L.V. Bogachev

Driven Diffusion in a Periodically Compartmentalized Tube: Homogeneity versus Intermittency of Particle Motion

We study the effect of a driving force F on drift and diffusion of a point Brownian particle in a tube formed by identical cylindrical compartments, which create periodic entropy barriers for the particle motion along the tube axis. The particle transport exhibits striking features: the effective mobility monotonically decreases with increasing F, and the effective diffusivity diverges as F→∞, which indicates that the entropic effects in diffusive transport are enhanced by the driving force. Our consideration is based on two different scenarios of the particle motion at small and large F, homogeneous and intermittent, respectively. The scenarios are deduced from the careful analysis of statistics of the particle transition times between neighboring openings. From this qualitative picture, the limiting small-F and large-F behaviors of the effective mobility and diffusivity are derived analytically. Brownian dynamics simulations are used to find these quantities at intermediate values of the driving force for various compartment lengths and opening radii. This work shows that the driving force may lead to qualitatively different anomalous transport features, depending on the geometry design.

On Bounded Solutions of the Balanced Generalized Pantograph Equation

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in the early 1970s by T. Kato in connection with the analysis of the pantograph equation, y′(x)=ay(qx)+by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a 2 y″(x + a 1 y′(x) + y(x) = ∫ 0 ∞ μ (dα), where a 1 ≥ 0, a 2 ≥ 0 a 1 2 + a 2 2 > 0, and μ is a probability measure. By setting K:=∫ 0 ∞ ln ± μ(dα), we prove that if K≦0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K > 0 then such a solution exists. The result in the critical case, K=0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with “multiplication” jumps. The paper also includes three “elementary” proofs for the simple prototype equation y′(x)+y(x)=1/2y(qx)+1/2y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.