Effect of the thermodynamic factor on the intrinsic and tracer diffusivities in binary mixtures.

One of the Darken equations gives a relationship between the intrinsic and the tracer diffusion coefficients, D_{A} and D_{A}^{*}, of species A in a solid binary mixture. In its original formulation, the equation reads D_{A}=D_{A}^{*}Γ, with Γ the thermodynamic factor. The question addressed in this paper is how D_{A} and D_{A}^{*} depend separately on Γ. Using a recent result for transition probabilities in terms of the excess chemical potential [M. Di Muro and M. Hoyuelos, Phys. Rev. E 104, 044104 (2021)2470-004510.1103/PhysRevE.104.044104], it is shown that the intrinsic diffusivity does not depend on Γ. This approach simplifies a previous theoretical analysis that reaches the same result. Experimental results of diffusion in Ni-Pd and Fe-Pd alloys [M. J. H. van Dal et al., Acta Mater. 48, 385 (2000)1359-645410.1016/S1359-6454(99)00375-4] are used to check the theory. Numerical simulations of Ni-Pd were performed to show that the migration energy is the main factor responsible for the increase in diffusivity at intermediate concentrations.

Dependence on the thermodynamic state of self-diffusion of pseudo-hard-sphere and Lennard-Jones potentials.

Self-diffusion D in a system of particles that interact with a pseudo-hard-sphere or a Lennard-Jones potential is analyzed. Coupling with a solvent is represented by a Langevin thermostat, characterized by the damping time t_{d}. The hypotheses that D=D_{0}φ is proposed, where D_{0} is the small concentration diffusivity and φ is a thermodynamic function that represents the effects of interactions as concentration is increased. Molecular dynamics simulations show that different values of the noise intensity modify D_{0}, but do not have an effect on φ. This result is consistent with the assumption that φ is a thermodynamic function since the thermodynamic state is not altered by the presence of damping and noise.

Diffusion on a lattice: Transition rates, interactions, and memory effects.

We analyze diffusion of particles on a two-dimensional square lattice. Each lattice site contains an arbitrary number of particles. Interactions affect particles only in the same site, and are macroscopically represented by the excess chemical potential. In a recent work, a general expression for transition rates between neighboring cells as functions of the excess chemical potential was derived. With transition rates, the mean-field tracer diffusivity, D^{MF}, is immediately obtained. The tracer diffusivity, D=D^{MF}f, contains the correlation factor f, representing memory effects. An analysis of the joint probability of having given numbers of particles at different sites when a force is applied to a tagged particle allows an approximate expression for f to be derived. The expression is applied to soft core interaction (different values for the maximum number of particles in a site are considered) and extended hard core.

Transition rates of noninteracting quantum particles from the Widom insertion formula.

Transition rates among different states in a system of noninteracting quantum particles in contact with a heat reservoir include the factor 1∓n[over ¯]_{i}, with a minus sign for fermions and a plus sign for bosons, where n[over ¯]_{i} is the average occupation number of the final state. It is shown that this factor can be related to the difference of the chemical potential from that of an ideal classical mixture; this difference is formally equivalent to the excess chemical potential in a classical system of interacting particles. Using this analogy, Widom's insertion formula is used in the calculation of transition rates. The result allows an alternative derivation of quantum statistics from the condition that transition rates depend only on the number of particles in the target energy level. Instead, if transition rates depend on the particle number only in the origin level, the statistics of ewkons is obtained; this is an exotic statistics that can be applied to the description of dark energy.

Application of the Widom insertion formula to transition rates in a lattice.

We consider diffusion of particles on a lattice in the so-called dynamical mean-field regime (memory effects are neglected). Interactions are local, that is, only among particles at the same lattice site. It is shown that a statistical mechanics analysis that combines detailed balance and Widom's insertion formula allows for the derivation of an expression for transition rates in terms of the excess chemical potential. The rates reproduce the known dependence of self-diffusivity as the inverse of the thermodynamic factor. Soft-core interactions and general forms of the excess chemical potential (linear, quadratic, and cubic with the density) are considered.

Out-of-equilibrium Monte Carlo simulations of a classical gas with Bose-Einstein statistics.

Algorithms to determine transition probabilities in Monte Carlo simulations are tested using a system of classical particles with effective interactions which reproduce Bose-Einstein statistics. The system is appropriate for testing different Monte Carlo simulation methods in out-of-equilibrium situations since nonequivalent results are produced. We compare mobility numerical results obtained with transition probabilities derived from Glauber and Metropolis algorithms. Then, we compare these with a recent method, the interpolation algorithm, appropriate for nonequilibrium systems in homogeneous substrata and without phase transitions. The results of mobility obtained from the interpolation algorithm are qualitatively verified with molecular dynamics simulations for low concentrations.

