### Nonreciprocal forces enable cold-to-hot heat transfer between nanoparticles

**É. Roldán**

Sci. Rep. **13**, 4517 (2023) [PDF]

We study the heat transfer between two nanoparticles held at different temperatures that interact through nonreciprocal forces, by combining molecular dynamics simulations with stochastic thermodynamics. Our simulations reveal that it is possible to construct nano refrigerators that generate a net heat transfer from a cold to a hot reservoir at the expense of power exerted by the nonreciprocal forces. Applying concepts from stochastic thermodynamics to a minimal underdamped Langevin model, we derive exact analytical expressions predictions for the fluctuations of work, heat, and efficiency, which reproduce thermodynamic quantities extracted from the molecular dynamics simulations. The theory only involves a single unknown parameter, namely an effective friction coefficient, which we estimate fitting the results of the molecular dynamics simulation to our theoretical predictions. Using this framework, we also establish design principles which identify the minimal amount of entropy production that is needed to achieve a certain amount of uncertainty in the power fluctuations of our nano refrigerator. Taken together, our results shed light on how the direction and fluctuations of heat flows in natural and artificial nano machines can be accurately quantified and controlled by using nonreciprocal forces.

### Taxis of cargo-carrying microswimmers in traveling activity waves

**É. Roldán**, A. Gambassi, A. Sharma

We study the behavior of a self-propelled particle carrying a passive cargo in a travelling activity wave and show that this active-passive dimer displays a rich, emergent tactic behavior. For cargoes with low mobility, the dimer always drifts in the direction of the wave propagation. For highly mobile cargoes, instead, the dimer can also drift against the traveling wave. The transition between these two tactic behaviors is controlled by the ratio between the frictions of the cargo and the microswimmer. In slow activity waves the dimer can perform an active surfing of the wave maxima, with an average drift velocity equal to the wave speed. These analytical predictions, which we confirm by numerical simulations, might be useful for the future efficient design of bio-hybrid microswimmers.

### Martingales for Physicists

*(300pp. Tutorial Review)*

**É. Roldán**, I. Neri, R. Chetrite, S. Gupta, S. Pigolotti, F. Julicher, K. Sekimoto

arXiv:2210.09983v3 (2022) [PDF]

We review the theory of martingales as applied to stochastic thermodynamics and stochastic processes in physics more generally.

Invited contribution for Advances in Physics (under Review).

### Modeling Active Non-Markovian Oscillations

**É. Roldán**, A. Gambassi , R. Belousov, F. Berger, R. G. Alonso, and A. J. Hudspeth

Phys. Rev. Lett. **129**, 030603 (2022) [PDF]

Modeling noisy oscillations of active systems is one of the current challenges in physics and biology. Because the physical mechanisms of such processes are often difficult to identify, we propose a linear stochastic model driven by a non-Markovian bistable noise that is capable of generating self-sustained periodic oscillation. We derive analytical predictions for most relevant dynamical and thermodynamic properties of the model. This minimal model turns out to describe accurately bistablelike oscillatory motion of hair bundles in bullfrog sacculus, extracted from experimental data. Based on and in agreement with these data, we estimate the power required to sustain such active oscillations to be of the order of 100 kBT per oscillation cycle.

### What to learn from few visible transitions’ statistics?

**É. Roldán**

arXiv 2203.07427 (2022)

We consider an observer who records a time series of occurrences of one or several transitions performed by a system, under the single assumption that its underlying dynamics is Markovian. We pose the question of how one can use the transitions’ information to make inferences of dynamical, thermodynamical, and biochemical properties. First, putting forward first-passage time techniques, we derive analytical expressions for the probabilities of consecutive transitions and for the time elapsed between them, which we call inter-transition times. Second, we develop an estimate lower bound to the entropy production rate which can be split into two non-negative contributions, one due to the statistics of transitions and a second due to the statistics of inter-transition times. We also show that when only one current is measured, our estimate still detects irreversibility even in the absence of net currents in the transition time series. While entropy production is entailed in the statistics of two successive transitions of the same type (i.e. repeated transitions), the statistics of two different successive transitions (i.e. alternated transitions) can probe the existence of an underlying disorder in the motion of e.g. molecular motors.

**É. Roldán**

Phys. Rev. Lett. **1 26 **(8), 080603 (2021) [PDF]

We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum nonstationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.

### Quantifying entropy production in active fluctuations of the hair-cell bundle from time irreversibility and uncertainty relations

**É. Roldán**, J. Barral, P. Martin, J.M.R. Parrondo and F. Jülicher

New J. Phys. **23**, 083013 (2021) [PDF]

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 recordings of spontaneous hair-bundle oscillations in mechanosensory hair cells from the ear of the bullfrog. By analyzing the fluctuations of only the tip position of hair bundles, we reveal irreversibility in active oscillations and estimate an associated rate of entropy production of at least ∼3*k*_{B}/*s*, on average. Applying thermodynamic uncertainty relations, we predict that measuring both the tip position of the hair bundle and the mechano-electrical transduction current that enters the hair cell leads to tighter lower bounds for the rate of entropy production, up to ∼10^{3}*k*_{B}/*s* in the oscillatory regime.

