Non-Markovian diffusion and the adsorption-desorption process E. K. Lenzi, 1,2 C. A. R. Yednak, 1,2 and L. R. Evangelista 1 1 Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, PR, Brazil 2 Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Received 16 November 2009; published 13 January 2010 The non-Markovian diffusion of dispersed particles in a semi-infinite cell of an isotropic fluid limited by an adsorbing-desorbing surface is theoretically investigated. The density of dispersed particles in the bulk is a time dependent function and the time dependent density of surface particles is governed by a modified kinetic equation with a time dependent kernel. In this framework, the densities of bulk and surface particles are analytically determined, taking into account the conservation of the number of particles immersed in the sample. This system exhibits anomalous diffusion behavior as well as memory effects in the adsorption- desorption process. The results obtained here are expected to be useful to investigate the adsorption-desorption phenomena of neutral as well as charged particles in an isotropic fluid in contact with a solid substrate when the anomalous diffusion is present. DOI: 10.1103/PhysRevE.81.011116 PACS numbers: 05.40.-a, 68.43.Mn, 66.10.C-, 47.57.J- I. INTRODUCTION One of the most common phenomena in the nature is the diffusion which may be describe in terms of an stochastic process. The kind of the diffusion process is related to the properties of the medium and depends on the conditions im- posed to the system. This manner, the diffusion may have an usual behavior where the main characteristic is the square mean displacement with a linear dependence on time, i.e., z - z 2  t 1, which reflects the Markovian nature of this process or an anomalous behavior 2. The last case usually occurs when the system present non-Markovian aspects such as, for example, memory effects 3,4, long-range correla- tions, and long-range interactions 5–7. A direct conse- quence is an anomalous spreading of the system manifested by the square mean displacement which has not linear de- pendence on time or, in the case of Lévy distributions 8, is not finite. These situations can be found, for instance, in atom deposition into a porous substrate 9, diffusion of high molecular weight polyisopropylacrylamide in nanopores 10, highly confined hard disk fluid mixtures 11, fluctuat- ing particle fluxes 12, diffusion on fractals 13, ferrofluid 14, and colloids 15. Different diffusive behaviors may also be exhibited by these systems such as the ones found in 6,7, for the case of long-range interaction, and in 16,17, for active intracellular transport. This rich variety of situa- tions may be investigated by several formalisms 2–4,18–20, in particular, by extensions of the diffusion equation such as the fractional diffusion equations 3,4,21 or by incorporating spatial and time dependence on the diffu- sion coefficient. The conditions imposed to the system may also lead us to an anomalous diffusion 22. A typical situa- tion may be found in adsorption-desorption process by a sur- face, governed by a typical balance equation characterizing a chemical reaction of first kind Langmuir’s approximation, when the conservation of the number of particles is imposed 22–24. For this case, the results for the momentum distri- bution 22 show that the system exhibits anomalous diffu- sion 2 behavior, according to the values of the characteris- tic times entering the problem. More precisely, not only a single subdiffusive or superdiffusive motion can be found, but the system presents a multiple behavior that includes both modes of subdiffusion and superdiffusion and, for large times, the normal diffusive behavior. A system presenting similar behavior can be found in the dynamics of vesicles driven by adhesion gradients of a Langmuir monolayer 25. In this scenario, the diffusion process of suspended or dis- persed particles in an anisotropic media like liquid crystals 26 may be found suitable description. In fact, dispersions of particles in an anisotropic host medium such as nematic liquid crystals are responsible for a series of different physi- cal scenarios deserving a more fundamental explanation 26. Among the systems recently considered, it is worth mentioning the suspension of magnetic grains 27, silica spheres 28, latex particles 29, oil droplets 30. All the bulk effects in these systems have been object of attention in the last few years. In this paper, we investigate the adsorption-desorption process of a system governed by an unusual diffusion equa- tion which may recover, by suitable choice, several situations such as the fractional diffusion equations 2–4, fractional diffusion equations of distributed order 31,32 and Cattaneo equation 33,34. We also consider here a modified kinetic equation in which by a suitable choice for a temporal kernel in the desorption rate can account for the relative importance of physisorption or of chemisorption 23,24, according to the time scale governing the adsorption phenomena. Indeed, more realistic descriptions of the kinetics at the interface in the framework of a first-order chemical reaction should be developed by taking into account both the chemisorption and the physisorption processes, because the actual position of the molecule on the surface can have a memory of its incom- ing state, eventually modifying the adsorption-desorption rates. For this context, we found exact solutions in the Laplace space and analyzed some representative situations. These developments are performed in Sec. II while in Sec. III our discussions and conclusions are presented. II. PROBLEM To mathematically formulate the problem we consider a typical geometry for the sample such that the Cartesian ref- PHYSICAL REVIEW E 81, 011116 2010 1539-3755/2010/811/0111167 ©2010 The American Physical Society 011116-1