Diffusion in binary mixtures: An analysis of the dependence on the thermodynamic factor.

We study the diffusion process in binary mixtures using transition probabilities that depend on a mean-field potential. This approach reproduces the Darken equation, a relationship between the intrinsic and the tracer diffusion coefficients, D_{A} and D_{A}^{*}, through the thermodynamic factor Φ (a function of the derivative of the activity coefficient against the molar fraction). The mean-field approach allows us to go beyond the Darken equation and separately specify the dependence of D_{A} and D_{A}^{*} on the thermodynamic factor. We obtain that Φ appears in the expression for D_{A}^{*}, but the intrinsic diffusivity D_{A} turns out to be independent of Φ. Experimental results taken from the literature on diffusion in metal alloys are consistent with this theoretical prediction.

Mean-field approach to diffusion with interaction: Darken equation and numerical validation.

A mean-field theory for diffusion with interaction was introduced in Phys. Rev. E 92, 062118 (2015)PLEEE81539-375510.1103/PhysRevE.92.062118. Interaction effects are represented with a mean-field potential. Here we show that the potential can be directly related to the activity coefficient. In this context, we obtain an alternative derivation of the Darken equation, that relates collective diffusion coefficient and single particle diffusion coefficient (generally different in the presence of interactions). We also carry out a validation test of the model using, as a case study, effective interactions that reproduce Bose-Einstein statistics.

Current of interacting particles inside a channel of exponential cavities: Application of a modified Fick-Jacobs equation.

Recently a nonlinear Fick-Jacobs equation has been proposed for the description of transport and diffusion of particles interacting through a hard-core potential in tubes or channels of varying cross section [Suárez et al., Phys. Rev. E 91, 012135 (2015)]PLEEE81539-375510.1103/PhysRevE.91.012135. Here we focus on the analysis of the current and mobility when the channel is composed by a chain of asymmetric cavities and a force is applied in one or the opposite direction, for both interacting and noninteracting particles, and compare analytical and Monte Carlo simulation results. We consider a cavity with a shape given by exponential functions; the linear Fick-Jacobs equation for noninteracting particles can be exactly solved in this case. The results of the current difference (when a force is applied in opposite directions) are more accurate for the modified Fick-Jacobs equation for particles with hard-core interaction than for noninteracting ones.

Quantum statistics of classical particles derived from the condition of a free diffusion coefficient.

We derive an equation for the current of particles in energy space; particles are subject to a mean-field effective potential that may represent quantum effects. From the assumption that noninteracting particles imply a free diffusion coefficient in energy space, we derive Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics. Other new statistics are associated to a free diffusion coefficient; their thermodynamic properties are analyzed using the grand partition function. A negative relation between pressure and energy density for low temperatures can be derived, suggesting a possible connection with cosmological dark energy models.

Transport of interacting particles in a chain of cavities: description through a modified Fick-Jacobs equation.

We study the transport process of interacting Brownian particles in a tube of varying cross section. To describe this process we introduce a modified Fick-Jacobs equation, considering particles that interact through a hard-core potential. We were able to solve the equation with numerical methods for the case of symmetric and asymmetric cavities. We focused in the concentration of particles along the direction of the tube. We also preformed Monte Carlo simulations to evaluate the accuracy of the results, obtaining good agreement between theory and simulations.

Mean-field approach for diffusion of interacting particles.

A nonlinear Fokker-Planck equation is obtained in the continuous limit of a one-dimensional lattice with an energy landscape of wells and barriers. Interaction is possible among particles in the same energy well. A parameter γ, related to the barrier's heights, is introduced. Its value is determinant for the functional dependence of the mobility and diffusion coefficient on particle concentration, but has no influence on the equilibrium solution. A relation between the mean-field potential and the microscopic interaction energy is derived. The results are illustrated with classical particles with interactions that reproduce fermion and boson statistics.

Dynamics of localized structures in vectorial waves

Dynamical properties of topological defects in a two dimensional complex vector field are considered. These objects naturally arise in the study of polarized transverse light waves. Dynamics is modeled by a vector complex Ginzburg-Landau equation with parameter values appropriate for linearly polarized laser emission. Creation and annihilation processes, and self-organization of defects in lattice structures, are described. We find "glassy" configurations dominated by vectorial defects and a melting process associated with topological-charge unbinding.