Jakub Spiechowicz

GPU accelerated Monte Carlo simulation of Brownian motors dynamics with CUDA

Abstract This work presents an updated and extended guide on methods of a proper acceleration of the Monte Carlo integration of stochastic differential equations with the commonly available NVIDIA Graphics Processing Units using the CUDA programming environment. We outline the general aspects of the scientific computing on graphics cards and demonstrate them with two models of a well known phenomenon of the noise induced transport of Brownian motors in periodic structures. As a source of fluctuations in the considered systems we selected the three most commonly occurring noises: the Gaussian white noise, the white Poissonian noise and the dichotomous process also known as a random telegraph signal. The detailed discussion on various aspects of the applied numerical schemes is also presented. The measured speedup can be of the astonishing order of about 3000 when compared to a typical CPU. This number significantly expands the range of problems solvable by use of stochastic simulations, allowing even an interactive research in some cases. Program Summary Program title: Poisson, dich Catalogue identifier: AEVP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEVP_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU Lesser General Public License. version 3 No. of lines in distributed program, including test data, etc.: 3338 No. of bytes in distributed program, including test data, etc.: 45009 Distribution format: tar.gz Programming language: CUDA C. Computer: Any with CUDA-compliant GPU. Operating system: No limits (tested on Linux). RAM: Hundreds of megabytes for typical case Classification: 4.3, 23. External routines: The NVIDIA CUDA Random Number Generation library (cuRAND) Nature of problem: Graphics processing unit accelerated numerical simulation of stochastic differential equation. Solution method: The jump-adapted simplified weak order 2.0 predictor–corrector method is employed to integrate the Langevin equation of motion. Ensemble-averaged quantities of interest are obtained through averaging over multiple independent realizations of the system generated by means of the Monte Carlo method. Unusual features: The actual numerical simulation runs exclusively on the graphics processing unit using the CUDA environment. This allows for a speedup as large as about 3000 when compared to a typical CPU. Running time: A few seconds

Brownian motors in the microscale domain: enhancement of efficiency by noise.

We study a noisy drive mechanism for efficiency enhancement of Brownian motors operating on the microscale domain. It was proven [J. Spiechowicz et al., J. Stat. Mech. (2013) P02044] that biased noise η(t) can induce normal and anomalous transport processes similar to those generated by a static force F acting on inertial Brownian particles in a reflection-symmetric periodic structure in the presence of symmetric unbiased time-periodic driving. Here, we show that within selected parameter regimes, noise η(t) of the mean value 〈η(t)〉=F can be significantly more effective than the deterministic force F: the motor can move much faster, its velocity fluctuations are much smaller, and the motor efficiency increases several times. These features hold true in both normal and absolute negative mobility regimes. We demonstrate this with detailed simulations by resource to generalized white Poissonian noise. Our theoretical results can be tested and corroborated experimentally by use of a setup that consists of a resistively and capacitively shunted Josephson junction. The suggested strategy to replace F by η(t) may provide a new operating principle in which micro- and nanomotors could be powered by biased noise.

Absolute negative mobility induced by white Poissonian noise

We study the transport properties of inertial Brownian particles which move in a symmetric periodic potential and are subjected to both a symmetric, unbiased time-periodic external force and a biased Poissonian white shot noise (of non-zero average F) which is composed of a random sequence of ?-shaped pulses with random amplitudes. Upon varying the parameters of the white shot noise, one can conveniently manipulate the transport direction and the overall nonlinear response behavior. We find that within tailored parameter regimes the response is opposite to the applied average bias F of such white shot noise. This particular transport characteristic thus mimics that of a nonlinear absolute negative mobility (ANM) regime. Moreover, such white shot noise driven ANM is robust with respect to the statistics of the shot noise spikes. Our findings can be checked and corroborated experimentally by the use of a setup that consists of a single resistively and capacitively shunted Josephson junction device.

Transient anomalous diffusion in periodic systems: ergodicity, symmetry breaking and velocity relaxation

We study far from equilibrium transport of a periodically driven inertial Brownian particle moving in a periodic potential. As detected for a SQUID ratchet dynamics, the mean square deviation of the particle position from its average may involve three distinct intermediate, although extended diffusive regimes: initially as superdiffusion, followed by subdiffusion and finally, normal diffusion in the asymptotic long time limit. Even though these anomalies are transient effects, their lifetime can be many, many orders of magnitude longer than the characteristic time scale of the setup and turns out to be extraordinarily sensitive to the system parameters like temperature or the potential asymmetry. In the paper we reveal mechanisms of diffusion anomalies related to ergodicity of the system, symmetry breaking of the periodic potential and ultraslow relaxation of the particle velocity towards its steady state. Similar sequences of the diffusive behaviours could be detected in various systems including, among others, colloidal particles in random potentials, glass forming liquids and granular gases.

Diffusion anomalies in ac-driven Brownian ratchets.