**É. Roldán**

EPL **132** (5), 50003 (2020) [PDF]

We study minimal mean-field models of viral drug resistance development in which the efficacy of a therapy is described by a one-dimensional stochastic resetting process with mixed reflecting-absorbing boundary conditions. We derive analytical expressions for the mean survival time for the virus to develop complete resistance to the drug. We show that the optimal therapy resetting rates that achieve a minimum and maximum mean survival times undergo a second- and first-order phase transition-like behaviour as a function of the therapy efficacy drift. We illustrate our results with simulations of a population dynamics model of HIV-1 infection.

### Energetics of critical oscillators in active bacterial baths

A. Gopal, **É. Roldán**, and S. Ruffo

arXiv:2011.00858 [PDF]

We investigate the nonequilibrium energetics near a critical point of a non-linear driven oscillator immersed in an active bacterial bath. At the critical point, we reveal a scaling exponent of the average power ⟨W˙⟩∼(Da/τ)^(1/4) where Da is the effective diffusivity and τ the correlation time of the bacterial bath described by a Gaussian colored noise. Other features that we investigate are the average stationary power and the variance of the work both below and above the saddle-node bifurcation. Above the bifurcation, the average power attains an optimal, minimum value for finite τ that is below its zero-temperature limit. Furthermore, we reveal a finite-time uncertainty relation for active matter which leads to values of the Fano factor of the work that can be below 2kTeff, with Teff the effective temperature of the oscillator in the bacterial bath. We analyze different Markovian approximations to describe the nonequilibrium stationary state of the system. Finally, we illustrate our results in the experimental context by considering the example of driven colloidal particles in periodic optical potentials within an E. Coli bacterial bath.

### First-passage Fingerprints of Water Diffusion near Glutamine Surfaces

R. Belousov, M. N. Qaisrani, A. Hassanali , and **É. Roldán
**Soft Matter

**16**, 9202 (2020) [PDF]

We study the dynamics of water near glutamine surfaces—a system of interest in studies of neurodegenerative diseases. Combining molecular-dynamics simulations and stochastic modelling, we study how the mean first- passage time and related statistics of water molecules escaping subnanometer-sized regions vary from the interface to the bulk. Our analysis reveals a dynamical complexity that reflects underlying chemical and geometrical properties of the glutamine surfaces. From the first-passage time statistics of water molecules, we infer their space-dependent diffusion coefficient in directions normal to the surfaces. Interestingly, our results suggest that the mobility of water varies over a longer length scale than the chemical potential associated with the water-protein interactions. The synergy of molecular dynamics and first-passage techniques opens the possibility for extracting space-dependent diffusion coefficients in more complex, inhomogeneous environments that are commonplace in living matter.

### Extreme-Value Statistics of Stochastic Transport Processes

A. Guillet, **É. Roldán **and F. Julicher

New J. Phys. **22**, 123038 (2020) [PDF]

We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks capturing key features of molecular motor motion along linear filaments. Our results generalize the infimum law for entropy production and reveal a symmetry of the distribution of its maxima and minima, which are confirmed by numerical simulations of stochastic models of molecular motors.

We also show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Mar{\v{c}}enko-Pastur distribution of random-matrix theory. Using this result, we obtain estimates for the extreme-value statistics of molecular motors from the eigenvalue distributions of suitable Wishart and Laguerre random matrices.

### Controlling Particle Currents with Evaporation and Resetting from an Interval

### Optimal Work Extraction and the Minimum Description Length Principle

L. Touzo , M. Marsili , N. Merhav , and **É. Roldán
**J. Stat. Mech.

**2020**(9), 093403 [PDF]

We discuss work extraction from classical information engines (e.g., Szilárd) with N-particles, q partitions, and initial arbitrary non-equilibrium states. In particular, we focus on their {\em optimal} behaviour, which includes the measurement of a set of quantities Φ with a feedback protocol that extracts the maximal average amount of work. We show that the optimal non-equilibrium state to which the engine should be driven before the measurement is given by the normalised maximum-likelihood probability distribution of a statistical model that admits Φ as sufficient statistics. Furthermore, we show that the minimax universal code redundancy R∗ associated to this model, provides an upper bound to the work that the demon can extract on average from the cycle, in units of kB T. We also find that, in the limit of N large, the maximum average extracted work cannot exceed one half times the Shannon entropy of the measurement. Our results establish a connection between optimal work extraction in stochastic thermodynamics and optimal universal data compression, providing design principles for optimal information engines. In particular, they suggest that: (i) optimal coding is thermodynamically efficient, and (ii) it is essential to drive the system into a critical state in order to achieve optimal performance.

