Generalized Einstein Relation Research Articles

Efficient single-run implementation of generalized Einstein relation to compute transport coefficients: A binary-based time sampling.

A robust and simple implementation of the generalized Einstein formulation using single equilibrium molecular dynamics simulation is introduced to compute diffusion and shear viscosity. The unique features underlying this framework are as follows: (1) The use of a simple binary-based method to sample time-dependent transport coefficients results in a uniform distribution of data on a logarithmic time scale. Although we sample "on-the-fly," the algorithm is readily applicable for post-processing analysis. Overlapping same-length segments are not sampled as they indicate strong correlations. (2) Transport coefficients are estimated using a power law fitting function, a generalization of the standard linear relation, that accurately describes the long-time plateau. (3) The use of a generalized least squares (GLS) fitting estimator to explicitly consider correlations between fitted data points results in a reliable estimate of the statistical uncertainties in a single run. (4) The covariance matrix for the GLS method is estimated analytically using the Wiener process statistics and computed variances. (5) We provide a Python script to perform the fits and automate the procedure to determine the optimal fitting domain. The framework is applied to two fluids, binary hard sphere and a Lennard-Jones near the triple point, and the validity of the single-run estimates is verified against multiple independent runs. The approach should be applicable to other transport coefficients since the diffusive limit is universal to all of them. Given its rigor and simplicity, this methodology can be readily incorporated into standard molecular dynamics packages using on-the-fly or post-processing analysis.

A broad class of nonlinear Langevin equations with drift and diffusion coefficients separable in time and space: generalized n-moment, ergodicity, Einstein relation, and fluctuations of the system

An extensive class of nonlinear Langevin equations with drift and diffusion coefficients separable in time and space driven by Gaussian white noise is analyzed in terms of a generalized n-moment. We show that the system may exhibit an ergodic property, a key property in statistical mechanics, for space-time-dependent drift and diffusion coefficients. A generalized Einstein relation is also obtained. We also demonstrate that the first two generalized moments and variances are useful to describe the drift and fluctuations of the system.

Decoding the hydrodynamic properties of microscale helical propellers from Brownian fluctuations

The complex motility of bacteria, ranging from single-swimmer behaviors such as chemotaxis to collective dynamics, including biofilm formation and active matter phenomena, is driven by their microscale propellers. Despite extensive study of swimming flagellated bacteria, the hydrodynamic properties of their helical-shaped propellers have never been directly measured. The primary challenges to directly studying microscale propellers are 1) their small size and fast, correlated motion, 2) the necessity of controlling fluid flow at the microscale, and 3) isolating the influence of a single propeller from a propeller bundle. To solve the outstanding problem of characterizing the hydrodynamic properties of these propellers, we adopt a dual statistical viewpoint that connects to the hydrodynamics through the fluctuation-dissipation theorem (FDT). We regard the propellers as colloidal particles and characterize their Brownian fluctuations, described by 21 diffusion coefficients for translation, rotation, and correlated translation-rotation in a static fluid. To perform this measurement, we applied recent advances in high-resolution oblique plane microscopy to generate high-speed volumetric movies of fluorophore-labeled, freely diffusing Escherichia coli flagella. Analyzing these movies with a bespoke helical single-particle tracking algorithm, we extracted trajectories, calculated the full set of diffusion coefficients, and inferred the average propulsion matrix using a generalized Einstein relation. Our results provide a direct measurement of a microhelix's propulsion matrix and validate proposals that the flagella are highly inefficient propellers, with a maximum propulsion efficiency of less than 3%. Our approach opens broad avenues for studying the motility of particles in complex environments where direct hydrodynamic approaches are not feasible.

Generalized Einstein relation in disordered organic semiconductors: Influence of the acoustic phonons–charge carriers scattering

In this work, we analyze the generalized Einstein relation for disordered organic semiconductors with a non-equilibrium Druyvesteyn-type distribution function. The Druyvesteyn behavior of hot electrons in a solid state is associated with the acoustic phonons–charge carriers scattering. Such a case has been experimentally demonstrated in electroluminescent inorganic rare–earth–doped zinc chalcogenides. Therefore, we can assume that, in a part of organic materials used in organic light-emitting diodes (OLEDs), we can also find the Druyvesteyn-type distribution of charge carriers under external electric fields. It looks that the electric-field-dependent diffusion coefficient, which is observed in the phonon-induced hopping electron transport, plays a key role here. The theoretical analysis of the Einstein relation shows that for lower concentrations of charge carriers, the diffusivity–mobility ratio (D/μ) reaches a value lower than 1 (in kT/q units). The temperature dependencies of D/μ are similar to the tendencies reported for the equilibrium conditions. The obtained satisfactory agreement between experimental results and calculations based on the Druyvesteyn-type distribution function confirms the usefulness of the presented model.