We study diffusion in ratchet systems. As a particular experimental realization we consider an asymmetric SQUID subjected to an external ac current and a constant magnetic flux. We analyze mean-square displacement of the Josephson phase and find that within selected parameter regimes it evolves in three distinct stages: initially as superdiffusion, next as subdiffusion, and finally as normal diffusion in the asymptotic long-time limit. We show how crossover times that separates these stages can be controlled by temperature and an external magnetic flux. The first two stages can last many orders longer than characteristic time scales of the system, thus being comfortably detectable experimentally. The origin of abnormal behavior is noticeable related to the ratchet form of the potential revealing an entirely new mechanism of emergence of anomalous diffusion. Moreover, a normal diffusion coefficient exhibits nonmonotonic dependence on temperature leading to an intriguing phenomenon of thermal noise suppressed diffusion. The proposed setup for experimental verification of our findings provides a new and promising testing ground for investigating anomalies in diffusion phenomena.

Efficiency of the SQUID ratchet driven by external current

We study theoretically the efficiency of an asymmetric superconducting quantum interference device (SQUID) which is constructed as a loop with three capacitively and resistively shunted Josephson junctions. Two junctions are placed in series in one arm and the remaining one is located in the other arm. The SQUID is threaded by an external magnetic flux and driven by an external current of both constant (dc) and time periodic (ac) components. This system acts as a nonequilibrium ratchet for the dc voltage across the SQUID with the external current as a source of energy. We analyze the power delivered by the external current and find that it strongly depends on thermal noise and the external magnetic flux. We explore a space of the system parameters to reveal a set for which the SQUID efficiency is globally maximal. We detect the intriguing feature of the thermal noise enhanced efficiency and show how the efficiency of the device can be tuned by tailoring the external magnetic flux.

Efficiency of transport in periodic potentials: dichotomous noise contra deterministic force

We study the transport of an inertial Brownian particle moving in a symmetric and periodic one-dimensional potential, and subjected to both a symmetric, unbiased external harmonic force as well as biased dichotomic noise also known as a random telegraph signal or a two state continuous-time Markov process. In doing so, we concentrate on the previously reported regime (Spiechowicz et al 2014 Phys. Rev. E 90 032104) for which non-negative biased noise in the form of generalized white Poissonian noise can induce anomalous transport processes similar to those generated by a deterministic constant force but significantly more effective than F, i.e. the particle moves much faster, the velocity fluctuations are noticeably reduced and the transport efficiency is enhanced several times. Here, we confirm this result for the case of dichotomous fluctuations which, in contrast to white Poissonian noise, can assume positive as well as negative values and examine the role of thermal noise in the observed phenomenon. We focus our attention on the impact of bidirectionality of dichotomous fluctuations and reveal that the effect of nonequilibrium noise enhanced efficiency is still detectable. This result may explain transport phenomena occurring in strongly fluctuating environments of both physical and biological origin. Our predictions can be corroborated experimentally by use of a setup that consists of a resistively and capacitively shunted Josephson junction.

Josephson junction ratchet: The impact of finite capacitances

We study transport in an asymmetric superconducting quantum interference device (SQUID) which is composed of a loop with three capacitively and resistively shunted Josephson junctions: two in series in one arm and the remaining one in the other arm. The loop is threaded by an external magnetic flux and the system is subjected to both a time-periodic and a constant current. We formulate the deterministic and, as well, the stochastic dynamics of the SQUID in terms of the Stewart-McCumber model and derive an equation for the phase difference across one arm, in which an effective periodic potential is of the ratchet type, i.e., its reflection symmetry is broken. In doing so, we extend and generalize an earlier study by Zapata et al. [Phys. Rev. Lett. 77, 2292 (1996)] and analyze directed transport in wide parameter regimes: covering the overdamped to the moderate damping regime up to its fully underdamped regime. As a result we detect the intriguing features of a negative (differential) conductance, repeated voltage reversals, noise-induced voltage reversals, and solely thermal noise-induced ratchet currents. We identify a set of parameters for which the ratchet effect is most pronounced and show how the direction of transport can be controlled by tailoring the external magnetic flux.

Two coupled Josephson junctions: dc voltage controlled by biharmonic current.

We study transport properties of two Josephson junctions coupled by an external shunt resistance. One of the junctions (say, the first) is driven by an unbiased ac current consisting of two harmonics. The device can rectify the ac current yielding a dc voltage across the first junction. For some values of coupling strength, controlled by an external shunt resistance, a dc voltage across the second junction can be generated. By variation of system parameters such as the relative phase or frequency of two harmonics, one can conveniently manipulate both voltages with high efficiency, e.g. changing the dc voltages across the first and second junctions from positive to negative values and vice versa.

Brownian ratchets: How stronger thermal noise can reduce diffusion

We study diffusion properties of an inertial Brownian motor moving on a ratchet substrate, i.e., a periodic structure with broken reflection symmetry. The motor is driven by an unbiased time-periodic symmetric force that takes the system out of thermal equilibrium. For selected parameter sets, the system is in a non-chaotic regime in which we can identify a non-monotonic dependence of the diffusion coefficient on temperature: for low temperature, it initially increases as the temperature grows, passes through its local maximum, next starts to diminish reaching its local minimum, and finally it monotonically increases in accordance with the Einstein linear relation. Particularly interesting is the temperature interval in which diffusion is suppressed by the thermal noise, and we explain this effect in terms of transition rates of a three-state stochastic model.