### Modelling strategies to organize healthcare workforce during pandemics: application to COVID-19

D. Sánchez-Taltavull, D. Candinas, **É. Roldán**, and G. Beldi

medRxiv 2020.03.23.20041863 [PDF]

Protection of healthcare workforce is of paramount relevance for the care of infected and non-infected patients in the setting of a pandemic such as coronavirus disease 2019 (COVID-19). Healthcare workers are at increased risk to become infected because of contact to infected patients, infected co-workers and their community outside the hospital.

We develop mathematical models that describe strategies for the employment of hospital workforce with the goal to simulate health and productivity of the workers. We apply our theory to the case of coronavirus disease 2019 (COVID-19). The results of the models reveal that a desynchronization strategy in which two medical teams work alternating for 7 days reduces the infection rate of the healthcare workforce. Moreover, we show that a minimum of 50% home office efficiency is required to maintain the overall productivity of the entire team of caregivers.

### Martingale theory for housekeeping heat

R. Chetrite, S. Gupta, I. Neri and **É. Roldán**

EPL **124**,60006 (2018) – Editor’s Choice

[ PDF ]

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

J. Stat. Mech. **2019** (10), 104006 (2019) [PDF]

A stopping time T is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation ⟨e^{−Stot(T)}⟩ = 1 for the stochastic entropy production Stot at stopping times T of a nonequilibrium stationary physical system. This fluctuation relation implies the second law of thermodynamics at stopping times ⟨Stot(T)⟩ ≥ 0: it is not possible to reduce entropy on average, even by stopping a stochastic process at a stopping time. Using simulation results we demonstrate that, although mesoscopic systems can extract heat from an isothermal environment when stopped at a cleverly chosen moment, the average extracted heat is bounded by the second law at stopping times. Analogously, we show that stationary stochastic heat engines, such as Feynman’s ratchet, can reach efficiencies above the Carnot limit when stopped at an intelligently chosen moment.

### 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

**É. Roldán** and P. Vivo

arXiv:1903.08271

[ PDF ]

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

G. Manzano, R. Fazio and **É**.** Roldán
**Physical Review Letters

**122**, 220602 (2019)

[ PDF ]

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)

[PDF]

(* 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

S. Singh,** É. Roldán**, I. Neri, I. M. Khaymovich, D. S. Golubev,

V. F. Maisi, J. T. Peltonen, F. Jülicher and J. P. Pekola

Physical Review B **99**, 115422 (2019)

[ **PDF** ]

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.

### Testing Optimality of Sequential Decision-Making

M. Dörpinghaus, I. Neri, **É. Roldán**, H. Meyr and F. Jülicher

arXiv:1801.01574 (2018)

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)

[ **PDF** ]

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

**É. Roldán**, L. Dinis, and R. A. Rica

[

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)

[ **PDF** ]

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

I. Neri, **É. Roldán**, and F. Jülicher

Physical Review X **7** , 011019 (2017)

[ **PDF** ]

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

I. A. Martínez*, **É. Roldán***, L. Dinis, J. M. R. Parrondo, D. Petrov and R. A. Rica

Nature Physics **12**, 67-70 (2016)

[ **PDF** ]

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

A. Lisica, C. Engel, M. Jahnel, **É. Roldán**, E. A. Galburt, P. Cramer and S. W. Grill

PNAS **113** (11), 2946-2951 (2016)

[ **PDF** ]

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

**É. Roldán**, A. Lisica, D. Sánchez-Taltavull, and S. W. Grill

[ **PDF** ]

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.

### Decision Making in the Arrow of time

**É. Roldán**, I. Neri, M. Dörpinghaus, H. Meyr, F. Jülicher

Physical Review Letters

**115**(25), 250602 (2015)

[ **PDF** ]

### Fluctuation theorem between non-equilibrium states in an RC circuit

**É. Roldán**, S. Ciliberto, and H. Kantz

[ **PDF** ]

### Adiabatic processes realized with a trapped Brownian particle

**É. Roldán**, L. Dinis, D. Petrov, and R. A. Rica

[ **PDF** ]

### Realization of nonequilibrium thermodynamic processes using external colored noise

**É. Roldán**

[ **PDF** ]

### Universal features in the energetics of symmetry breaking

**É. Roldán***, I. A. Martínez*, J. M. R. Parrondo, and D. Petrov

### Irreversibility and dissipation in microscopic systems

**É. Roldán** (Springer-Nature, Berlin, 2014)

Springer Theses Prize

[ **Link** ]

### Measuring kinetic energy changes in the mesoscale with low acquisition rates

**É. Roldán**, I. A. Martínez, L. Dinis, and R. A. Rica

[ **PDF** ]

### Effective heating to several thousand kelvins of an optically trapped sphere in a liquid

**É. Roldán**, J. M. R. Parrondo, and D. Petrov

[ **PDF** ]

### Time series irreversibility: a visibility graph approach

**É. Roldán**, J. M. R. Parrondo, and B. Luque

[ **PDF** ]

### Entropy production and Kullback-Leibler divergence between stationary trajectories of discrete systems

**É. Roldán**and J. M. R. Parrondo

[ **PDF** ]