^{1}, P. Reimann

^{1}, B. Cleuren

^{2}and C. Van den Broeck

^{2}

### Abstract

We discuss the fundamental physical differences and the mathematical interconnections of counterintuitive transport and response properties in Brownian motion far from equilibrium. After reviewing the ubiquity of such effects in physical and other systems, we illustrate the general properties on paradigmatic models for both individually and collectively acting Brownian particles.

Nanotechnology and biotechnology thrive on our growing ability to observe, measure, control, and even manufacture on a very small scale. As one moves down in size, the molecular nature of matter becomes more apparent and the concomitant thermal agitation of molecules and the associated fluctuations in the properties of the system can no longer be ignored. Historically speaking, fluctuations and noise were considered a nuisance, to be avoided in technological applications. Over the last decade, the awareness has grown to the fact that noise can actually become the motor to achieve a specific goal. For example, the energy present in thermal noise can under specific circumstances, involving nonequilibrium and asymmetry, be harvested to transport small scale objects. This “moving forward noisily”

^{1}or “Brownian motor” mechanism can possibly explain transport in the cell or contraction of muscle fiber. In this review, we develop a related and maybe even more surprising theme of “moving backward noisily” or “Brownian donkeys.” These phenomena rest on the observation that the thermal energy can be harvested in a special way upon application of a force or bias on the Brownian motor. Instead of moving in the direction of the force, the particle will under the influence of the thermal agitation be sent in the opposite direction, akin to a donkey that steps backward when being pulled.

We thank P. Hänggi, R. Kawai, J. M. R. Parrondo, B. Jimenez de Cisneros, and M. Kambon for very fruitful discussions and collaborations. Parts of this work have been completed during a visit at the Centro de Ciencias Matematicas, Universidade da Madeira. We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft under SFB 613 and RE 1344/3-1, the Program on Inter University Attraction Poles of the Belgian Government, NSF Grant No. PHY-9970699, and the ESF-program STOCHDYN.

I. INTRODUCTION

II. ANM, DNM, AND BEYOND: EXAMPLES AND CONCEPTUAL CONNECTIONS

III. MATHEMATICAL MAPPING BETWEEN SINGLE PARTICLE ANM IN MEANDERING GEOMETRIES AND RATCHETS

A. The system class

B. Mapping to ratchet systems

C. Time-dependent forcing

D. Temperature anisotropy

E. Time-dependent temperature

IV. COLLECTIVE ANM BY CROWDING IN AN ISING MODEL

V. CONCLUSIONS

### Key Topics

- Anisotropy
- 5.0
- Brownian motion
- 4.0
- Collective effects
- 4.0
- Semiconductor device modeling
- 4.0
- Statics
- 4.0

## Figures

Schematic illustration of several fundamentally different current–force characteristics addressed in this work (see also Fig. 3 ): (a) “Usual” response behavior, e.g., of an equilibrium system. (b) Absolute negative mobility (ANM). (c) Differential negative mobility (DNM). (d) Ratchet effect.

Schematic illustration of several fundamentally different current–force characteristics addressed in this work (see also Fig. 3 ): (a) “Usual” response behavior, e.g., of an equilibrium system. (b) Absolute negative mobility (ANM). (c) Differential negative mobility (DNM). (d) Ratchet effect.

Schematic current–load curve for a system exhibiting DNM in the vicinity of and symmetrically around .

Schematic current–load curve for a system exhibiting DNM in the vicinity of and symmetrically around .

Schematic illustration of two further paradoxical current–load response curves: (a) Usual linear but unusual nonlinear response (extreme case of DNM). (b) Anomalous hysteresis as a collective effect.

Schematic illustration of two further paradoxical current–load response curves: (a) Usual linear but unusual nonlinear response (extreme case of DNM). (b) Anomalous hysteresis as a collective effect.

Sketch of the model: The “meandering” path with negligible width is represented by a one-dimensional curve in the -plane (the black curve). It is periodic in -direction with period and confined in -direction. The Brownian particle moves along this path under the action of an externally applied force .

