Kubo Relations Research Articles

Ionic self-diffusion coefficient and shear viscosity of high-Z materials in the hot dense regime

High-Z materials exhibit a broad range of variation of the charge state in the hot dense regime, and so ionic structures become complex with increasing density and temperature owing to ionization. Taking high-Z uranium as example, we study its electronic and ionic structures in the hot dense regime by combining an average-atom model with the hypernetted chain approximation. The electronic structure is described by solving the Dirac equation, taking account of relativistic effects, including broadening of the energy levels, and the effect of other ions via correlation functions. On the basis of the electronic distribution around a nucleus, the ion pair potential is constructed using the modified Gordon–Kim model in the frame of temperature-dependent density functional theory. Because of the presence of ion–ion strong coupling, the bridge function is included in the hypernetted chain approximation, which is used to calculate the correlation functions. To take account of the influence on transport properties of the strong correlation of electrons with highly charged ions, we perform both classical and Langevin molecular dynamics simulations to determine ion self-diffusion coefficients and the shear viscosity, using the Green–Kubo relation and an ion–ion pair potential with good convergence. We show that the influence of electron–ion collisions on transport properties becomes more important as the free electron density increases owing to thermal ionization.

Transport phenomena in electrolyte solutions: Nonequilibrium thermodynamics and statistical mechanics

AbstractThe theory of transport phenomena in multicomponent electrolyte solutions is presented here through the integration of continuum mechanics, electromagnetism, and nonequilibrium thermodynamics. The governing equations of irreversible thermodynamics, including balance laws, Maxwell's equations, internal entropy production, and linear laws relating the thermodynamic forces and fluxes, are derived. Green–Kubo relations for the transport coefficients connecting electrochemical potential gradients and diffusive fluxes are obtained in terms of the flux–flux time correlations. The relationship between the derived transport coefficients and those of the Stefan–Maxwell and infinitely dilute frameworks are presented, and the connection between the transport matrix and experimentally measurable quantities is described. To exemplify the application of the derived Green–Kubo relations in molecular simulations, the matrix of transport coefficients for lithium and chloride ions in dimethyl sulfoxide is computed using classical molecular dynamics and compared with experimental measurements.

Investigation of the relationship between nanochannel width and mass transfer characteristics for dense methane nanofluidics

In this paper, we analytically investigate the influence of the nanochannel width on mass transfer characteristics through the rough nanochannel by using non–equilibrium multiscale molecular dynamics simulation. The mass transfer characteristics are analyzed by fitting the curve correlation between the diffusion coefficient and nanochannel width. The diffusion coefficient is calculated using the revised Green–Kubo relation in different nanochannels. The numerical results demonstrate that the relationship between diffusion coefficient and the inverse of nanochannel width is parabolic. Moreover, the diffusion coefficients show significant variations in the small rough nanochannel, and gradually approach the values of bulk system with increasing the width of nanochannel. In addition, to explore its mechanism, the local atom number density, velocity profile, velocity autocorrelation function, the escape rate of molecule, local diffusion coefficient and projection radial distribution function are given in different widths of nanochannel under given boundary condition. Simulation results show that the local structure and number density of fluid influence the transfer mass characteristics. These results of this work may be conducive to further understand the mass transfer behaviors of nanofluidics, and guide the optimal design of nano–devices.

Generalization of the Kubo relation for confined motion and ergodicity breakdown.

A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.

Shear viscosity of n-pentane from diffusivity based molecular dynamics methods

The molecular dynamics methods for calculation of shear viscosity based on liquid diffusivity are tested against the classical Green–Kubo relation for n-pentane at 330 K and 0.601 g/cm3. The D-based method is shown to be as accurate as calculation of viscosity from the Green–Kubo formalism for pentane liquid. Stokes–Einstein relation is also in agreement with simulation results. However, it has much bigger uncertainty. The results of the simulation are in agreement with experimental data.

