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Frenesy (physics)

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Frenesy is a concept in statistical physics that measures the dynamical activity or "business" of a system's microscopic trajectories, especially under nonequilibrium conditions.[1] It complements the notion of entropy production, which measures time-antisymmetric aspects associated with irreversibility. Frenesy captures the dynamical activity or (unoriented) traffic of a system, reflecting how frequently states are visited or how many transitions occur over time. It relates to reactivities, escape rates and residence times of a physical state.

Origin and Context

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The notion of frenesy was introduced in 2006 in the study of nonequilibrium processes by Christian Maes and collaborators, and it has appeared in some works since then.[2][3][4][5][6] In systems described by trajectory ensembles or path-space measures (e.g. originating in Markov processes or Langevin dynamics), frenesy is associated with the time-symmetric part of the action functional, containing trajectory-dependent terms such as escape rates, undirected traffic and the total number of configuration changes. As with many other physical observables, it is the change in frenesy that makes the relevant quantity.

The role of dynamical activity in trajectory ensembles was explored in the study of large deviations.[7][8] The specific need for dealing with the time-symmetric fluctuation sector was explained in an early influential paper.[9] For some time, it was discussed under the name of "traffic", for example, in several studies on macroscopic fluctuations.[10][11][12] A year later, in the context of response theory, the term "frenetic" appeared.[13]

Mathematically, in a stochastic trajectory under local detailed balance, entropy production is tied to the asymmetry between forward and time-reversed paths, whereas frenesy quantifies the symmetric part invariant under time reversal. As such, it measures changes in dynamical activity or quiescence depending on the reference process and on the level of description.

Role in Fluctuation-Response Relations

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Frenesy is used in the generalization of fluctuation-dissipation relations beyond equilibrium. In nonequilibrium steady states, the linear response of an observable depends not only on the correlation with entropy production but also on correlations with frenesy. This correction has been proposed to describe response phenomena when systems are driven far from equilibrium.

As an extension of Kubo and Green-Kubo formulas, nonequilibrium linear response theory allows the response to be decomposed into an "entropic" term and a "frenetic" term. The frenetic component is absent in equilibrium but becomes significant under external driving forces. This is evident in nonequilibrium modifications of the Sutherland-Einstein relation, where mobility is no longer determined solely by the diffusion matrix of the unperturbed system but also includes force–current correlations.[14] The frenetic contribution can lead to negative responses, such as for differential mobility or non-equilibrium specific heats. This phenomenon—often described as "pushing more for getting less"[15]—is supported by similar theoretical considerations.[16][17] Frenetic effects also appear in second-order and higher-order nonlinear response expansions around equilibrium.[18][19]

A frenetic contribution also appears in corrections to the fluctuation-dissipation relation of the second kind, known as the Einstein relation. The (linear) friction has an entropic and frenetic part, where the entropic part connects with the noise in the usual equilibrium way. The frenetic part may be negative and dominating to the extent of rendering the friction negative.[20][21]

Applications

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The concept of frenesy is used in various areas of modern statistical physics. In the presence of dissipation, kinetic aspects in the form of increased or decreased dynamical activity and reactivities can determine a system's behavior. For example, jamming, localization, or glassy behavior are induced by dynamical heterogeneities where traps become important under driving or relaxation. Relaxation behavior is indeed another instance where kinetic aspects matter and where frenesy influences and shapes the landscape of possible pathways.[22] Kinetic phase transitions are governed by large deviations in frenesy that signal transitions between dynamical phases. In active matter, persistent motion is triggered by switches (discrete or continuous) introducing high frenetic activity.

Other applications concern selection and steering.[23] Kinetic proofreading and biological error correction are examples. The presence of driving allows changes in parameters governing dynamical activity to promote certain conditions of occupation and current. When those parameters depend on and receive feedback about the actual state, the system may evolve into a different phase or develop a dynamical pattern, as witnessed in active matter. As frenesy captures kinetic fluctuations and dynamical activity, its potential role has also been assessed in explaining nonequilibrium processes within the climate system.[24]

The concept of time-symmetric dynamical activity in nonequilibrium statistical mechanics has also been explored in the study of dynamical fluctuation symmetries. It deviates from stochastic thermodynamics by stressing kinetic aspects.

