We use the mesoscopic nonequilibrium thermodynamics theory to derive the general kinetic equation of a system in the presence of potential barriers. The result is applied to a description of the evolution of systems whose dynamics is influenced by entropic barriers. We analyze in detail the case of diffusion in a domain of irregular geometry in which the presence of the boundaries induces an entropy barrier when approaching the exact dynamics by a coarsening of the description. The corresponding kinetic equation, named the Fick-Jacobs equation, is obtained, and its validity is generalized through the formulation of a scaling law for the diffusion coefficient which depends on the shape of the boundaries. The method we propose can be useful to analyze the dynamics of systems at the nanoscale where the presence of entropy barriers is a common feature.

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 ...

Concepts of everyday use such as energy, heat, and temperature have acquired a precise meaning after the development of thermodynamics. Thermodynamics provides the basis for understanding how heat and work are related and the general rules that the macroscopic properties of systems at equilibrium follow. Outside equilibrium and away from macroscopic regimes, most of those rules cannot be applied directly. Here we present recent developments that extend the applicability of thermodynamic concepts deep into mesoscopic and irreversible regimes. We show how the probabilistic interpretation of thermodynamics together with probability conservation laws can be used to obtain Fokker−Planck equations for the relevant degrees of freedom. This approach provides a systematic method to obtain the stochastic dynamics of a system directly from its equilibrium properties. A wide variety of situations can be studied in this way, including many that were thought to be out of reach of thermodynamic theories, such as nonlinear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules, such as translocation and stretching.

An exact analytical expression for the effective diffusion coefficient of an overdamped Brownian particle in a tilted periodic potential is derived for arbitrary potentials and arbitrary strengths of the thermal noise. Near the critical tilt ͑threshold of deterministic running solutions͒ a scaling behavior for weak thermal noise is revealed and various universality classes are identified. In comparison with the bare ͑potential-free͒ thermal diffusion, the effective diffusion coefficient in a critically tilted periodic potential may be, in principle, arbitrarily enhanced. For a realistic experimental setup, an enhancement by 14 orders of magnitude is predicted so that thermal diffusion should be observable on a macroscopic scale at room temperature.