Juan M. R. Parrondo

New Paradoxical Games Based on Brownian Ratchets

Based on Brownian ratchets, a counterintuitive phenomenon has recently emerged-namely, that two losing games can yield, when combined, a paradoxical tendency to win. A restriction of this phenomenon is that the rules depend on the current capital of the player. Here we present new games where all the rules depend only on the history of the game and not on the capital. This new history-dependent structure significantly increases the parameter space for which the effect operates.

Dissipation: The Phase-Space Perspective

We show, through a refinement of the work theorem, that the average dissipation, upon perturbing a Hamiltonian system arbitrarily far out of equilibrium in a transition between two canonical equilibrium states, is exactly given by =W-DeltaF=kTD(rho||rho[over ])=kT, where rho and rho[over ] are the phase-space density of the system measured at the same intermediate but otherwise arbitrary point in time, for the forward and backward process. D(rho||rho[over ]) is the relative entropy of rho versus rho[over ]. This result also implies general inequalities, which are significantly more accurate than the second law and include, as a special case, the celebrated Landauer principle on the dissipation involved in irreversible computations.

Criticism of Feynman’s analysis of the ratchet as an engine

The well‐known discussion on an engine consisting of a ratchet and a pawl in [R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison‐Wesley, Reading, MA, 1963), Vol. 1, pp. 46.1–46.9] is shown to contain some misguided aspects: Since the engine is simultaneously in contact with reservoirs at different temperatures, it can never work in a reversible way. As a consequence, the engine can never achieve the efficiency of a Carnot cycle, not even in the limit of zero power (infinitely slow motion), in contradiction with the conclusion reached in the Lectures.

Time series irreversibility: a visibility graph approach

We propose a method to measure real-valued time series irreversibility which combines two different tools: the horizontal visibility algorithm and the Kullback-Leibler divergence. This method maps a time series to a directed network according to a geometric criterion. The degree of irreversibility of the series is then estimated by the Kullback-Leibler divergence (i.e. the distinguishability) between the inand outdegree distributions of the associated graph. The method is computationally efficient and does not require any ad hoc symbolization process. We find that the method correctly distinguishes between reversible and irreversible stationary time series, including analytical and numerical studies of its performance for: (i) reversible stochastic processes (uncorrelated and Gaussian linearly correlated), (ii) irreversible stochastic processes (a discrete flashing ratchet in an asymmetric potential), (iii) reversible (conservative) and irreversible (dissipative) chaotic maps, and (iv) dissipative chaotic maps in the presence of noise. Two alternative graph functionals, the degree and the degree-degree distributions, can be used as the Kullback-Leibler divergence argument. The former is simpler and more intuitive and can be used as a benchmark, but in the case of an irreversible process with null net current, the degree-degree distribution has to be considered to identify the irreversible nature of the series.

Efficiency of Brownian motors

The efficiency of different types of Brownian motors is calculated analytically and numerically. We find that motors based on flashing ratchets present a low efficiency and an unavoidable entropy production. On the other hand, a certain class of motors based on adiabatically changing potentials, named reversible ratchets, exhibit a higher efficiency and the entropy production can be arbitrarily reduced.

Entropy production and the arrow of time

We present an exact relationship between the entropy production and the distinguishability of a process from its time-reverse, quantified by the relative entropy between forward and backward states. The relationship is shown to remain valid for a wide family of initial conditions, such as canonical, constrained canonical, multi-canonical and grand canonical distributions, as well as both for classical and quantum systems.

Brownian motion and gambling: from ratchets to paradoxical games

Two losing gambling games, when alternated in a periodic or random fashion, can produce a winning game. This paradox has been inspired by certain physical systems capable of rectifying fluctuations: the so-called Brownian ratchets. In this paper we review this paradox, from Brownian ratchets to the most recent studies on collective games, providing some intuitive explanations of the unexpected phenomena that we will find along the way.

Brownian Carnot engine

The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths1. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors2 and some artificial micro-engines3–5 operate. As described by stochastic thermodynamics6,7, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit8. 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 temperatures9–11. 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.

Noise-induced spatial patterns

By modifying the spatial coupling a la Swift-Hohenberg in the model introduced in Phys. Rev. Lett. 73 (1994) 3395, we obtain a system that displays noise-induced spatial patterns. We present a mean field theory of this phenomenon and verify some of its predictions by numerical simulations.

Thermodynamic reversibility in feedback processes

The sum of the average work dissipated plus the information gained during a thermodynamic process with discrete feedback must exceed zero. We demonstrate that the minimum value of zero is attained only by feedback-reversible processes that are indistinguishable from their time-reversal, thereby extending the notion of thermodynamic reversibility to feedback processes. In addition, we prove that in every realization of a feedback-reversible process the sum of the work dissipated and change in uncertainty is zero.