F. Marchesoni

Diffusion in Confined Geometries

Diffusive transport of particles or, more generally, small objects, is a ubiquitous feature of physical and chemical reaction systems. In configurations containing confining walls or constrictions, transport is controlled both by the fluctuation statistics of the jittering objects and the phase space available to their dynamics. Consequently, the study of transport at the macro- and nanoscales must address both Brownian motion and entropic effects. Herein we report on recent advances in the theoretical and numerical investigation of stochastic transport occurring either in microsized geometries of varying cross sections or in narrow channels wherein the diffusing particles are hindered from passing each other (single-file diffusion). For particles undergoing biased diffusion in static suspension media enclosed by confining geometries, transport exhibits intriguing features such as 1) a decrease in nonlinear mobility with increasing temperature or also 2) a broad excess peak of the effective diffusion above the free diffusion limit. These paradoxical aspects can be understood in terms of entropic contributions resulting from the restricted dynamics in phase space. If, in addition, the suspension medium is subjected to external, time-dependent forcing, rectification or segregation of the diffusing Brownian particles becomes possible. Likewise, the diffusion in very narrow, spatially modulated channels is modified via contact particle-particle interactions, which induce anomalous sub-diffusion. The effective sub-diffusion constant for a driven single file also develops a resonance-like structure as a function of the confining coupling constant.

The Virgo interferometer

The Virgo gravitational wave detector is an interferometer with 3 km long arms in construction near Pisa to be commissioned in the year 2000. Virgo has been designed to achieve a strain sensitivity of a few times at 200 Hz. A large effort has gone into the conception of the mirror suspension system, which is expected to reduce noise to the level of at 10 Hz. The expected signals and main sources of noise are briefly discussed; the choices made are illustrated together with the present status of the experiment.

Introduction: 100 years of Brownian motion

In the year 1905 Albert Einstein published four pap that raised him to a giant in the history of science. Th works encompass the photo-electric effect sfor which he ob tained the Nobel prize in 1921 d, his first two papers onsspeciald relativity theory, and his first paper on Brownian m tion, entitled “Über die von der molekularkinetisch Theorie der Wärme geforderte Bewegung von in ruhen Flüssigkeiten suspendierten Teilchen” ssubmitted on 11 Ma 1905d. Thanks to Einstein’s intuition, the phenomenon served by the Scottish botanist Robert Brown 2 i 1827—a little more than a naturalist’s curiosity—becomes the k stone of a fully probabilistic formulation of statistical m chanics and a well-established subject of physical inves tion which we celebrate in this Focus Issue entitled—for reason—“100 Years of Brownian Motion.” Although written in a dated language, Einstein’s first per on Brownian motion already contains the cornerston the modern theory of stochastic processes. The author out using arguments of thermodynamics and the conce osmotic pressure of suspended particles to evaluate a p diffusion constant by balancing a diffusion current wit drift currentsthrough Stokes’ lawd. In doing so, he obtains relation between two transport coefficients: the particle fusion constant and the fluid viscosity, or friction. This re tion, known as the Einstein relation, 3 was later generalized terms of the famous fluctuation-dissipation theorem Callen and Welton 4 and by the linear response theory Kubo. A much clearer discussion of Einstein’s argume can be found in his thesis work, accepted by the Unive of Zurich in July 1905, which he submitted for publicatio 6

Self-Propelled Janus Particles in a Ratchet: Numerical Simulations

Brownian transport of self-propelled overdamped microswimmers (like Janus particles) in a two-dimensional periodically compartmentalized channel is numerically investigated for different compartment geometries, boundary collisional dynamics, and particle rotational diffusion. The resulting time-correlated active Brownian motion is subject to rectification in the presence of spatial asymmetry. We prove that ratcheting of Janus particles can be orders of magnitude stronger than for ordinary thermal potential ratchets and thus experimentally accessible. In particular, autonomous pumping of a large mixture of passive particles can be induced by just adding a small fraction of Janus particles.

On the extension of the Kramers theory of chemical relaxation to the case of nonwhite noise

The first step of our approach consists of relating the generalized Brownian motion in a double‐well potential to a suitable time‐independent Fokker–Planck operator implying that an arbitrary large number of ‘‘virtual’’ variables be used. Then, to simplify the solution of this multidimensional Fokker–Planck equation, we develop a procedure of adiabatic elimination of the fastly relaxing variables. As a significant feature of this reduction scheme, we point out that no limitation on the number of the virtual variables is implied. The explicit form of the first correction term to the Smoluchowski equation is also shown to depend on whether or not the stochastic force is white. Via a comparison with the analytical results of Grote and Hynes’ theory [J. Chem. Phys. 73, 2715 (1980)] it is argued that the ‘‘exact’’ approach and the ‘‘reduction’’ procedure can be regarded as being complementary to one another.

Geometric Stochastic Resonance

A Brownian particle moving across a porous membrane subject to an oscillating force exhibits stochastic resonance with properties which strongly depend on the geometry of the confining cavities on the two sides of the membrane. Such a manifestation of stochastic resonance requires neither energetic nor entropic barriers, and can thus be regarded as a purely geometric effect. The magnitude of this effect is sensitive to the geometry of both the cavities and the pores, thus leading to distinctive optimal synchronization conditions.

Recycled noise rectification: An automated Maxwell's daemon

The one-dimensional motion of a massless Brownian particle on a symmetric periodic substrate can be rectified by reinjecting its driving noise through a realistic recycling procedure. If the recycled noise is multiplicatively coupled to the substrate, the ensuing feedback system works like a passive Maxwells daemon, capable of inducing a net current that depends on both the delay and the autocorrelation times of the noise signals. Extensive numerical simulations show that the underlying rectification mechanism is a resonant nonlinear effect: The observed currents can be optimized for an appropriate choice of the recycling parameters with immediate application to the design of nanodevices for particle transport.

Brownian motors: Current fluctuations and rectification efficiency

With this work, we investigate an often neglected aspect of Brownian motor transport, namely, the role of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time-, and noise-averaged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet potential and appropriate drive parameters, we can identify a drastic enhancement of the rectification efficiency. This regime is marked by persistent, unidirectional motion of the Brownian motor with few back-turns only. The corresponding asymmetric velocity distribution is then rather narrow, with a support that predominantly favors only one sign for the velocity.

Time behaviour of non-linear stochastic processes in the presence of multiplicative noise: From Kramers' to Suzuki's decay

The diffusional regime of a Brownian particle in a double-well potential in the presence of both additive and multiplicative noise is explored. As a relevant effect of the multiplicative noise, the escape rate from a well is shown to change from the small value of the Kramers theory into the large relaxation rate of the Suzuki regime. It is shown, furthermore, that the time required to get equilibrium in a well after sudden application of multiplicative noise (the activation time) is very much shorter than the Kramers relaxation time. We envisage therefore an operational scheme making available multiplicative noise for a short interval of time (for example using a light pulse) as an efficient tool to get a fast process of escape from a well. These results are obtained by using a continued-fraction algorithm which makes it possible even to successfully deal with the decay of an unstable state at the critical point.

Mobility in periodic channels formed by cylindrical cavities

Received 15 December 2009; accepted 30 March 2010; published online 26 April 2010 doi:10.1063/1.3402779Let us consider a system of fluid filled cavities con-nected by narrow pores so as to form a channel network. Asuspended Brownian particle freely diffuses in such a make-shift porous medium by overcoming the entropic barriersassociated with the pores.