In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of ``Brownian motors.'' In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications.

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Artificial Brownian motors: Controlling transport on the nanoscale

Numerical simulations of homogeneous crystal nucleation with a model for globular proteins with short-range attractive interactions showed that the presence of a metastable fluid-fluid critical point drastically changes the pathway for the formation of a crystal nucleus. Close to this critical point, the free-energy barrier for crystal nucleation is strongly reduced and hence, the crystal nucleation rate increases by many orders of magnitude. Because the location of the metastable critical point can be controlled by changing the composition of the solvent, the present work suggests a systematic approach to promote protein crystallization.

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Enhancement of Protein Crystal Nucleation by Critical Density Fluctuations

Advanced Mathematical Methods for Scientists and Engineers

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

A new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.

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Calculating free energies using average force

We report a computer‐simulation study of the rate of homogeneous crystal nucleation and the structure of crystal nuclei in a Lennard‐Jones system at moderate undercooling. The height of the nucleation barrier has been determined using umbrella sampling, whereas the barrier crossing rate is calculated using molecular dynamics simulation. The simulations clearly show that the barrier crossing is a diffusive process. Nevertheless, the kinetic prefactor in the nucleation rate is found to be some two orders of magnitude larger than predicted by classical nucleation theory. The height of the barrier is in good agreement with the theoretical prediction. Although the Lennard‐Jones system has a stable face‐centered cubic (fcc) phase below the melting line, the precritical nuclei are found to be mainly body‐centered cubic (bcc) ordered. As they grow to their critical size, they become more fcc ordered in the core. However, the critical and postcritical nuclei retain a high degree of bcc ordering in the interface. Furthermore it is found that in the interface the density falls off faster than the structural order parameter, which is in agreement with the predictions of density functional calculations. [P. Harrowell and D. W. Oxtoby, J. Chem. Phys. 80, 1639 (1984)].

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Numerical calculation of the rate of crystal nucleation in a Lennard‐Jones system at moderate undercooling

We report a computer-simulation study of the crystal nucleation barrier and the structure of crystal nuclei in a Lennard-Jones system at moderate supercooling. The stable structure of the Lennard-Jones solid is known to be face-centered cubic. We find that the precritical nuclei are predominantly body-centered cubic ordered. But, as the nucleus grows to its critical size, the core becomes fcc ordered. Surprisingly, however, the interface of the critical nucleus retains a high degree of bcc-like ordering.

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Numerical evidence for bcc ordering at the surface of a critical fcc nucleus

We discuss a numerical scheme to study homogeneous crystal nucleation. Using this approach, it is possible to compute the height of the free energy barrier that separates the solid from the liquid phase and the rate at which this barrier is crossed. We point out that there is a fundamental difference between the use of a global- and a local-order parameter to measure the degree of crystallinity. Using a global-order parameter, precritical nuclei may break up spontaneously for entropic reasons. Near the top of the barrier the nuclei combine to form a relatively large cluster. The transition from many small clusters to one large cluster is discussed in some detail. Finally we present a new method that allows us to avoid this entropic cluster break up.

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Simulation of homogeneous crystal nucleation close to coexistence

This research was partially supported by Grant No. PB96- 0534 from the Direccion General de Investigacion Cientifica y Tecnica (Spain)

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Homogeneous cooling state of a low-density granular flow

We report a computer-simulation study of the absolute rate of homogeneous gas–liquid nucleation in a Lennard-Jones system. The height of the barrier has been computed using umbrella sampling, whereas the kinetic prefactor is calculated using molecular dynamics simulations. The simulations show that the nucleation process is highly diffusive. We find that the kinetic prefactor is a factor of 10 larger than predicted by classical nucleation theory.

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M. J. Ruiz-Montero

Papers

Numerical calculation of the rate of crystal nucleation in a Lennard‐Jones system at moderate undercooling

Numerical evidence for bcc ordering at the surface of a critical fcc nucleus

Simulation of homogeneous crystal nucleation close to coexistence

Homogeneous cooling state of a low-density granular flow

Numerical calculation of the rate of homogeneous gas–liquid nucleation in a Lennard-Jones system