Sketch of the model: The “meandering” path with negligible width is represented by a one-dimensional curve in the -plane (the black curve). It is periodic in -direction with period and confined in -direction. The Brownian particle moves along this path under the action of an externally applied force .

(a) Piecewise linear meandering path parametrized by the angle and the lengths , . (b) The corresponding one-dimensional effective potential with for or and for or . The encircled numbers mark corresponding positions in (a) and (b).

(a) Piecewise linear meandering path parametrized by the angle and the lengths , . (b) The corresponding one-dimensional effective potential with for or and for or . The encircled numbers mark corresponding positions in (a) and (b).

Mechanism for the occurrence of ANM in the system (1) and (2) , or equivalently (3) , when driven away from thermal equilibrium by a time-dependent forcing . The dashed arrows represent jumps of the total external force (8) between its different states . The solid arrows illustrate the different routes the particle can follow, when starting in the potential minimum marked by the little “particle.” Due to the short sojourn time (“fast” driving) for the states thermally induced escapes out of the potential minima can be neglected.

Mechanism for the occurrence of ANM in the system (1) and (2) , or equivalently (3) , when driven away from thermal equilibrium by a time-dependent forcing . The dashed arrows represent jumps of the total external force (8) between its different states . The solid arrows illustrate the different routes the particle can follow, when starting in the potential minimum marked by the little “particle.” Due to the short sojourn time (“fast” driving) for the states thermally induced escapes out of the potential minima can be neglected.

Current (7) versus load for the piecewise linear meandering path of Fig. 5 [with , , in (a), (b) and in (c)] under different nonequilibrium situations: (a) Time-dependent forcing as defined in (8) . Shown are the results of numerical simulations of (3) and (8) for dimensionless parameters , , , and . (b) Temperature anisotropy resulting in a state-dependent temperature (9) in (3) . The shown curve is obtained from the analytical result for the current detailed in Ref. ^{ 42 } . The dimensionless parameters are , , and . (c) Time-dependent temperature as defined in (10) . The dots represent the results of numerical simulations of (3) and (10) for dimensionless parameters , , , , and .

Current (7) versus load for the piecewise linear meandering path of Fig. 5 [with , , in (a), (b) and in (c)] under different nonequilibrium situations: (a) Time-dependent forcing as defined in (8) . Shown are the results of numerical simulations of (3) and (8) for dimensionless parameters , , , and . (b) Temperature anisotropy resulting in a state-dependent temperature (9) in (3) . The shown curve is obtained from the analytical result for the current detailed in Ref. ^{ 42 } . The dimensionless parameters are , , and . (c) Time-dependent temperature as defined in (10) . The dots represent the results of numerical simulations of (3) and (10) for dimensionless parameters , , , , and .

(a) Sketch of the basic collective model. A collection of particles move along a circle, which is divided into two parts. The particles can move between the two parts by crossing the gates, located at the North and South Pole. (b) Sketch of the four possible situations for the gate at the South Pole. The gate is characterized by the critical density . The particle density on a particular part of the circle is represented by the symbol . Depending on the value of , the gate is either open or closed for particles coming from that part.

(a) Sketch of the basic collective model. A collection of particles move along a circle, which is divided into two parts. The particles can move between the two parts by crossing the gates, located at the North and South Pole. (b) Sketch of the four possible situations for the gate at the South Pole. The gate is characterized by the critical density . The particle density on a particular part of the circle is represented by the symbol . Depending on the value of , the gate is either open or closed for particles coming from that part.

The mobility from (11) and (20) as a function of the particle number for different values of the critical density in units with . [Each discrete gives rise to exactly one point in each plot. The apparent “patterns” or “multiple curves” are only an artifact of the jumps of when going from one -value to the next.]

The mobility from (11) and (20) as a function of the particle number for different values of the critical density in units with . [Each discrete gives rise to exactly one point in each plot. The apparent “patterns” or “multiple curves” are only an artifact of the jumps of when going from one -value to the next.]

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