×

zbMATH — the first resource for mathematics

A large deviation perspective on ratio observables in reset processes: robustness of rate functions. (English) Zbl 1437.60016
Summary: We study large deviations of a ratio observable in discrete-time reset processes. The ratio takes the form of a current divided by the number of reset steps and as such it is not extensive in time. A large deviation rate function can be derived for this observable via contraction from the joint probability density function of current and number of reset steps. The ratio rate function is differentiable and we argue that its qualitative shape is ‘robust’, i.e. it is generic for reset processes regardless of whether they have short- or long-range correlations. We discuss similarities and differences with the rate function of the efficiency in stochastic thermodynamics.

MSC:
60F10 Large deviations
82B26 Phase transitions (general) in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
Software:
zoverw
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barato, A.; Chetrite, R.; Faggionato, A.; Gabrielli, D., A unifying picture of generalized thermodynamic uncertainty relations, J. Stat. Mech: Theory Exp., 2019, 8, 084017 (2019)
[2] Bénichou, O.; Loverdo, C.; Moreau, M.; Voituriez, R., Intermittent search strategies, Rev. Mod. Phys., 83, 1, 81 (2011)
[3] Bercu, B.; Richou, A., Large deviations for the Ornstein-Uhlenbeck process with shift, Adv. Appl. Prob., 47, 3, 880-901 (2015) · Zbl 1326.60031
[4] Bercu, B.; Rouault, A., Sharp large deviations for the Ornstein-Uhlenbeck process, Theory Prob. Appl., 46, 1, 1-19 (2002) · Zbl 1101.60320
[5] Bouchet, F.; Barre, J., Classification of phase transitions and ensemble inequivalence, in systems with long range interactions, J. Stat. Phys., 118, 5-6, 1073-1105 (2005) · Zbl 1070.82502
[6] Brockwell, PJ, The extinction time of a birth, death and catastrophe process and of a related diffusion model, Adv. Appl. Prob., 17, 1, 42-52 (1985) · Zbl 0551.92013
[7] Chavanis, PH, Phase transitions in self-gravitating systems: self-gravitating fermions and hard-sphere models, Phys. Rev. E, 65, 5, 056123 (2002)
[8] Cyranoski, D., Swimming against the tide, Nature, 408, 6814, 764-6 (2000)
[9] Danskin, JM, The theory of max-min, with applications, SIAM J. Appl. Math., 14, 4, 641-664 (1966) · Zbl 0144.43301
[10] Danskin, JM, The Theory of Max-Min and Its Application to Weapons Allocation Problems (2012), New York: Springer, New York
[11] Dembo, A.; Zeitouni, O., Large Deviations Techniques and Applications (2010), Berlin: Springer, Berlin · Zbl 1177.60035
[12] Den Hollander, F., Large Deviations (2008), Providence: American Mathematical Society, Providence
[13] Di Crescenzo, A.; Giorno, V.; Nobile, AG; Ricciardi, LM, On the M/M/1 queue with catastrophes and its continuous approximation, Queueing Syst., 43, 4, 329-347 (2003) · Zbl 1016.60080
[14] Di Terlizzi, I.; Baiesi, M., Kinetic uncertainty relation, J. Phys. A, 52, 2, 02LT03 (2018) · Zbl 1422.82023
[15] Ellis, RS, An overview of the theory of large deviations and applications to statistical mechanics, Scand. Actuarial J., 1995, 1, 97-142 (1995) · Zbl 0838.60027
[16] Evans, MR; Majumdar, SN, Diffusion with stochastic resetting, Phys. Rev. Lett., 106, 16, 160601 (2011)
[17] Feller, W., An introduction to probability theory and its applications (2008), New York: Wiley, New York · Zbl 0158.34902
[18] Gherardini, S.; Gupta, S.; Cataliotti, FS; Smerzi, A.; Caruso, F.; Ruffo, S., Stochastic quantum Zeno by large deviation theory, New J. Phys., 18, 1, 013048 (2016)
[19] Gingrich, TR; Rotskoff, GM; Vaikuntanathan, S.; Geissler, PL, Efficiency and large deviations in time-asymmetric stochastic heat engines, New J. Phys., 16, 10, 102003 (2014)
[20] Glynn, PW; Whitt, W., Large deviations behavior of counting processes and their inverses, Queueing Syst., 17, 1-2, 107-128 (1994) · Zbl 0805.60023
[21] Gradenigo, G.; Majumdar, SN, A first-order dynamical transition in the displacement distribution of a driven run-and-tumble particle, J. Stat. Mech. Theory Exp., 2019, 5, 053206 (2019)
[22] Grimmett, G.; Stirzaker, D., Probability and Random Processes (2001), Oxford: Oxford University Press, Oxford · Zbl 1015.60002
[23] Gross, D.: The microcanonical entropy is multiply differentiable. No dinosaurs in microcanonical gravitation: No special ‘microcanonical phase transitions’. arXiv:cond-mat/0403582 (2004)
[24] Gupta, D.; Sabhapandit, S., Stochastic efficiency of an isothermal work-to-work converter engine, Phys. Rev. E, 96, 4, 042130 (2017)
[25] Harris, RJ; Touchette, H., Phase transitions in large deviations of reset processes, J. Phys. A, 50, 10, 10LT01 (2017) · Zbl 1362.82040