Linear Response and Fluctuation-Dissipation Relations for Brownian Motion under Resetting.

We consider fluctuation-dissipation relations (FDRs) for a Brownian motion under renewal resetting with arbitrary waiting time distribution between the resetting events. We show that if the distribution of waiting times of the resetting process possesses the second moment, the usual (generalized) FDR and the equivalent generalized Einstein's relation (GER) apply for the response function of the coordinate. If the second moment of waiting times diverges but the first one stays finite, the static susceptibility diverges, the usual FDR breaks down, but the GER still applies. In any of these situations, the fluctuation dissipation relations define the effective temperature of the system which is twice as high as the temperature of the medium in which the Brownian motion takes place.

Different effects of external force fields on aging Lévy walk.

Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time ta for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of the Lévy walk and the generalized Green-Kubo formula, we investigate the quantities that include the ensemble- and time-averaged mean-squared displacements in both weak aging ta≪t and strong aging ta≫t cases and compare them to the ones in the absence of any force field. Two typical force fields, constant force F and time-dependent periodic force F(t)=f0sin⁡(ωt), are considered for comparison. The generalized Einstein relation is also discussed in the case with the constant force. We find that the constant force is the key to causing the aging phenomena and enhancing the diffusion behavior of the aging Lévy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging Lévy walk are verified by both theoretical analyses and numerical simulations.

Tunable metallic-like transport in polypyrrole

Abstract Conjugated polymers (CPs), organic macromolecules with a linear backbone of alternating C–C and C=C bonds, possess unique semiconductive properties, providing new opportunities for organic electronics, photonics, information, and energy devices. Seeking the metallic or metallic-like, even superconducting properties beyond semiconductivity in CPs is always one of the ultimate goals in polymer science and condensed matter. Only two metallic and semi-metallic transport cases—aniline-derived polyaniline and thiophene-derived poly(3,4-ethylenedioxythiophene)—have been reported since the development of CPs for four decades. Controllable synthesis is a key challenge in discovering more cases. Here we report the metallic-like transport behavior of another CP, polypyrrole (PPy). We observe that the transport behavior of PPy changes from semiconductor to insulator-metal transition, and gradually realizes metallic-like performance when the crystalline degree increases. Using a generalized Einstein relation model, we rationalized the mechanism behind the observation. The metallic-like transport in PPy demonstrates electron strong correlation and phonon–electron interaction in soft condensation matter, and may find practical applications of CPs in electrics and spintronics.

Generalized Einstein Relation in Disordered Organic Semiconductors: Influence of the Charge Carriers Scattering on Acoustic Phonons

Generalized Einstein Relation in Disordered Organic Semiconductors: Influence of the Charge Carriers Scattering on Acoustic Phonons

Diffusion coefficient of charge carriers in disordered semiconductors retaining a combination of exponential and Gaussian mobility-gap states: Application to amorphous selenium

Charge carrier transport in disordered semiconductors is highly influenced by the defect states near the mobility edges. A theoretical model for the generalized Einstein relation, namely, the diffusivity-mobility ratio, for disordered semiconductors retaining a combination of exponential and Gaussian mobility-gap states with square-root distribution of extended states, is presented in this article. The conditions for determining the diffusion coefficient of charge carriers in disordered semiconductors from the Einstein relation are described in the article. The effects of various parameters constituting the density of states (DOS) distribution on the Einstein relation are examined. The results show that the diffusivity-mobility ratio for such DOS distribution substantially deviates from the traditional constant value for carrier concentration larger than 1010 cm−3. The value of diffusivity-mobility ratio strongly depends on the amount, energy position, and the shape of the Gaussian peaks. The additional diffusion coefficient due to multiple trapping in disordered semiconductors (namely, field diffusion) under quasi-equilibrium transport is also examined as a function of electric field and carrier concentration.

Physics of Selective Conduction and Point Mutation in Biological Ion Channels.

We introduce a statistical and linear response theory of selective conduction in biological ion channels with multiple binding sites and possible point mutation. We derive an effective grand-canonical ensemble and generalized Einstein relations for the selectivity filter, assuming strongly coordinated ionic motion, and allowing for ionic Coulomb blockade. The theory agrees well with data from the KcsA K^{+} channel and a mutant. We show that the Eisenman relations for thermodynamic selectivity follow from the condition for fast conduction and find that maximum conduction requires the binding sites to be nearly identical.

Relations between absorption, emission, and excited state chemical potentials from nanocrystal 2D spectra.

