Showing papers on "Einstein relation published in 2018"

Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity.

Abstract: How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

Diffusion for Holographic Lattices

Abstract: We consider black hole spacetimes that are holographically dual to strongly coupled field theories in which spatial translations are broken explicitly. We discuss how the quasinormal modes associated with diffusion of heat and charge can be systematically constructed in a long wavelength perturbative expansion. We show that the dispersion relation for these modes is given in terms of the thermoelectric DC conductivity and static susceptibilities of the dual field theory and thus we derive a generalised Einstein relation from Einstein’s equations. A corollary of our results is that thermodynamic instabilities imply specific types of dynamical instabilities of the associated black hole solutions.

Continuous-time random walks and Fokker-Planck equation in expanding media

Abstract: We consider a separable continuous-time random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates the probability distribution function (pdf) of finding a particle at a given position and time to the single-step jump length and waiting time pdfs is provided. The equation takes the form of a generalized Fokker-Planck equation when the jump length pdf of the particle has a finite variance. This generalized equation becomes a fractional Fokker-Planck equation in the case of a heavy-tailed waiting time pdf. These equations allow us to study the relationship between expansion, diffusion, and external force. We establish the conditions under which the dominant contribution to transport stems from the diffusive transport rather than from the drift due to the medium expansion. We find that anomalous diffusion processes under a constant external force in an expanding medium described by means of our continuous-time random walk model violate the generalized Einstein relation and lead to propagators that are qualitatively different from the ones found in a static medium. Our results are supported by numerical simulations.

Fick and Fokker--Planck diffusion law in inhomogeneous media.

Abstract: We discuss diffusion of particles in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un--oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker--Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart of reversibility is encoded into a tensor metrics and a positive function. The Fick's law with inhomogeneous diffusion matrix is obtained in the case when the spatial inhomogeneity is associated exclusively with the edge weights. We discuss also some related properties of the systems like a non-homogeneous Einstein relation and the possibility of uphill diffusion.

Sudden removal of a static force in a disordered system: Induced dynamics, thermalization, and transport

Abstract: We study the real-time dynamics of local occupation numbers in a one-dimensional model of spinless fermions with a random on-site potential for a certain class of initial states. The latter are thermal (mixed or pure) states of the model in the presence of an additional static force, but become non-equilibrium states after a sudden removal of this static force. For this class and high temperatures, we show that the induced dynamics is given by a single correlation function at equilibrium, independent of the initial expectation values being prepared close to equilibrium (by a weak static force) or far away from equilibrium (by a strong static force). Remarkably, this type of universality holds true in both, the ergodic phase and the many-body localized regime. Moreover, it does not depend on the specific choice of a unit cell for the local density. We particularly discuss two important consequences. First, the long-time expectation value of the local density is uniquely determined by the fluctuations of its diagonal matrix elements in the energy eigenbasis. Thus, the validity of the eigenstate thermalization hypothesis is not only a sufficient but also a necessary condition for thermalization. Second, the real-time broadening of density profiles is always given by the current autocorrelation function at equilibrium via a generalized Einstein relation. In the context of transport, we discuss the influence of disorder for large particle-particle interactions, where normal diffusion is known to occur in the disorder-free case. Our results suggest that normal diffusion is stable against weak disorder, while they are consistent with anomalous diffusion for stronger disorder below the localization transition. Particularly, for weak disorder, Gaussian density profiles can be observed for single disorder realizations, which we demonstrate for finite lattices up to 31 sites.

Two-tag correlations and nonequilibrium fluctuation-response relation in ageing single-file diffusion.

Abstract: Spatiotemporally correlated motions of interacting Brownian particles, confined in a narrow channel of infinite length, are studied in terms of statistical quantities involving two particles. A theoretical framework that allows analytical calculation of two-tag correlations is presented on the basis of the Dean-Kawasaki equation describing density fluctuations in colloidal systems. In the equilibrium case, the time-dependent Einstein relation holds between the two-tag displacement correlation and the response function corresponding to it, which is a manifestation of the fluctuation-dissipation theorem for the correlation of density fluctuations. While the standard procedure of closure approximation for nonlinear density fluctuations is known to be obstructed by inconsistency with the fluctuation-dissipation theorem, this difficulty is naturally avoided by switching from the standard Fourier representation of the density field to the label-based Fourier representation of the vacancy field. In the case of ageing dynamics started from equidistant lattice configuration, the time-dependent Einstein relation is violated, as the two-tag correlation depends on the waiting time for equilibration while the response function is not sensitive to it. Within linear approximation, however, there is a simple relation between the density (or vacancy) fluctuations and the corresponding response function, which is valid even if the system is out of equilibrium. This non-equilibrium fluctuation-response relation can be extended to the case of nonlinear fluctuations by means of closure approximation for the vacancy field.

