Martingale theory for housekeeping heat
The housekeeping heat is the energy exchanged between a system and its environment in a nonequilibrium process that results from the violation of detailed balance.
We describe fluctuations of the housekeeping heat in mesoscopic systems using the theory of martingales, a mathematical framework widely used in probability theory and finance.
We show that the exponentiated housekeeping heat (in units of kT, with k the Boltzmann constant and T the temperature) of a Markovian nonequilibrium process under arbitrary time-dependent driving is a martingale process. From this result, we derive universal equalities and inequalities for the statistics of stopping-times and suprema of the housekeeping heat.
Integral fluctuation relation for entropy production at stopping times
I Neri, E Roldán, S Pigolotti, F Jülicher
A stopping time is the first time when a trajectory of a stochastic process satisfies a certain criterion, which does not anticipate future events. In this paper, we derive using martingale theory the integral fluctuation relation for the stochastic entropy production at stopping times of a non-equilibrium steady-state physical system. This fluctuation relation implies a law of thermodynamics at stopping times, which is similar to the second law and states that it is not possible to reduce entropy on average, even when stopping a stochastic process at a cleverly chosen moment. Applying the integral fluctuation relation to different examples of stopping times we derive universal equalities and inequalities for the fluctuations of entropy production. For example, we bound the probability of entropy decreases by deriving bounds on the statistics of negative records of entropy production and we derive generic bounds for splitting probabilities of entropy production. For continuous processes we derive equalities for fluctuation properties of entropy production, such as, splitting probabilities and the statistics of negative records of entropy production.
Universal First-Passage-Time Distribution of Non-Gaussian Currents
S. Singh, P. Menczel, D. S. Golubev, I. M. Khaymovich, J. T. Peltonen, C. Flindt, K. Saito, E. Roldan, and J. P. Pekola
We investigate the fluctuations of the time elapsed until the electric charge transferred through a conductor reaches a given threshold value. For this purpose, we measure the distribution of the first-passage times for the net number of electrons transferred between two metallic islands in Coulomb blockade regime. Our experimental results are in excellent agreement with numerical calculations based on a recent theory describing the exact first-passage-time distributions for any non-equilibrium stationary Markov process. We also derive a simple analytical approximation for the first-passage-time distribution, which takes into account the non-Gaussian statistics of the electron transport, and show that it describes the experimental distributions with high accuracy. This universal approximation describes a wide class of stochastic processes, and can be used beyond the context of mesoscopic charge transport. In addition, we verify experimentally a fluctuation relation between the first-passage-time distributions for positive and negative thresholds.
Exact distributions of currents and frenesy in 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.
Quantum martingale theory and entropy production
We employ martingale theory to describe fluctuations of entropy production for open quantum systems in nonequilbrium steady states. Using the formalism of quantum jump trajectories, we identify a decomposition of entropy production into an exponential martingale and a purely quantum term, both obeying integral fluctuation theorems. An important consequence of this approach is the derivation of a set of genuine universal results for stopping-time and infimum statistics of stochastic entropy production. Finally, we complement the general formalism with numerical simulations of a qubit system.
Arcsine Laws in Stochastic Thermodynamics
A. C. Barato*, É. Roldán*, I. A. Martínez and S. Pigolotti
Physical Review Letters 121, 090601 (2018)
(* equal contribution)
We demonstrate with theory and experiment that the fraction of time a thermodynamic current elapses above its average value follows the arcsine law, a prominent result obtained by Levy for independent random variables.
Stochastic currents with long streaks above or below their average are much more likely than those that spend similar fraction of times above and below their average.
We demonstrate this result with simulations of molecular motors, quantum dots and colloidal systems, and with experimental data of a Brownian Carnot engine.
Extreme reductions of entropy in an electronic double dot
We measure single-electron fluctuations in a nanoelectronic device, a double dot, sealed at very low temperature 50mK.
Statistics of millions of records in the double dot confirm that entropy production’s average negative record cannot be below minus the Boltzmann constant. Moreover the data reveals a new bound for the maximal heat absorption of a nanoscopic system from its environment.
Multiplex Decomposition of Non-Markovian Dynamics and the Hidden Layer Reconstruction Problem
L. Lacasa, I. P. Mariño, J. Miguez, V. Nicosia, É. Roldán, A. Lisica, S. W. Grill, and J. Gómez-Gardeñes
Physical Review X 8, 031038 (2018)
We show that by using local information provided by a random walker navigating the aggregated network, it is possible to determine, in a robust manner, whether these dynamics can be more accurately represented by a single layer or if they are better explained by a (hidden) multiplex structure. In the latter case, we also provide Bayesian methods to estimate the most probable number of hidden layers and the model parameters, thereby fully reconstructing its architecture. We apply our method to experimental recordings from (i) the mobility dynamics of human players in an online multiplayer game and (ii) the dynamics of RNA polymerases at the single-molecule level.
Arrow of Time in Active Fluctuations
É. Roldán, J. Barral, P. Martin, J.M.R. Parrondo and F. Jülicher
We introduce lower bounds for the rate of entropy production of an active stochastic process by quantifying the irreversibility of stochastic traces obtained from mesoscopic degrees of freedom. Our measures of irreversibility reveal signatures of time’s arrow and provide bounds for entropy production even in the case of active fluctuations that have no drift. We apply these irreversibility measures to experimental spontaneous hair-bundle oscillations from the ear of the bullfrog.