[26] Hinkley, DV, On the ratio of two correlated normal random variables, Biometrika, 56, 3, 635-639 (1969) · Zbl 0183.48101
[27] Hinkley, DV, Correction: on the ratio of two correlated normal random variables, Biometrika, 57, 683 (1970)
[28] Hogan, W., Directional derivatives for extremal-value functions with applications to the completely convex case, Oper. Res., 21, 1, 188-209 (1973) · Zbl 0278.90062
[29] Hovhannisyan, V.; Ananikian, N.; Campa, A.; Ruffo, S., Complete analysis of ensemble inequivalence in the Blume-Emery-Griffiths model, Phys. Rev. E, 96, 6, 062103 (2017)
[30] Jack, RL, Large deviations in models of growing clusters with symmetry-breaking transitions, Phys. Rev. E, 100, 1, 012140 (2019)
[31] Jülicher, F.; Ajdari, A.; Prost, J., Modeling molecular motors, Rev. Mod. Phys., 69, 4, 1269 (1997)
[32] Kitamura, K.; Tokunaga, M.; Iwane, AH; Yanagida, T., A single myosin head moves along an actin filament with regular steps of 5.3 nanometres, Nature, 397, 6715, 129 (1999)
[33] Kyriakidis, E., Stationary probabilities for a simple immigration-birth-death process under the influence of total catastrophes, Stat. Prob. Lett., 20, 3, 239-240 (1994) · Zbl 0801.60073
[34] Lifson, S., Partition functions of linear-chain molecules, J. Chem. Phys., 40, 12, 3705-3710 (1964)
[35] Lo, AW, The statistics of Sharpe ratios, Financ. Anal. J., 58, 4, 36-52 (2002)
[36] Mahmoud, H., Pólya Urn Models (2008), Boca Raton: Chapman and Hall/CRC, Boca Raton
[37] Marsaglia, G., Ratios of normal variables and ratios of sums of uniform variables, J. Am. Stat. Assoc., 60, 309, 193-204 (1965) · Zbl 0126.35302
[38] Marsaglia, G., Ratios of normal variables, J. Stat. Softw., 16, 4, 1-10 (2006)
[39] Martínez, IA; Roldán, É.; Dinis, L.; Petrov, D.; Parrondo, JM; Rica, RA, Brownian Carnot engine, Nat. Phys., 12, 1, 67-70 (2016)
[40] Mehl, J.; Speck, T.; Seifert, U., Large deviation function for entropy production in driven one-dimensional systems, Phys. Rev. E, 78, 1, 011123 (2008)
[41] Merrikh-Bayat, F.: Two methods for numerical inversion of the \(z\)-transform. arXiv:1409.1727 (2014)
[42] Meylahn, J.M.: Biofilament interacting with molecular motors. Ph.D. thesis, Stellenbosch University (2015)
[43] Meylahn, JM; Sabhapandit, S.; Touchette, H., Large deviations for Markov processes with resetting, Phys. Rev. E, 92, 6, 062148 (2015)
[44] Mukherjee, B.; Sengupta, K.; Majumdar, SN, Quantum dynamics with stochastic reset, Phys. Rev. B, 98, 10, 104309 (2018)
[45] Nickelsen, D.; Touchette, H., Anomalous scaling of dynamical large deviations, Phys. Rev. Lett., 121, 9, 090602 (2018)
[46] Nyawo, PT; Touchette, H., Large deviations of the current for driven periodic diffusions, Phys. Rev. E, 94, 3, 032101 (2016)
[47] Poland, D.; Scheraga, HA, Occurrence of a phase transition in nucleic acid models, J. Chem. Phys., 45, 5, 1464-1469 (1966)
[48] Polettini, M.; Verley, G.; Esposito, M., Efficiency statistics at all times: Carnot limit at finite power, Phys. Rev. Lett., 114, 5, 050601 (2015)
[49] Proesmans, K.; Cleuren, B.; Van den Broeck, C., Stochastic efficiency for effusion as a thermal engine, Europhys. Lett., 109, 2, 20004 (2015)
[50] Proesmans, K.; Derrida, B., Large-deviation theory for a Brownian particle on a ring: a WKB approach, J. Stat. Mech. Theory Exp., 2019, 2, 023201 (2019)
[51] Richard, C.; Guttmann, AJ, Poland-Scheraga models and the DNA denaturation transition, J. Stat. Phys., 115, 3-4, 925-947 (2004) · Zbl 1052.92501
[52] Rose, DC; Touchette, H.; Lesanovsky, I.; Garrahan, JP, Spectral properties of simple classical and quantum reset processes, Phys. Rev. E, 98, 2, 022129 (2018)
[53] Shreshtha, M.; Harris, RJ, Thermodynamic uncertainty for run-and-tumble-type processes, Europhys. Lett., 126, 4, 40007 (2019)
[54] Touchette, H.: Legendre-Fenchel transforms in a nutshell. http://www.maths.qmul.ac.uk/ ht/archive/lfth2.pdf (2005)
[55] Touchette, H., Simple spin models with non-concave entropies, Am. J. Phys., 76, 1, 26-30 (2008)
[56] Touchette, H., The large deviation approach to statistical mechanics, Phys. Rep., 478, 1-3, 1-69 (2009)
[57] Verley, G.; Esposito, M.; Willaert, T.; Van den Broeck, C., The unlikely Carnot efficiency. Nature, Communications, 5, 4721 (2014)
[58] Verley, G.; Willaert, T.; Van den Broeck, C.; Esposito, M., Universal theory of efficiency fluctuations, Phys. Rev. E, 90, 5, 052145 (2014)
[59] Vroylandt, H., Esposito, M., Verley, G.: Efficiency fluctuations of stochastic machines undergoing a phase transition. arXiv:1912.06528 (2019) · Zbl 1412.82024
[60] Zamparo, M.: Large deviations in discrete-time renewal theory. arXiv:1903.03527 (2019)
[61] Zamparo, M.: Large deviations in renewal models of statistical mechanics. arXiv:1904.04602 (2019)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.