Steady States, Fluctuation-Dissipation Theorems and Homogenization for Reversible Diffusions in a Random Environment

Abstract: Prolongating our previous paper on the Einstein relation, we study the motion of a particle diffusing in a random reversible environment when subject to a small external forcing. In order to describe the long time behavior of the particle, we introduce the notions of steady state and weak steady state. We establish the continuity of weak steady states for an ergodic and uniformly elliptic environment. When the environment has finite range of dependence, we prove the existence of the steady state and weak steady state and compute its derivative at a vanishing force. Thus we obtain a complete ‘fluctuation–dissipation Theorem’ in this context as well as the continuity of the effective variance.

Investigation of a chaotic thermostat

Abstract: A numerical study is presented of a free particle interacting with a deterministic thermostat in which the usual friction force is supplemented with a fluctuating force that depends on the self-consistent damping coefficient associated with coupling to the heat bath. It is found that this addition results in a chaotic environment in which a particle self-heats from rest and moves in positive and negative directions, exhibiting a characteristic diffusive behavior. The frequency power spectrum of the dynamical quantities displays the exponential frequency dependence ubiquitous to chaotic dynamics. The velocity distribution function approximates a Maxwellian distribution, but it does show departures from perfect thermal equilibrium, while the distribution function for the damping coefficient shows a closer fit. The behavior for the classic Nose-Hoover (NH) thermostat is compared to that of the enlarged Martyna-Klein-Tuckerman (MKT) model. Over a narrow amplitude range, the application of a constant external force results quantitatively in the Einstein relation for the NH thermostat, and for the MKT model it differs by a factor of 2.

Macroscopic magnetic self-assembly

Abstract: Exploring the macroscopic scale's similarities to the microscale is part and parcel of this thesis as reflected in the research question: what can we learn about the microscopic scale by studying the macroscale? Investigations of the environment in which the self-assembly takes place, and the self-assembly itself helps to answer this question. We mimicked the microscale and identified several analogue parameters. Instead of heat we use turbulence, instead of microscopic we use centimeter-sized particles. Gravity was counteracted by anupward directed water flow since its influence on macroscopic particles is considerable but has only a minor influence on microscopic particles. Likewise heat has a great influence on the microscopic scale but a minor influence on macroscopic particles. Turbulence proved to be an accurate representation for heat and was modelled as if on a microscopic scale, applying thermodynamical concepts such as Brownian motion, diffusion, kinetics and the Einstein relation. Those concepts proved suitable also on the macroscopic scale. Particle velocity is Maxwell-Boltzmann distributed and the average squared displacement is in agreement with a confined random walk. The diffusion coefficient and velocity is independent on particle size. This leads to the interpretation that the motion of a single centimeter-size sphere resembles the motion of a microscopic particle in that it conducts a random walk and Brownian motion. To visualize micro- or nanoscopic particles electron- or light-microscopy is often used. Instead of microscopes we used video cameras to record the experiments with centimeter sized particles. A swimmingpool pump and asymmetric inflow is used to create upward flow and turbulence. The asymmetric inflow causes large macroscopic swirls representing the applied heat level at the microscale. In the microscopic case the Brownian motion of particles is result of propagating heat originating at its source whereas at the macroscopic scale the vortice propagation originating in the asymmetry of flow cause the Brownian motion of large particles. Despite of those analogies between heat and turbulence the values for the disturbing energy varies considerably depending on if they were determined via single sphere diffusion (Einstein relation) and velocity (kinetic energy) or via two sphere interactions over distance. The latter case is an order of magnitude lower, approximately 6.5mJ compared to 80mJ. This suggests that the heat or turbulence energy spectra may differ with respect to its action on the particle(s). There is a directional dependency of particle velocity, diffusion and disturbing energy. The horizontal dimensions are similar but the vertical component show a stronger dependency with respect to flow asymmetry and turbulence. The directional dependency can most likely be counteracted via future technical adjustments. It can also be interpreted as a temperature gradient. Self-assembly was studied via structure formation of multiple magnetic spheres or twelve heptagonal magnetic platelets by systematic variation of turbulence and asymmetry. The multiple magnetic spheres form lines and rings and their occurance were in accordance with theory, however the absolute energies of the structures deviated from theory. For experiments with increasing number of spheres, four spheres represents a transition between lines and rings. The system proved to seak for the minimum energy structure which again makes the our macroscopic system behave similar to the microscale. Turbulence acted in a similar way as heat since almost only individual particles were observed at high turbulence whereas lines and rings formed as turbulence decreased which resembles a phase transition between a liquid and a solid or a gas and a liquid. Self-assembly of twelve centimeter-sized pentagonal platelets showed the same energy minimum seaking behavior. A complete self-assembly of the dodecahedron was not achieved. Predominantley intermediate structures with maximum contacts to each particle formed (trimer and tetramer etc.) which is the minimum energy structure. Also in this more complex case the system prove to behave similar to the microscale. The two examples of self-assembly represent on the one hand formation of simple structures (rings and lines) and on the other hand a more complex case of self-assembly (a hollow dodecahedron). The later example can be interpreted as self-assembly of geometrical construct or as a representation of self-assembly of a spherical virus. This underlines the potential of macroscopic self-assembly; it can be used in the investigation of general largely scale-independent problems or as an analogue representation in the investigations of natural occuring phenomena.