Testing Optimality of Sequential Decision-Making
M. Dörpinghaus, I. Neri, É. Roldán, H. Meyr and F. Jülicher
We introduce a statistical method to test whether a system that performs a binary sequential hypothesis test is optimal in the sense of minimizing the average decision times while taking decisions with given reliabilities. The proposed method requires samples of the decision times, the decision outcomes, and the true hypotheses, but does not require knowledge on the statistics of the observations or the properties of the decision-making system. We illustrate these tests with numerical simulations and discuss potential applications in machine learning and biology.
Generic properties of stochastic entropy production
S. Pigolotti, I. Neri, É. Roldán and F. Jülicher
Physical Review Letters 119 (14), 140604 (2017)
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Introducing a random-time transformation, entropy production obeys a one-dimensional Ito drift-diffusion equation, independent of the underlying physical model.
This transformation leads to an exact uncertainty equality between the Fano factor of entropy production and the Fano factor of the entropic time.
Colloidal heat engines: A review
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We review recent experimental constructions of small heat engines which are the colloidal equivalents to the macroscopic Stirling, Carnot and steam engines, whose fluctuations lead to unique phenomena that have no equivalent in the macroscopic world. We also discuss the work extraction from bacterial reservoirs.
Finally, we provide some guidance on how the work extracted from colloidal heat engines can be used to generate net particle or energy currents, proposing a new generation of experiments with colloidal systems.
Path-integral formalism for stochastic resetting:
Exactly solved examples and shortcuts to confinement
É. Roldán and S. Gupta
Physical Review E 96 (2), 022130 (2017)
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We present a path-integral approach to derive analytical expressions for a variety of statistics of the dynamics of overdamped Brownian particles under stochastic resetting.
We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential.
Statistics of Infima and Stopping Times of Entropy production and Applications to Active Molecular Processes
We study the statistics of infima, stopping times, and passage probabilities of entropy production in nonequilibrium steady states, and we show that they are universal. We consider two examples of stopping times: first-passage times of entropy production and waiting times of stochastic processes, which are the times when a system reaches a given state for the first time. Our main results are as follows: (i) The distribution of the global infimum of entropy production is exponential with mean equal to minus Boltzmann’s constant; (ii) we find exact expressions for the passage probabilities of entropy production; (iii) we derive a fluctuation theorem for stopping-time distributions of entropy production. These results have interesting implications for stochastic processes that can be discussed in simple colloidal systems and in active molecular processes. In particular, we show that the timing and statistics of discrete chemical transitions of molecular processes, such as the steps of molecular motors, are governed by the statistics of entropy production. We also show that the extreme-value statistics of active molecular processes are governed by entropy production; for example, we derive a relation between the maximal excursion of a molecular motor against the direction of an external force and the infimum of the corresponding entropy-production fluctuations. Using this relation, we make predictions for the distribution of the maximum backtrack depth of RNA polymerases, which follow from our universal results for entropy-production infima.
Brownian Carnot engine
The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors and some artificial micro-engines operate. As described by stochastic thermodynamics, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit. Here we report an experimental realization of a Carnot engine with a single optically trapped Brownian particle as the working substance. We present an exhaustive study of the energetics of the engine and analyse the fluctuations of the finite-time efficiency, showing that the Carnot bound can be surpassed for a small number of non-equilibrium cycles. As its macroscopic counterpart, the energetics of our Carnot device exhibits basic properties that one would expect to observe in any microscopic energy transducer operating with baths at different temperatures. Our results characterize the sources of irreversibility in the engine and the statistical properties of the efficiency—an insight that could inspire new strategies in the design of efficient nano-motors.
Mechanisms of backtrack recovery in RNA polymerases I and II
Transcription of the genetic information from DNA into RNA is the central process of gene expression, and it is performed by enzymes called RNA polymerases (Pol). Transcription is interspersed with a proofreading mechanism called backtracking, during which the polymerase moves backward on the DNA template and displaces the RNA 3′ end from its active site. Backtrack recovery can happen by diffusion of the enzyme along the DNA or cleavage of the backtracked RNA. Using single-molecule optical tweezers and stochastic theory, we quantified distinct diffusion and cleavage rates of Pol I and Pol II and described distinct backtrack recovery strategies of these essential enzymes.
Stochastic resetting in backtrack recovery by RNA polymerases
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Transcription is a key process in gene expression, in which RNA polymerases produce a complementary RNA copy from a DNA template. RNA polymerization is frequently interrupted by backtracking, a process in which polymerases perform a random walk along the DNA template. Recovery of polymerases from the transcriptionally inactive backtracked state is determined by a kinetic competition between one-dimensional diffusion and RNA cleavage. Here we describe backtrack recovery as a continuous-time random walk, where the time for a polymerase to recover from a backtrack of a given depth is described as a first-passage time of a random walker to reach an absorbing state. We represent RNA cleavage as a stochastic resetting process and derive exact expressions for the recovery time distributions and mean recovery times from a given initial backtrack depth for both continuous and discrete-lattice descriptions of the random walk. We show that recovery time statistics do not depend on the discreteness of the DNA lattice when the rate of one-dimensional diffusion is large compared to the rate of cleavage.