Exact distributions of currents and frenesy for Markov bridges.

We consider discrete-time Markov bridges, chains whose initial and final states coincide. We derive exact finite-time formulae for the joint probability distributions of additive functionals of trajectories. We apply our theory to time-integrated currents and frenesy of enzymatic reactions, which may include absolutely irreversible transitions. We discuss the information that frenesy carries about the currents and show that bridges may violate known uncertainty relations in certain cases. Numerical simulations are in perfect agreement with our theory.

[1]  Francesco Zerbetto,et al.  Synthetic molecular motors and mechanical machines. , 2007, Angewandte Chemie.

[2]  S. Majumdar,et al.  Area distribution and the average shape of a Lévy bridge , 2010, 1004.5046.

[3]  J. P. Garrahan,et al.  Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems. , 2018, Physical review letters.

[4]  H. Touchette.  The large deviation approach to statistical mechanics , 2008, 0804.0327.

[5]  Udo Seifert.  Entropy production along a stochastic trajectory and an integral fluctuation theorem. , 2005, Physical review letters.

[6]  A. Dechant,et al.  Current fluctuations and transport efficiency for general Langevin systems , 2017, Journal of Statistical Mechanics: Theory and Experiment.

[7]  J. Carrascosa,et al.  Mechano-chemical kinetics of DNA replication: identification of the translocation step of a replicative DNA polymerase , 2015, Nucleic acids research.

[8]  I M Sokolov,et al.  Random Search with Resetting: A Unified Renewal Approach. , 2018, Physical review letters.

[9]  Javier Villarroel,et al.  Continuous-time random walks with reset events , 2017, 1706.04812.

[10]  Arnab Pal,et al.  First Passage under Restart. , 2016, Physical review letters.

[11]  C. Pérez-Espigares,et al.  Sampling rare events across dynamical phase transitions. , 2019, Chaos.

[12]  C. Micheletti,et al.  Exact and Efficient Sampling of Conditioned Walks , 2018 .

[13]  J. P. Garrahan.  Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables. , 2017, Physical review. E.

[14]  U. Seifert,et al.  Stochastic thermodynamics of resetting , 2016, 1603.01141.

[15]  Y. Murashita,et al.  Gibbs Paradox Revisited from the Fluctuation Theorem with Absolute Irreversibility. , 2015, Physical review letters.

[16]  M. Esposito,et al.  Effective Fluctuation and Response Theory , 2018, Journal of Statistical Physics.

[17]  Uttam Bhat,et al.  Stochastic search with Poisson and deterministic resetting , 2016, 1605.08812.

[18]  R. Fazio,et al.  Quantum Martingale Theory and Entropy Production. , 2019, Physical review letters.

[19]  F. Jülicher,et al.  Exactly solvable dynamics of forced polymer loops , 2018, New Journal of Physics.

[20]  Shamik Gupta,et al.  Two Refreshing Views of Fluctuation Theorems Through Kinematics Elements and Exponential Martingale , 2010, 1009.0707.

[21]  M. Baiesi,et al.  Kinetic uncertainty relation , 2018, Journal of Physics A: Mathematical and Theoretical.

[22]  U. Seifert,et al.  A generalization of the thermodynamic uncertainty relation to periodically driven systems , 2018, Journal of Physics A: Mathematical and Theoretical.

[23]  Arnab K. Pal,et al.  Local time of diffusion with stochastic resetting , 2019, Journal of Physics A: Mathematical and Theoretical.

[24]  J. Horowitz,et al.  Hysteretic thermodynamic uncertainty relation for systems with broken time-reversal symmetry , 2019, Journal of Statistical Mechanics: Theory and Experiment.

[25]  Simone Pigolotti,et al.  Mapping of uncertainty relations between continuous and discrete time. , 2017, Physical review. E.

[26]  S. Ciliberto,et al.  Experiments in Stochastic Thermodynamics: Short History and Perspectives , 2017 .

[27]  Physikalische Gesellschaft in der Deutschen Demokratischen Republik.  Fortschritte der Physik = Progress of physics , 1953 .

[28]  Édgar Roldán,et al.  Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement. , 2017, Physical review. E.

[29]  Daniel Sánchez-Taltavull,et al.  Stochastic resetting in backtrack recovery by RNA polymerases. , 2016, Physical review. E.

[30]  K. Sekimoto,et al.  Progressive quenching: Globally coupled model. , 2017, Physical review. E.

[31]  S. Majumdar,et al.  Effective Langevin equations for constrained stochastic processes , 2015, 1503.02639.

[32]  Hugo Touchette,et al.  Nonequilibrium Markov Processes Conditioned on Large Deviations , 2014, 1405.5157.

[33]  Udo Seifert,et al.  Thermodynamic uncertainty relation for biomolecular processes. , 2015, Physical review letters.

[34]  Todd R. Gingrich,et al.  Proof of the finite-time thermodynamic uncertainty relation for steady-state currents. , 2017, Physical review. E.

[35]  J. L. Doob,et al.  Conditional brownian motion and the boundary limits of harmonic functions , 1957 .

[36]  Satya N Majumdar,et al.  Diffusion with stochastic resetting. , 2011, Physical review letters.

[37]  C. Jarzynski.  Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale , 2011 .

[38]  H. Qian,et al.  Stochastic theory of nonequilibrium steady states. Part II: Applications in chemical biophysics , 2012 .

[39]  C. Maes,et al.  Statistical forces from close-to-equilibrium media , 2015, 1505.04259.

