Transport phenomena in spatially periodic systems far from thermal equilibrium are considered. The main emphasize is put on directed transport in so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the presence of thermal noise and some prototypical perturbation that drives the system out of equilibrium without introducing a priori an obvious bias into one or the other direction of motion. Symmetry conditions for the appearance (or not) of directed current, its inversion upon variation of certain parameters, and quantitative theoretical predictions for specific models are reviewed as well as a wide variety of experimental realizations and biological applications, especially the modeling of molecular motors. Extensions include quantum mechanical and collective effects, Hamiltonian ratchets, the influence of spatial disorder, and diffusive transport.Comment: Revised version (Aug. 2001), accepted for publication in Physics Report

We demonstrate the equilibration of isolated macroscopic quantum systems, prepared in nonequilibrium mixed states with significant population of many energy levels, and observed by instruments with a reasonably bound working range compared to the resolution limit. Both properties are fulfilled under many, if not all, experimentally realistic conditions. At equilibrium, the predictions and limitations of Statistical Mechanics are recovered.PACS numbers: 05.30.Ch, Fundamental aspects of equilibrium Statistical Mechanics (ESM) are intensely reconsidered at present in the context of (almost) integrable many-body quantum systems [1,2,3,4], bringing back to our attention that very basic issues are still not satisfactorily understood [5,6,7,8,9,10,11,12,13]. As in every theory, we are faced with the three sub-problems to realistically model preparation, time evolution, and measurement of a given system. It is well know and will be worked out in detail below that the question of experimentally realistic initial conditions and observables is much more urgent in the "derivation" of ESM than in most other fields [5,7,8,9,11,14]. Regarding time evolution, we take the widely (yet not unanimously) accepted viewpoint that standard Quantum Mechanics without any additional "postulate" or "hypothesis" must do [15,16]. The two key questions are then: In how far does a nonequilibrium seed evolve to a stationary long-time behavior ("equilibration")? In how far is this steady state in agreement with the corresponding ESM ensemble ("thermalization")?Since open systems (interacting and entangled with an environment) are not directly tractable by standard Quantum Mechanics, the starting point must be a closed (autonomous) system (microcanonical framework), incorporating all relevant thermal baths, reservoirs etc. [15,16]. Accordingly, the system "lives" in some Hilbert space H and is at any time instant t ≥ 0 in a mixed state (including pure states as special case) ρ(t) = U t ρ(0)U † t with propagator U t := exp{−iHt/ }, seed ρ(0), and time-independent Hamiltonian H. Denoting its eigenfunctions and eigenvalues by |n and E n (n = 0, 1, 2, ...) and the matrix elements m|ρ(t)|n by ρ mn (t) we thus obtainwhere the sum runs over all m, n ≥ 0. As usual, observables are represented by Hermitean operators A with expectation values Tr{ρ(t)A} and, without loss of generality, are assumed not to depend explicitly on time.Generically, the ensemble ρ(t) is not stationary right from the beginning, in particular for an initial condition ρ(0) out of equilibrium. But if the right hand side of (1) depends on t initially, it cannot approach for large t any time-independent "equilibrium ensemble" whatsoever. In fact, any mixed state ρ(t) returns arbitrarily "near" to its seed ρ(0) for certain, sufficiently large time-points t, and similarly for the expectation values Tr{ρ(t)A}, see Appendix D in Ref. [17]. More specifically, consider any ρ(0) with at least one ρ mn (0) = 0 and ω := [E n − E m ]/ = 0. Chosingit follows that Tr{ρ(t)A} = 2 cos(ωt). It is thus...

The effective diffusion coefficient for the overdamped Brownian motion in a tilted periodic potential is calculated in closed analytical form. Universality classes and scaling properties for weak thermal noise are identified near the threshold tilt where deterministic running solutions set in. In this regime the diffusion may be greatly enhanced, as compared to free thermal diffusion with, for a realistic experimental setup, an enhancement of up to 14 orders of magnitude.

