Institut for Theoretical Physics
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Noise induced phenomena
The interaction of a small classical system with its
environment can often be described by a stochastic force.
A thermal environment is usually described by a white noise.
Models that contain time-correlated noise or a white noise
and some additional periodic forces show interesting new
phenomena. Typical examples are stochastic resonance,
noise induced transport, and noise induced stability.
We obtained some interesting results for the last two
classes of systems. Most of these models are motivated
by biological systems like motor proteins or cell surface
Selected publications in this field
Sandro Breuer, Andreas Mielke: Multi player Parrondo games with rigid coupling
In a Parrondo game, a single player combines two losing strategies to a winning strategy. In this paper we investigate the question what happens, if two or more players play Parrondo games in a coordinated way. We introduce a strong coupling between the players such that the gain or loss of all players in one round is the same. We investigate two possible realizations of such a coupling. For both we show that the coupling increases the gain per player. The dependency of the gain on the various parameters of the games is determined. The coupling can not only lead to a larger gain, but it can also dominate the driving mechanism of the uncoupled games. Which driving mechanism dominates, depends on the type of coupling. Both couplings are set side by side and the main similarities and differences are emphasised.
Andreas Mielke: Quantum Parrondo Games in Low-Dimensional Hilbert Spaces
We consider quantum variants of Parrondo games on low-dimensional Hilbert spaces. The two games which form the Parrondo game are implemented as quantum walks on a small cycle of length M. The dimension of the Hilbert space is 2M. We investigate a random sequence of these two games which is realized by a quantum coin, so that the total Hilbert space dimension is 4M. We show that in the quantum Parrondo game constructed in this way a systematic win or loss occurs in the long time limit. Due to entaglement and self-interference on the cycle, the game yields a rather complex structure for the win or loss depending on the parameters.
Andreas Mielke: Effective rate equations for the over-damped motion in fluctuating potentials
Phys. Rev. E64,
We discuss physical and mathematical aspects of the over-damped motion of a Brownian particle in fluctuating potentials. It is shown that such a system can be described quantitatively by fluctuating rates if the potential fluctuations are slow compared to relaxation within the minima of the potential, and if the position of the minima does not fluctuate. Effective rates can be calculated; they describe the long-time dynamics of the system. Furthermore, we show the existence of a stationary solution of the Fokker-Planck equation that describes the motion within the fluctuating potential under some general conditions. We also show that a stationary solution of the rate equations with fluctuating rates exists.
Stefan Klumpp, Andreas Mielke, Christian Wald: Noise induced transport of two coupled particles
Phys. Rev. E63,
We study the motion of two harmonically coupled particles in a sawtooth potential. The particles are subject to temporally correlated multiplicative noise. The stationary current is calculated in an expansion about the limit of rigid coupling. For two coupled particles a driving mechanism occurs which is different from the one occurring in the case of a single particle. In particular this mechanism does not need diffusion. Depending on the equilibrium distance of the particles, a current reversal occurs. Possible relevance as a model for motor proteins is discussed.
Andreas Mielke: Noise induced stability in fluctuating, bistable potentials
Phys. Rev. Lett.84,
The over-damped motion of a Brownian particle in an asymmetric, bistable, fluctuating potential shows noise induced stability: For intermediate fluctuation rates the mean occupancy of minima with an energy above the absolute minimum is enhanced. The model works as a detector for potential fluctuations being not too fast and not too slow. This effect occurs due to the different time scales in the problem. We present a detailed analysis of this effect using the exact solution of the Fokker-Planck equation for a simple model. Further we show that for not too fast fluctuations the system can be well described by effective rate equations. The results of the rate equations agree quantitatively with the exact results.
Enrique Abad, Andreas Mielke: Brownian motion in fluctuating periodic potentials
Ann. Physik (Leipzig)7,
This work deals with the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. If the potential has no inversion symmetry and its fluctuations are asymmetric and correlated in time, a net flow can be generated at finite temperatures. We present results for the stationary current for the case of a piecewise linear potential, especially for potentials being close to the case with inversion symmetry. The aim is to study the stationary current as a function of the potential. Depending on the form of the potential, the current changes sign once or even twice as a function of the correlation time of the potential fluctuations. To explain these current reversals, several mechanisms are proposed. Finally, we discuss to what extent the model is useful to understand the motion of biomolecular motors.
We consider a particle in the over-damped regime at zero temperature under the influence of a sawtooth potential and of a noisy force, which is correlated in time. A current occurs, even if the mean of the noisy force vanishes. We calculate the stationary probability distribution and the stationary current. We discuss, how these items depend on the characteristic parameters of the underlying stochastic process. A formal expansion of the current around the white-noise limit not always gives the correct asymptotic behaviour. We improve the expansion for some simple but representative cases.
Andreas Mielke: Noise induced transport
Ann. Physik (Leipzig)4,
We study the overdamped motion of a particle in a one-dimensional periodic potential driven by a stochastic force. If the force is correlated in time (non-white), and if the potential has no inversion symmetry, a current is generated. In the case of a piecewise linaer potential we obtain a closed form for the current as a ratio of two determinants. This allows us to calculate the current as a function of the noise strength, the correlation time and the temperature of the system for several stochastic processes. We examin several limiting situations. Depending on the statistics of the noise process, the direction of the current may change. Two different mechanisms for this effect are discussed.
Andreas Mielke: Transport in a fluctuating potential
Ann. Physik (Leipzig)4,
We study the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. The potential has no inversion symmetry, and the fluctuations are correlated in time. At finite temperatures, a stationary current is induced. The amplitude and the direction of the current depend on the details of the noise process that is responsible for the potential fluctuations. We discuss several limiting situations for a general case. Furthermore we calculate the current in the case of a piecewise linear potential for different noise processes and parameters. A detailed discussion of the results is given, including a discussion of the mechanism that is responsible for the current reversal. We compare the present results with results for transport in a ratchet-like potential due to a fluctuating force. We also discuss the biological relevance of the present models for molecular motors. We present a model for the motion of molecular motors that explains why similar molecular motors can move in different directions.
Last changes: 21.8.2023.