We discuss anomalous relaxation processes in Davydov one-dimensional chain molecule that consists of an exciton and an acoustic phonon field as a thermal reservoir in the chain. We derive a kinetic equation for the exciton using the complex spectral representation of the Liouville-von Neumann operator. Due to the one-dimensionality, the momentum space separates into infinite sets of disjoint irreducible subspaces dynamically independent of one another. Hence, momentum relaxation occurs only within each subspace toward the Maxwell distribution. We obtain a hydrodynamic mode with transport coefficients, a sound velocity and a diffusion coefficient, defined in each subspace. Moreover, because the sound velocity has momentum dependence, phase mixing affects the broadening of the spatial distribution of the exciton in addition to the diffusion process. Due to the phase mixing the increase rate of the mean-square displacement of the exciton increases linearly with time and diverges in the long-time limit.
By employing the Dirac-Frenkel time-dependent variational principle, we study the dynamical properties of the Holstein molecular crystal model with diagonal and off-diagonal exciton-phonon coupling. A linear combination of the Davydov D$_1$ (D$_2$) Anstaz, referred to as the multi-D$_1$ Ansatz (multi-D$_2$ Ansatz), is used as the trial state with enhanced accuracy but without sacrificing efficiency. The time evolution of the exciton probability is found to be in perfect agreement with that of the hierarchy equations of motion, demonstrating the promise the multiple Davydov trial states hold as…
We propose a computational method to simulate anomalous self-diffusion in a simple liquid. The method is based on a molecular dynamics simulation on which we impose the following two conditions: firstly, the inter-particle interaction is described by a soft-core potential and secondly, the system is forced out of equilibrium. The latter can be achieved by subjecting the system to changes in the length scale at intermittent times. In many respects, our simulation system bears resemblance to slowly driven sandpile models displaying self-organised criticality. We find non-Gaussian single time ste…
The nonlinear theory of anomalous diffusion is based on particle interactions giving an explicit microscopic description of diffusive processes leading to sub-, normal, or super-diffusion as a result competitive effects between attractive and repulsive interactions. We present the explicit analytical solution to the nonlinear diffusion equation which we then use to compute the correlation function which is experimentally measured by correlation spectroscopy. The theoretical results are applicable in particular to the analysis of fluorescence correlation spectroscopy of marked molecules in biol…
We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement ($<r^2(t)>\sim t^γ$ with $0 <γ\leq 1.5$) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a p…
We study by molecular dynamics computer simulation a binary soft-sphere mixture that shows a pronounced decoupling of the species' long-time dynamics. Anomalous, power-law-like diffusion of small particles arises, that can be understood as a precursor of a double-transition scenario, combining a glass transition and a separate small-particle localization transition. Switching off small-particle excluded-volume constraints slows down, rather than enhances, small-particle transport. The data are contrasted with results from the mode-coupling theory of the glass transition.
The self-diffusion coefficient of a granular gas in the homogeneous cooling state is analyzed near the shearing instability. Using mode-coupling theory, it is shown that the coefficient diverges logarithmically as the instability is approached, due to the coupling of the diffusion process with the shear modes. The divergent behavior, which is peculiar of granular gases and disappears in the elastic limit, does not depend on any other transport coefficient. The theoretical prediction is confirmed by molecular dynamics simulation results for two-dimensional systems.
In a freely cooling granular material fluctuations in density and temperature cause position dependent energy loss. Due to strong local dissipation, pressure and energy drop rapidly and material moves from `hot' to `cold' regions, leading to even stronger dissipation and thus causing the density instability. The assumption of `molecular chaos' is valid only in the homogeneous cooling regime. As soon as the density instability occurs, the impact parameter is not longer uniformly distributed. The pair-correlation and the structure functions show that the molecular chaos assumption --- together w…
In this paper we study kinetically rough surfaces which display anomalous scaling in their local properties such as roughness, or height-height correlation function. By studying the power spectrum of the surface and its relation to the height-height correlation, we distinguish two independent causes for anomalous scaling. One is super-roughening (global roughness exponent larger than or equal to one), even if the spectrum behaves non anomalously. Another cause is what we term an intrinsically anomalous spectrum, in whose scaling an independent exponent exists, which induces different scaling p…
From three-dimensional linearized hydrodynamic equations, it is found that the heat conductivity is proportional to $(L_x/(L_y^2 L_z^2))^{1/3}$, where $L_x$, $L_y$ and $L_z$ are the lengths of the system along the $x$, $y$ and $z$ directions, and we consider the case in which $L_x \gg L_y, L_z$. The necessary condition for such a size dependence is derived as $ϕ\equiv L_x/(n^{1/2} L_y^{5/4} L_z^{5/4}) \gg 1$, where $ϕ$ is the critical condition parameter and $n$ is the number density. This size dependence of the heat conductivity has been confirmed by molecular dynamics simulation.
The silo discharge process is studied by molecular dynamics simulations. The development of the velocity profile and the probability density function for the displacements in the horizontal and vertical axis are obtained. The PDFs obtained at the beginning of the discharge reveal non-Gaussian statistics and superdiffusive behaviors. When the stationary flow is developed, the PDFs at shorter temporal scales are non-Gaussian too. For big orifices a well defined transition between ballistic and diffusive regime is observed. In the case of a small outlet orifice, no well defined transition is obse…
We provide molecular dynamics simulation of heat transport in one-dimensional molecular chains with different interparticle pair potentials. We show that the thermal conductivity is finite in the thermodynamic limit in the chains with the potential, which allows for bond dissociation. The Lennard-Jones, Morse and Coulomb potentials belong to such type of potentials. The convergence of the thermal conductivity is provided by phonon scattering on the locally stretched loose interatomic bonds at low temperature and by the many-particle scattering at high temperature. On the other hand, the chains…
There is a long-standing question about the molecular configuration of interfacial water molecules in the proximity of solid surfaces, particularly carbon atoms which play a crucial role in electrochemistry and biology. In this study, the dielectric, structural and dynamical properties of confined water placed between two parallel graphene walls at different inter distances from the Angstrom scale to few tens of nanometer have been investigated using molecular dynamics. For dielectric properties of water, we show that the perpendicular component of water dielectric constant drastically decreas…
The way tension propagates along a chain is a key to govern many of anomalous dynamics in macromolecular systems. After introducing the weak and the strong force regimes of the tension propagation, we focus on the latter, in which the dynamical fluctuations of a segment in a long polymer during its stretching process is investigated. We show that the response, i.e., average drift, is anomalous, which is characterized by the nonlinear memory kernel, and its relation to the fluctuation is nontrivial. These features are discussed on the basis of the generalized Langevin equation, in which the rol…
Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. A…
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also deriv…