[40]  Carlos Bustamante,et al.  Exact solutions for kinetic models of macromolecular dynamics. , 2008, The journal of physical chemistry. B.

[41]  C. Maes.  Frenetic Bounds on the Entropy Production. , 2017, Physical review letters.

[42]  C. Maes.  Non-Dissipative Effects in Nonequilibrium Systems , 2016, 1603.05147.

[43]  F. Ritort,et al.  Finite-time generalization of the thermodynamic uncertainty relation. , 2017, Physical review. E.

[44]  S. Wadia.  BOOK REVIEW: The Legacy of Albert Einstein: A Collection of Essays in Celebration of the Year of Physics , 2006 .

[45]  J. Doob.  What is a Martingale , 1971 .

[46]  Christoph F. Schmidt,et al.  Direct observation of kinesin stepping by optical trapping interferometry , 1993, Nature.

[47]  P. Hurtado,et al.  Structure of the optimal path to a fluctuation. , 2016, Physical review. E.

[48]  Jorge Kurchan,et al.  Direct evaluation of large-deviation functions. , 2005, Physical review letters.

[49]  A. C. Barato,et al.  Dispersion of the time spent in a state: general expression for unicyclic model and dissipation-less precision , 2017, 1703.04386.

[50]  A. Mazzolo.  Constrained Brownian processes and constrained Brownian bridges , 2017 .

[51]  Full counting statistics of interacting electrons , 2006, cond-mat/0605263.

[52]  Todd R. Gingrich,et al.  Dissipation Bounds All Steady-State Current Fluctuations. , 2015, Physical review letters.

[53]  R. J. Harris,et al.  Thermodynamic uncertainty for run-and-tumble–type processes , 2019, EPL (Europhysics Letters).

[54]  U. Seifert.  Stochastic thermodynamics, fluctuation theorems and molecular machines , 2012, Reports on progress in physics. Physical Society.

[55]  C. Bustamante,et al.  Complete dissection of transcription elongation reveals slow translocation of RNA polymerase II in a linear ratchet mechanism , 2013, eLife.

[56]  Christian Maes,et al.  Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions , 2009, 0909.5306.

[57]  I. Z. Steinberg,et al.  On the time reversal of noise signals. , 1986, Biophysical journal.

[58]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[59]  Deepak Gupta,et al.  Stochastic resetting in underdamped Brownian motion , 2018, Journal of Statistical Mechanics: Theory and Experiment.

[60]  J. Parrondo,et al.  Intermittent transcription dynamics for the rapid production of long transcripts of high fidelity. , 2013, Cell reports.

[61]  Jaume Masoliver,et al.  Telegraphic processes with stochastic resetting. , 2019, Physical review. E.

[62]  Stephen M. Krone,et al.  Analyzing animal movements using Brownian bridges. , 2007, Ecology.

[63]  N. Straumann.  The Legacy of Albert Einstein: A Collection of Essays in Celebration of the Year of Physics , 2007 .

[64]  G. Parisi.  Brownian motion , 2005, Nature.

[65]  A. Nandi,et al.  First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate. , 2019, Physical review. E.

[66]  Jordan M. Horowitz,et al.  Quantifying dissipation using fluctuating currents , 2019, Nature Communications.

[67]  Frank Julicher,et al.  Statistics of Infima and Stopping Times of Entropy Production and Applications to Active Molecular Processes , 2016, 1604.04159.

[68]  Y. Nazarov,et al.  Full counting statistics of charge transfer in Coulomb blockade systems , 2002, cond-mat/0207624.

[69]  W. Marsden.  I and J , 2012 .

[70]  K. Funo,et al.  Speed Limit for Classical Stochastic Processes. , 2018, Physical review letters.

[71]  Veikko F. Geyer,et al.  Broken detailed balance at mesoscopic scales in active biological systems , 2016, Science.

[72]  Udo Seifert,et al.  Universal bounds on current fluctuations. , 2015, Physical review. E.

[73]  K. Funo,et al.  Nonequilibrium equalities in absolutely irreversible processes. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[74]  Massimiliano Esposito,et al.  Ensemble and trajectory thermodynamics: A brief introduction , 2014, 1403.1777.

[75]  Shamik Gupta,et al.  Martingale theory for housekeeping heat , 2018, EPL (Europhysics Letters).

[76]  C. Maes,et al.  Nonequilibrium Response and Frenesy , 2015, 1507.01228.

[77]  J. Lebowitz,et al.  A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics , 1998, cond-mat/9811220.

[78]  A. Auffèves,et al.  An autonomous quantum machine to measure the thermodynamic arrow of time , 2018, npj Quantum Information.

[79]  N. H. Bingham,et al.  Seminar on Stochastic Processes , 1993 .

[80]  Édgar Roldán,et al.  Colloidal heat engines: a review. , 2017, Soft Matter.

[81]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[82]  J. Szavits-Nossan,et al.  Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges , 2015, 1508.04969.

[83]  Unifying approach for fluctuation theorems from joint probability distributions. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[84]  P. Krapivsky,et al.  Joint distributions of partial and global maxima of a Brownian bridge , 2016, 1608.02161.

[85]  M. Mézard,et al.  Journal of Statistical Mechanics: Theory and Experiment , 2011 .

[86]  M. Qian,et al.  Cycle symmetries and circulation fluctuations for discrete-time and continuous-time Markov chains , 2014, 1407.1263.

[87]  C. Broeck,et al.  Discrete-time thermodynamic uncertainty relation , 2017, 1708.07032.