See also

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References

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  1. ^ Maes, Christian (2016). "Frenesy: Time-symmetric dynamical activity in nonequilibria". Physical Review E. 94 (3): 032108. arXiv:1605.06130. doi:10.1103/PhysRevE.94.032108. PMID 27739726.
  2. ^ Roldán, Édgar; Vivo, Pierpaolo (2019). "Exact Distributions of Currents and Frenesy for Markov Bridges". Physical Review E. 100 (4): 042108. doi:10.1103/PhysRevE.100.042108.
  3. ^ Prech, Kacper; Landi, Gabriel T.; Meier, Florian; Nurgalieva, Nuriya; Potts, Patrick P.; Silva, Ralph; Mitchison, Mark T. (2024). "Optimal time estimation and the clock uncertainty relation for stochastic processes". arXiv:2406.19450v1 [cond-mat.stat-mech].
  4. ^ Monnai, Takaaki (December 2024). "Kinetic equality for susceptibility and dynamical activity". Physical Review E. 110 (6). American Physical Society: L062101. doi:10.1103/PhysRevE.110.L062101.
  5. ^ Mihashi, Ryo; Hasegawa, Yoshihiko (2022). "Uncertainty relation for indirect measurement". arXiv:2208.03341v1 [quant-ph].
  6. ^ Gaspard, Pierre (2022). The Statistical Mechanics of Irreversible Phenomena. Cambridge University Press. ISBN 9781108563055.
  7. ^ Garrahan, J. P.; Jack, R. L.; Lecomte, V.; Pitard, E.; Van Duijvendijk, K.; Van Wijland, F. (2007). "Dynamical First-Order Phase Transition in Kinetically Constrained Models of Glasses". Physical Review Letters. 98 (19): 195702. doi:10.1103/PhysRevLett.98.195702.
  8. ^ Garrahan, Juan P.; Jack, Robert L.; Lecomte, Vivien; Pitard, Estelle; Van Duijvendijk, Kristina; Van Wijland, Frédéric (2009). "First-order dynamical phase transition in models of glasses: An approach based on ensembles of histories". Journal of Physics A: Mathematical and Theoretical. 42 (7). doi:10.1088/1751-8113/42/7/075007.
  9. ^ Maes, Christian; Van Wieren, Maarten H. (2006). "Time-Symmetric Fluctuations in Nonequilibrium Systems". Physical Review Letters. 96 (24): 240601. doi:10.1103/PhysRevLett.96.240601.
  10. ^ Maes, Christian; Netočný, Karel; Wynants, Bram (2008). "On and beyond entropy production: The case of Markov jump processes". Markov Processes and Related Fields. 14: 445–464.
  11. ^ Maes, Christian; Netočný, Karel; Wynants, Bram (2008). "Steady state statistics of driven diffusions". Physica A: Statistical Mechanics and Its Applications. 387 (12): 2675–2689. doi:10.1016/j.physa.2008.01.097.
  12. ^ Maes, C.; Netočný, K. (2008). "Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states". Epl (Europhysics Letters). 82 (3): 30003. doi:10.1209/0295-5075/82/30003.
  13. ^ Baiesi, Marco; Maes, Christian; Wynants, Bram (2009). "Fluctuations and Response of Nonequilibrium States". Physical Review Letters. 103 (1): 010602. doi:10.1103/PhysRevLett.103.010602.
  14. ^ Baiesi, Marco; Maes, Christian; Wynants, Bram (2011). "The modified Sutherland–Einstein relation for diffusive non-equilibria". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 467 (2134): 2792–2809. doi:10.1098/rspa.2011.0046.
  15. ^ Zia, R. K. P.; Praestgaard, E. L.; Mouritsen, O. G. (2002). "Getting more from pushing less: Negative specific heat and conductivity in nonequilibrium steady states". American Journal of Physics. 70 (4): 384–392. doi:10.1119/1.1427088.
  16. ^ Baerts, Pieter; Basu, Urna; Maes, Christian; Safaverdi, Soghra (2013). "Frenetic origin of negative differential response". Physical Review E. 88 (5): 052109. doi:10.1103/PhysRevE.88.052109.
  17. ^ Bogers, Lander; Khodabandehlou, Faezeh; Maes, Christian (2025). "Negative specific heats: Where Clausius and Boltzmann entropies separate". arXiv:2503.15999 [cond-mat.stat-mech].
  18. ^ Basu, Urna; Krüger, Matthias; Lazarescu, Alexandre; Maes, Christian (2015). "Frenetic aspects of second order response". Physical Chemistry Chemical Physics. 17 (9): 6653–6666. doi:10.1039/C4CP04977B.
  19. ^ Müller, Fenna; Basu, Urna; Sollich, Peter; Krüger, Matthias (2020). "Coarse-grained second-order response theory". Physical Review Research. 2 (4): 043123. doi:10.1103/PhysRevResearch.2.043123.
  20. ^ Pei, Ji-Hui; Maes, Christian (2025). "Transfer of active motion from medium to probe via the induced friction and noise". arXiv:2504.03210 [cond-mat.stat-mech].
  21. ^ Pei, Ji-Hui; Maes, Christian (2025). "Induced friction on a probe moving in a nonequilibrium medium". Physical Review E. 111 (3): L032101. doi:10.1103/PhysRevE.111.L032101.
  22. ^ Maes, Christian (2017). "Frenetic Bounds on the Entropy Production". Physical Review Letters. 119 (16): 160601. doi:10.1103/PhysRevLett.119.160601.
  23. ^ Lefebvre, Bram; Maes, Christian (2024). "Frenetic steering: Nonequilibrium-enabled navigation". Chaos: An Interdisciplinary Journal of Nonlinear Science. 34 (6). arXiv:2309.09227. doi:10.1063/5.0177223. PMID 38848269.
  24. ^ Benincasa, Roberta; Weiss, Jeffrey B.; Du, Danni; Duane, Gregory S.; Pinardi, Nadia. "Non-Equilibrium Thermodynamics and Climate Predictability: Investigating Entropy Production and Frenesy". EGU General Assembly 2025, Vienna, Austria, 27 Apr–2 May 2025, EGU25-3266. doi:10.5194/egusphere-egu25-3266.

Further reading

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  • Maes, Christian (2020). "Frenesy: Time-symmetric dynamical activity in nonequilibria". Physics Reports. 850: 1–33. doi:10.1016/j.physrep.2020.01.002.
  • Baiesi, Marco; Maes, Christian; Wynants, Bram (2009). "Fluctuations and Response of Nonequilibrium States". Physical Review Letters. 103 (1): 010602. doi:10.1103/PhysRevLett.103.010602.
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