Anomalous charged-particle transport in organic and soft-condensed matter

Abstract: A general phase-space kinetic model for non-equilibrium charged particle transport through combined localised and delocalised states is presented that accounts for scattering, trapping/detrapping and recombination loss processes in organic and soft-condensed matter. The model takes the form of a generalised Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. A Chapman-Enskog-type perturbative solution technique is also applied, confirming the analytical results and highlighting the emergence of a density gradient series representation in the weak-gradient hydrodynamic regime. This representation validates Fick's law for this model, providing expressions for the flux transport coefficients of drift velocity and diffusion. By applying Fick's law, a generalised diffusion equation with a unique global time operator is shown to arise that coincides with both the standard diffusion equation and the Caputo fractional diffusion equation in the respective limits of normal and dispersive transport. A subordination transformation is used to efficiently solve the generalised diffusion equation by mapping from the solution of a corresponding classical diffusion equation. From the aforementioned density gradient expansion, we extend Fick's law to consider also the third-order transport coefficient of skewness. This extension is in turn applied to yield a corresponding generalised advection-diffusion-skewness equation. Negative skewness is observed and a physical interpretation is provided in terms of the processes of trapping and detrapping. By analogy with the generalised Einstein relation, a relationship between skewness, diffusion, mobility and temperature is also formed. The phase-space model is generalised further by introducing energy-dependence in the collision, trapping and loss frequencies. The solution of this resulting model is explored indirectly through balance equations for particle continuity, momentum and energy. Generalised Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field-dependence of the mobility. Interesting phenomena such as negative differential conductivity (NDC) and recombination heating/cooling are shown to arise from recombination loss processes and the localised and delocalised nature of transport. Fractional generalisations of the GER and mobility are also explored. Finally, a planar organic semiconductor device simulation is presented that makes use of the aforementioned generalised advection-diffusion equation to account for the trapping and detrapping of charge carriers. In this simulation, we use Poisson's equation to account for space-charge effects and Kirchhoff's circuit laws to account for RC effects. These considerations allow for a variety of charge transport experiments to be simulated in a planar geometry, including time of flight (TOF), charge extraction by linearly increasing voltage (CELIV) and resistance-dependent photovoltage (RPV) experiments. The simulation is used to explore a proposed experimental technique for the characterisation of the recombination coefficient, as well as to study what effects traps would have on the measured current.

Diffusion of a particle in the Gaussian random-energy landscape: Einstein relation and analytical properties of average velocity and diffusivity as functions of driving force

Abstract: We demonstrate that the Einstein relation for the diffusion of a particle in the random-energy landscape with the Gaussian density of states is an exclusive one-dimensional property and does not hold in higher dimensions. We also consider the analytical properties of the particle velocity and diffusivity for the limit of weak driving force and establish a connection between these properties and dimensionality and spatial correlation of the random-energy landscape.

Diffusion of a particle in the spatially correlated exponential random energy landscape: Transition from normal to anomalous diffusion.

Abstract: Diffusive transport of a particle in a spatially correlated random energy landscape having exponential density of states has been considered. We exactly calculate the diffusivity in the nondispersive quasi-equilibrium transport regime for the 1D transport model and found that for slow decaying correlation functions the diffusivity becomes singular at some particular temperature higher than the temperature of the transition to the true non-equilibrium dispersive transport regime. It means that the diffusion becomes anomalous and does not follow the usual ∝ t1/2 law. In such situation, the fully developed non-equilibrium regime emerges in two stages: first, at some temperature there is the transition from the normal to anomalous diffusion, and then at lower temperature the average velocity for the infinite medium goes to zero, thus indicating the development of the true dispersive regime. Validity of the Einstein relation is discussed for the situation where the diffusivity does exist. We provide also some ...