For quantum-confined nanomaterials, size dispersion causes a static broadening of spectra that has been difficult to measure and invalidates all-optical methods for determining the maximum photovoltage that an excited state can generate. Using femtosecond two-dimensional (2D) spectroscopy to separate size dispersion broadening of absorption and emission spectra allows a test of single-molecule generalized Einstein relations between such spectra for colloidal PbS quantum dots. We show that 2D spectra and these relations determine the thermodynamic standard chemical potential difference between the lowest excited and ground electronic states, which gives the maximum photovoltage. Further, we find that the static line broadening from many slightly different quantum dot structures allows single-molecule generalized Einstein relations to determine the average single-molecule linewidth from Stokes' frequency shift between ensemble absorption and emission spectra.

Revisiting Einstein's diffusion-mobility relation for universal quantum materials: A generalized paradigm

We have revisited the Einstein diffusion-mobility ratio, on the basis of the generalized Einstein relation, which quantifies both the quantum and classical levels of understanding on electron-hole dynamics in ordered and disordered materials. While the Einstein relation (diffusion-mobility ratio, D/) works only for classical systems under equilibrium cases, our generalized paradigm is quite general and works for systems ranging from classical to quantum systems in any dimensions. The proposed D/ formalism relies on the chemical potential which provides the cooperative nature between electronic and temperature counterparts on fundamental transport. Our generalized paradigm is applicable to both degenerate and non-degenerate cases, and explains the transition from linear to nonlinear relationship between the charge density and chemical potential. The symmetrical nature of electron-hole transport in highly degenerate two-dimensional quantum materials gives linear dispersion, preserving the time-reversal invariance property, while the symmetry is found to be broken in the nonlinear regime. Our generalized diffusion-mobility relation explains both the relativistic (Dirac particles) and non-relativistic regimes (Schrödinger particles) from low temperature to high temperature, and reproduces the classical Einstein relation in an appropriate limit.

Alignment of linear polymeric grains for highly stable N-type thin-film transistors

Summary Temperature-insensitive properties are attractive for most electronics, including polymeric semiconducting devices. Especially, polymeric field-effect transistors (FETs) with high mobility have been important research targets due to their broad applications. However, polymeric FETs with stable charge transport operating at extremely cold or hot zones are faced with enormous challenges. In this study, the polyacrylonitrile was found to significantly tune the sizes of pre-aggregates of polymers in solutions and the crystallinity of the polymeric films. The orientation of 5–25 μm linear grains in the films were prepared through the bar-coating process with polyacrylonitrile as an additive, which stabilized the electron mobility over a wide range of temperatures. The linear-grain morphology of the film contributed to reducing the holes and grain boundaries in the transport paths of carriers. Typically, the top-gate FETs based on P(NDI2OD-T2) displayed a stable electron transporting behavior from 200 to 460 K, with mobility greater than 3.5 cm2V−1s−1.

(Invited) Parametrization of Gaussian Disordered Model with Fully Spatial and Energetic Disorders for Organic Thin Film Transistors

So far, the most widely addressed hopping transport model for organic disordered semiconductors (ODSs) is the so-called extended Gaussian disordered model (EGDM) [1] based on numerical simulations within the cubic lattice without spatial disorder. As known since 1973 [2] and confirmed recently [3], the EGDM is based on the parametrization of transport coefficients in the framework of the irrelevant set of parameters. Parameters responsible for the effects are not in the equations, while parameters in the equations are not responsible for the effects [3]. The deficiencies of the EGDM are particularly pronounced for systems with spatial disorder. Therefore, the study of Gaussian disordered model (GDM) with spatial disorder is necessary to provide a simulation tool for transport coefficients of charge carriers in real organic thin film transistors (OTFTs) and organic light-emitting diodes (OLEDs) based on materials with spatial disorder.In this study, we examine the GDM which contains both spatial and energetic disorders and propose a physical parametrization for OTFTs. We determine all the parameters by adjusting the localization length and the attempt-to-escape frequency to satisfy experimentally measured mobility. For the localization length, the concept of effective localization length [4] is adopted which has the order of a molecular diameter, because assumption of the very low localization length (commonly order of 10-8 cm) induces very low mobility due to the strong localization of the each state. The large effective localization length can alleviate the strong localization and so it can predict the similar scale of mobility with experimentally measured one. In addition, the proper value of the attempt-to-escape frequency is estimated from comparison between a direct calculation of Miller-Abrahams (MA) transition rate and theoretical estimations of the hopping transition rate based on MA and Marcus theories. We show that its value should be increased by more than two orders of magnitude than conventional value to simulate high mobility device.Then, we validate our parametrization of GDM through comparison between experimentally measured and numerically calculated transfer curves and charge carrier mobility of OTFTs at various temperature (Fig. 1). The staggered pentacene-based OTFT device was fabricated and measured at different temperatures T = 329, 293, 258 and 233 K. The charge carrier mobility was extracted by the transmission line method (TLM). For numerical simulation, we implemented the GDM by Baranovskii [3] into the commercial TCAD simulation tool, Silvaco ATLAS. Then, the numerical solver calculated Poisson’s, continuity and drift-diffusion equations in a self-consistent manner with characteristics of ODSs such as Gaussian density-of-states, generalized Einstein relation and GDM by Baranovskii. Numerical simulation with a single set of parameters makes an accurate fit with experimentally measured transfer curves (Fig. 1a) and charge carrier mobility (Fig. 1b) at every temperature condition.In conclusion, we propose physically based parametrization of GDM with fully spatial and energetical disorder for OTFTs. It is confirmed that the localization length and the attempt-to-escape frequency are key parameters for the practical fitting with experimental result. In addition, we can expect that the electronic design automate (EDA) industry can be promoted by providing rigorous transport model and its parametrization for OTFTs within commercial simulation software.[1] W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, M. De Leeuw, and M. A. J. Michels, Phys. Rev. Lett. 94, 206601 (2005).[2] B.I. Shklovskii Sov. Phys.-Semicond. 6, 1964 (1973).[3] S. D. Baranovskii, Phys. Status Solidi Appl. Mater. Sci. 215, 1700676 (2018).[4] A. V Nenashev, J. O. Oelerich, and S. D. Baranovskii, J. Phys. Condens. Matter 27, 093201 (2015). Figure 1

