Using molecular dynamics simulations and theoretical analysis of velocity-autocorrelation functions, we study ion transport mechanisms in typical room-temperature ionic liquids. We show that ions may reside in two states: free and bound with an interstate exchange. We investigate quantitatively the exchange process and reveal new qualitative features of this process. To this end, we propose a dynamic criterion for free and bound ions based on the ion trajectory density and demonstrate that this criterion is consistent with a static one based on interionic distances. Analyzing the trajectories of individual cations and anions, we estimate the time that ions spend in bound “clustered states” and when they move quasifreely. Using this method, we evaluate the average portion of “free” ions as approximately 15%–25%, increasing with temperature in the range of 300–600 K. The ion diffusion coefficients and conductivities as a function of the temperature calculated from the velocity and electrical-current autocorrelation functions reproduce the reported experimental data very well. The experimental data for the direct-current conductivity (constant ionic current) is in good agreement with theoretical predictions of the Nernst-Einstein equation based on the concentrations and diffusion coefficients of free ions obtained in our simulations. In analogy with electronic semiconductors, we scrutinize an “ionic semiconductor” model for ionic liquids, with valence and conduction “bands” for ions separated by an energy gap. The obtained band gap for the ionic liquid is small, around 26 meV, allowing for easy interchange between the two dynamic states. Moreover, we discuss the underscreening paradox in the context of the amount of free charge carriers, showing that the obtained results do not yet approve its simplistic resolution.

We discuss the situations under which Brownian yet non-Gaussian (BnG) diffusion can be observed in the model of a particle’s motion in a random landscape of diffusion coefficients slowly varying in space (quenched disorder). Our conclusion is that such behavior is extremely unlikely in the situations when the particles, introduced into the system at random at t = 0, are observed from the preparation of the system on. However, it indeed may arise in the case when the diffusion (as described in Ito interpretation) is observed under equilibrated conditions. This paradigmatic situation can be translated into the model of the diffusion coefficient fluctuating in time along a trajectory, i.e. into a kind of the ‘diffusing diffusivity’ model.

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized Lévy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to nonspectral relaxation can be considered exotic, for generalized OUPs driven by Lévy noise, such initial conditions are the rule.