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This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or \emph{"piecewise"}) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein-Uhlenbeck stochastic process (which is \emph{globally} stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager-Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear, and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager-Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the \emph{non-equilibrium self-consistent generalized Langevin equation} (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and the gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

We present a first-principles formalism for studying dynamical heterogeneities in glass forming liquids. Based on the Non-Equilibrium Self-Consistent Generalized Langevin Equation theory, we were able to describe the time-dependent local density profile during the particle interchange among small regions of the fluid. The final form of the diffusion equation contains both, the contribution of the chemical potential gradient written in terms of a coarse-grained density and a collective diffusion coefficient as well as the effect of a history-dependent mobility factor. With this diffusion equation we captured interesting phenomena in glass forming liquids such as the cases when a strong density gradient is accompanied with a very low mobility factor attributable to the denser part: in such circumstances the density profile falls into an arrested state even in the presence of a density gradient. On the other hand, we also show that above a certain critical temperature,which depends on the volume fraction, any density heterogeneity relaxes to a uniform state in a finite time, known as equilibration time. We further show that such equilibration time varies little with the temperature in diluted systems but can change drastically with temperature in concentrated systems.

We explore an idealized theoretical model for the transport of ions within highly asymmetric ionic liquid mixtures. A primitive model (PM)-inspired system serves as a representative for asymmetric ionic materials (such as liquid crystalline salts) which quench to form disordered, partially-arrested phases. Self-Consistent Generalized Langevin Equation (SCGLE) Theory is applied to understand the connection between the size ratio of charge-matched salts and their average mobility. Within this model, we identify novel glassy states where one of the two charged species (either the macro-cation or the micro-anion) are arrested, while the other retains mobility. We discuss how this result is useful in the development of novel single-ion conducting phases in ionic liquid based materials., Comment: 11 pages, 5 figures

The principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E \textbf{68}, 011405 (2003)] is employed to describe the interplay of the effects of varying the density n, the temperature T, and the softness (characterized by a softness parameter {\nu}^{-1}) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is that the dynamic parameters of these systems, such as the {\alpha}-relaxation time and the long-time self-diffusion coefficient, depend on n, T, and {\nu} only through the reduced density n^\ast \equiv n{\sigma}^{3}_{HS}(T, {\nu}),where the effective hard-sphere diameter {\sigma}_{HS}(T, {\nu}) is determined, for example, by the Andersen-Weeks-Chandler condition for soft-sphere-hard-sphere structural equivalence. A number of scaling properties observed in recent simulations involving glass-forming fluids with repulsive short range interactions are found to be a direct manifestation of this general dynamic equivalence principle. The self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics is shown to accurately capture these scaling rules

We report a systematic molecular dynamics study of the isochoric equilibration of hard-sphere fluids in their metastable regime close to the glass transition. The thermalization process starts with the system prepared in a non-equilibrium state with the desired final volume fraction {\phi} but with a prescribed non-equilibrium static structure factor S_0(k; {\phi}). The evolution of the {\alpha}- relaxation time {\tau}{\alpha} (k) and long-time self-diffusion coefficient DL as a function of the evolution time tw is then monitored for an array of volume fractions. For a given waiting time the plot of {\tau}{\alpha} (k; {\phi}, tw) as a function of {\phi} exhibits two regimes corresponding to samples that have fully equilibrated within this waiting time ({\phi} \leq {\phi}(c) (tw)), and to samples for which equilibration is not yet complete ({\phi} \geq {\phi}(c) (tw)). The crossover volume fraction {\phi}(c) (tw) increases with tw but seems to saturate to a value {\phi}(a) \equiv {\phi}(c) (tw \rightarrow \infty) \approx 0.582. We also find that the waiting time t^(eq)_w({\phi}) required to equilibrate a system grows faster than the corresponding equilibrium relaxation time, t^(eq)({\phi}) \approx 0.27 \times [{\tau}{\alpha} (k; {\phi})]^1.43, and that both characteristic times increase strongly as {\phi} approaches {\phi}^(a), thus suggesting that the measurement of equilibrium properties at and above {\phi}(a) is experimentally impossible.