We show that transport in the presence of entropic barriers exhibits peculiar characteristics which makes it distinctly different from that occurring through energy barriers. The constrained dynamics yields a scaling regime for the particle current and the diffusion coefficient in terms of the ratio between the work done to the particles and available thermal energy. This interesting property, genuine to the entropic nature of the barriers, can be utilized to effectively control transport through quasi-one-dimensional structures in which irregularities or tortuosity of the boundaries cause entropic effects. The accuracy of the kinetic description has been corroborated by simulations. Applications to different dynamic situations involving entropic barriers are outlined. Transport through quasi-one-dimensional structures as pores, ion channels, and zeolites is ubiquitous in biological and physicochemical systems and constitute a basic mechanism in processes as catalysis, osmosis, and particle separation [1][2][3][4][5][6]. A common characteristic of these systems is the confinement arising from the presence of boundaries which very often exhibit an irregular geometry. Variations of the shape of the structure along the propagation direction implies changes in the number of accessible states of the particles. Consequently, entropy is spatially varying, and the system evolves through entropic barriers, which controls the transport, promoting or hampering the transfer of mass and energy to certain regions. Motion in the system can be induced by the presence of external driving forces supplying the particles with the energy necessary to proceed. The study of the kinetics of the entropic transport, the properties of transport coefficients in far from equilibrium situations, and the possibility for transport control mechanisms are objectives of major importance in the dynamical characterization of those systems.Our purpose in this Letter is to demonstrate that entropic transport exhibits striking features, sometimes counterintuitive, which are different from those observed in the more familiar case with energy barriers [7]. We propose a general scenario describing the dynamics through entropic barriers and show the existence of a scaling regime for the current of particles and the effective diffusion coefficient. The presence of this regime might have important implications in the control of transport.Entropic transport.-The origin of the entropic barriers can be inherent to the intimate nature of the system or may emerge as a consequence of a coarsening of the description employed. A typical example presents the motion of a Brownian particle in an enclosure of varying cross section. This basic situation constitutes the starting point in the study of transport processes in the type of confined systems that are very often encountered at subcellular level, nanoporous materials, and in microfluidic applications. As shown in Ref. [8], the complicated boundary conditions of the diffusion equation in irregular channels can be ...

Reimann replies:In the preceding Comment [1], Brody raises some points of criticism regarding the recent Letter [2], most of which in fact apply equally well to the closely related works [3][4][5].The first main result in [2] is the inequality (13) therein and is clearly identified as such. All of Brody's concerns regard the conclusions drawn in [2] out of this inequality. Hence, (13) continues to represent a main result rather than a ''claim.'' Ad (a): Assumptions (i) and (ii), under which (13) is derived, are justified for macroscopic equilibrium systems in the three paragraphs above Eq. (14) in [2]. In particular, it is shown that the assumptions made in [3][4][5] are special cases of (i) and (ii) [6]. It is well know that Trf is time independent for any function f of the density operator of an isolated system. For fx x lnx one recovers the famous unsolved problem that the entropy does not increase. Brody points out that this problem was not solved for fx x 2 in [2] either. While all this is mathematically correct, it remains true that entropy does increase in physical reality and that equilibrium ensembles do exhibit extremely small purities Tr 2 (see the paragraph below (21) in [2] and the standard ensembles considered in [3][4][5]). The resolution of the paradox is part of the open question, how and in which sense a general initial converges towards an equilibrium ensemble. In a very similar context, the prominent last words of van Kampen in his book [7] are: ''it is better to say something that is true although not proven, than to prove something that is not true.'' Ad (b): I must admit that the term ''observable'' has not been specified sufficiently clearly in [2]. In [1], it is tacitly assumed that any Hermitian operator A is an observable. In contrast, the term observable in the spirit of [2] refers only to operators A corresponding to realistic experimental measurement devices. Clearly, any given real experimental apparatus has a finite range and hence the corresponding Hermitian operator A has a spectrum which is bounded by finite upper and lower limits a max and a min . Given a specific measurement device, these limits and hence their difference A : a max ÿ a min do not change when the size of the observed system is increasing. For instance, the energy of a harmonic oscillator is not an observable in the sense of [2]: in principle, the energy of the oscillator may become arbitrarily large (albeit with extremely small probability), but no real device would be able to display its value beyond a certain upper limit. Rather, all energies beyond this limit will yield one and the same measurement result (e.g., a blow up of the device), and hence only the corresponding ''truncated'' energy operator would be an admissible observable. Again, it is van Kampen which in the last chapter of his book [7] stresses the importance of properly characterizing admissible observables when dealing with basic problems of statistical mechanics; see also [5].