifying perspective from transition-based waiting time distributions, Phys. Rev. X 12, 031025 (2022).
  17. P. E. Harunari, A. Garilli, and M. Polettini, Beat of a current, Phys. Rev. E 107, L042105 (2023).
  18. A. Garilli, P. E. Harunari, and M. Polettini, Fluctuation relations for a few observable currents at their own beat, J. Phys. A: Math. Theor. 57, 455003 (2024).
  19. P. Pietzonka and F. Coghi, Thermodynamic cost for precision of general counting observables, Phys. Rev. E 109, 064128 (2024).
  20. P. E. Harunari, A. Dutta, M. Polettini, and E. Roldán, What to learn from a few visible transitions' statistics? Phys. Rev. X 12, 041026 (2022).
  21. A. Garilli and D. Frezzato, Interrelation between precisions on integrated currents and on recurrence times in Markov jump processes, Phys. Rev. E 112, 044141 (2025).
  22. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  23. As explained in the main text, S is obtained from the rate matrix R by setting the elements (α,β) and (β,α) to zero, without modifying the diagonal entries. Therefore, throughout the paper, we indicate with the symbol ∖ the removal of off-diagonal elements of a rate matrix associated to observable transitions. This should not be confused with stalling matrices [30, 31], where the rates kαβ and kβα are also removed from the diagonal of R.
  24. Let us introduce the “flux matrix” W=RΠ, with R the rate matrix and Π=diag(pss). We have that [M]ij=−Fji for i≠j, and [W]ii=∑n≠iFin. Thus, W depends only on the full set of probability fluxes. Then, let S̃=SΠ, from which we get S−1=ΠS̃−1. The matrix S̃ is the analogue of a survival flux matrix built with W, hence S̃−1 depends only on the full set of fluxes. By using Eq.  (7) we get p(ℓ|ℓ′)=kℓ∑i[W]s(ℓ)i[S̃−1]it(ℓ′)=kℓps(ℓ)ss[S̃−1]s(ℓ)t(ℓ′)=Fℓ[S̃−1]s(ℓ)t(ℓ′), which reveals that p(ℓ|ℓ′) depends only on the set of fluxes.
  25. M. J. Schnitzer and S. Block, Statistical kinetics of processive enzymes, in Cold Spring Harbor Symposia on Quantitative Biology (Cold Spring Harbor Laboratory Press, New York, 1995), Vol. 60, pp. 793–802.
  26. J. R. Moffitt and C. Bustamante, Extracting signal from noise: Kinetic mechanisms from a Michaelis–Menten-like expression for enzymatic fluctuations, FEBS J. 281, 498 (2014).
  27. J. R. Moffitt, Y. R. Chemla, and C. Bustamante, Methods in statistical kinetics, Methods Enzymol. 475, 221 (2010).
  28. H. Wierenga, P. R. ten Wolde, and N. B. Becker, Quantifying fluctuations in reversible enzymatic cycles and clocks, Phys. Rev. E 97, 042404 (2018).
  29. From the randomly generated networks used for Fig.  2, it is possible to find some examples where the bound Eq.  (16) is close to saturation despite not being unicyclic. For instance, let us enumerate the sites clockwise starting from the right-bottom one, and assign rates k12=2,k21=78,k13=40,k31=0.25,k14=4,k41=0.4,k23=0.25,k32=47,k34=2, and k43=40. This network is not unicyclic and is found to be far from equilibrium (σss∼230). When observed along the channel α↔β, with α=2 and β=1, it provides (σss−σℓ)/σss∼10−4, so that it can be considered very close to saturation. For a comparison, the same set of rate constant provides the estimate (σss−σTUR)/σss∼0.62 when using the asymptotic TUR.
  30. M. Polettini and M. Esposito, Effective thermodynamics for a marginal observer, Phys. Rev. Lett. 119, 240601 (2017).
  31. M. Polettini and M. Esposito, Effective fluctuation and response theory, J. Stat. Phys. 176, 94 (2019).
  32. A. Raghu and I. Neri, Effective affinity for generic currents in Markov processes, J. Stat. Phys. 192, 50 (2025).
  33. For the very special case where ΔPτ=0 and |JT∞|>1, the bound Eq.  (16) is always tighter than the TUR, since 2Jcoth−1(JT∞)>2/T∞.
  34. R. Yasuda, H. Noji, K. Kinosita, Jr., and M. Yoshida, F1−ATPase is a highly efficient molecular motor that rotates with discrete 120∘ steps, Cell 93, 1117 (1998).
  35. T. Watanabe-Nakayama, S. Toyabe, S. Kudo, S. Sugiyama, M. Yoshida, and E. Muneyuki, Effect of external torque on the ATP-driven rotation of F1−ATPase, Biochem. Biophys. Res. Commun. 366, 951 (2008).
  36. K. Hayashi, H. Ueno, R. Iino, and H. Noji, Fluctuation theorem applied to F1−ATPase, Phys. Rev. Lett. 104, 218103 (2010).
  37. T. Mishima, D. Gupta, Y. Nakayama, W. C. Wareham, T. Ohyama, D. A. Sivak, and S. Toyabe, Efficiently driving F1 molecular motor in experiment by suppressing nonequilibrium variation, Phys. Rev. Lett. 135, 148402 (2025).
  38. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Direct observation of kinesin stepping by optical trapping interferometry, Nature (London) 365, 721 (1993).
  39. C. M. Coppin, D. W. Pierce, L. Hsu, and R. D. Vale, The load dependence of kinesin's mechanical cycle, Proc. Natl. Acad. Sci. USA 94, 8539 (1997).
  40. M. Nishiyama, H. Higuchi, and T. Yanagida, Chemomechanical coupling of the forward and backward steps of single kinesin molecules, Nat. Cell Biol. 4, 790 (2002).
  41. N. J. Carter and R. Cross, Mechanics of the kinesin step, Nature (London) 435, 308 (2005).
  42. J. O. Wirth, L. Scheiderer, T. Engelhardt, J. Engelhardt, J. Matthias, and S. W. Hell, MINFLUX dissects the unimpeded walking of kinesin-1, Science 379, 1004 (2023).
  43. T. Deguchi, M. K. Iwanski, E.-M. Schentarra, C. Heidebrecht, L. Schmidt, J. Heck, T. Weihs, S. Schnorrenberg, P. Hoess, S. Liu, et al., Direct observation of motor protein stepping in living cells using MINFLUX, Science 379, 1010 (2023).
  44. S. Liepelt and R. Lipowsky, Kinesin's network of chemomechanical motor cycles, Phys. Rev. Lett. 98, 258102 (2007).
  45. A. B. Kolomeisky and M. E. Fisher, Molecular motors: A theorist's perspective, Annu. Rev. Phys. Chem. 58, 675 (2007).
  46. D. Frezzato, Stationary Markov jump processes in terms of average transition times: Setup and some inequalities of kinetic and thermodynamic kind, J. Phys. A: Math. Theor. 53, 365003 (2020).

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Phys. Rev. E 113, 054127– Published 18 May, 2026
  • Received 1 December 2025
  • Accepted 28 April 2026
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