Generalized Einstein relations and conditions for anomalous relaxation.

The generalized Einstein relation (GER) for nonergodic processes is investigated within the framework of the generalized Langevin equation. The conditions for anomalous relaxation such as long-tail decay and non-vanishing velocity autocorrelation function (VAF) are proposed and distinguished. For the stationary nonergodic process, if the initial preparation of the particle velocity is non-thermal, an asymptotic GER occurs in a departure from the usual result. It is shown that the GER holding is a necessary condition rather than a full condition for the system being close to equilibrium. For the nonergodic process of the second type due to cutoff of high frequencies, the VAF oscillates with time, the GER holds but the equilibrium fails in the long-time limit. Applications to some practical examples confirm the present theoretical findings.

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson–Peltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume Scharfetter–Gummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two non-isothermal generalizations of the Scharfetter–Gummel scheme for degenerate semiconductors (Fermi–Dirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme.

Generalized Einstein relation for nonparabolic multiple energy-band degenerate semiconductors

A generalized Einstein relation for electron gases of degenerate semiconductors with a system of typically two nonparabolic conduction band structures is derived and formulated in near-equilibrium condition based on using the semiclassical drift-diffusion equation. The result of this derivation shows that the Einstein relation for carrier gases in multiple energy bands generally depends on their mobilities unless the energy band structures are parabolic. The formulated results (a full formula and an approximate single-carrier-gas formula) are typically applied to GaAs and GaSb to calculate the values of the ratio of the diffusion coefficient to mobility for carriers in the lowest and upper conduction bands, and an in-depth investigation is made for the Einstein relation for conduction electrons in these semiconductors. It is shown, in particular, that highly degenerate GaAs exhibits the uncommon, peculiar property of the ratio of the diffusion coefficient to mobility for conduction electrons saturating in two different electron concentration, or Fermi energy, conditions, with increasing electron concentration or Fermi energy. It is also shown that for GaSb, the effect of the upper conduction band is so large that the use of any formula obtained for carriers in a single conduction band is generally unsuitable for describing the Einstein relation for conduction electrons in this semiconductor. In addition, a more generalized Einstein relation extended to a system of three nonparabolic energy band structures is also formulated and proposed in the paper, so as to be used to more accurately describe the Einstein relation for hole gases in degenerate semiconductors.

Einstein Relation for a Driven Disordered Quantum Chain in the Subdiffusive Regime

A quantum particle propagates subdiffusively on a strongly disordered chain when it is coupled to itinerant hard-core bosons. We establish a generalized Einstein relation (GER) that relates such subdiffusive spread to an unusual time-dependent drift velocity, which appears as a consequence of a constant electric field. We show that GER remains valid much beyond the regime of the linear response. Qualitatively, it holds true up to strongest drivings when the nonlinear field effects lead to the Stark-like localization. Numerical calculations based on full quantum evolution are substantiated by much simpler rate equations for the boson-assisted transitions between localized Anderson states.

Subdiffusion in an external force field.

The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time-dependent force, we study the subdiffusion described by the subordinated Langevin equation with white Gaussian noise or, equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as the ensemble- and time-averaged mean squared displacements, position autocorrelation function, correlation coefficient, and generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are nonstationary, nonergodic, and aging.

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

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.