Fluctuation\u2013dissipation relation from anomalous stress tensor and Hawking effect

We show a direct connection between Kubo’s fluctuation–dissipation relation and Hawking effect that is valid in any dimensions for any stationary or static black hole. The relevant correlators corresponding to the fluctuating part of the force, computed from the known expressions for the anomalous stress tensor related to gravitational anomalies, are shown to satisfy the Kubo relation, from which the temperature of a black hole as seen by an observer at an arbitrary distance is abstracted. This reproduces the Tolman temperature and hence the Hawking temperature as that measured by an observer at infinity.

Coherent Transport in Periodically Driven Mesoscopic Conductors: From Scattering Amplitudes to Quantum Thermodynamics

Abstract Scattering theory is a standard tool for the description of transport phenomena in mesoscopic systems. Here, we provide a detailed derivation of this method for nano-scale conductors that are driven by oscillating electric or magnetic fields. Our approach is based on an extension of the conventional Lippmann–Schwinger formalism to systems with a periodically time-dependent Hamiltonian. As a key result, we obtain a systematic perturbation scheme for the Floquet scattering amplitudes that describes the transition of a transport carrier through a periodically driven sample. Within a general multi-terminal setup, we derive microscopic expressions for the mean values and time-integrated correlation functions, or zero-frequency noise, of matter and energy currents, thus recovering the results of earlier studies in a unifying framework. We show that this framework is inherently consistent with the first and the second law of thermodynamics and prove that the mean rate of entropy production vanishes only if all currents in the system are zero. As an application, we derive a generalized Green–Kubo relation, which makes it possible to express the response of any mean currents to small variations of temperature and chemical potential gradients in terms of time integrated correlation functions between properly chosen currents. Finally, we discuss potential topics for future studies and further reaching applications of the Floquet scattering approach to quantum transport in stochastic and quantum thermodynamics.

The dependence of transport coefficient on spatial dimensions and grid shape in the direct simulation Monte Carlo based on Green–Kubo relation

Since the essence of the direct simulation Monte Carlo program is to decouple the motion and collision of molecules, this will cause errors of gas transport properties or the collision separation error (CSE). Based on the Green–Kubo relation, this paper studies the dependence of the CSE on the spatial dimension and grid shape, especially the anisotropy of gas transport properties for various collision schemes in multi-dimensional spaces. The results show that the CSEs caused by the traditional no time counter and the simplified Bernoulli-Trials collision schemes in different dimensional spaces are consistent, and they are isotropic for some grid shapes, while the CSE caused by the nearest neighbor method depends on the spatial dimension, and it is anisotropic. These results are verified by simulation of one-dimensional Fourier flow.

Calculation of electron-ion temperature equilibration rates and friction coefficients in plasmas and liquid metals using quantum molecular dynamics.

We discuss a method to calculate with quantum molecular dynamics simulations the rate of energy exchanges between electrons and ions in two-temperature plasmas, liquid metals, and hot solids. Promising results from this method were recently reported for various materials and physical conditions [Simoni and Daligault, Phys. Rev. Lett. 122, 205001 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.205001]. Like other ab initio calculations, the approach offers a very useful comparison with the experimental measurements and permits an extension into conditions not covered by the experiments. The energy relaxation rate is related to the friction coefficients felt by individual ions due to their nonadiabatic interactions with electrons. Each coefficient satisfies a Kubo relation given by the time integral of the autocorrelation function of the interaction force between an ion and the electrons. These Kubo relations are evaluated using the output of quantum molecular dynamics calculations in which electrons are treated in the framework of finite-temperature density functional theory. The calculation presents difficulties that are unlike those encountered with the Kubo formulas for the electrical and thermal conductivities. In particular, the widely used Kubo-Greenwood approximation is inapplicable here. Indeed, the friction coefficients and the energy relaxation rate diverge in this approximation since it does not properly account for the electronic screening of electron-ion interactions. The inclusion of screening effects considerably complicates the calculations. We discuss the physically motivated approximations we applied to deal with these complications in order to investigate a widest range of materials and physical conditions. Unlike the standard method used for the electronic conductivities, the Kubo formulas are evaluated directly in the time domain and not in the energy domain, which spares one from needing to introduce an extraneous undetermined numerical parameter to account for the discrete character of the numerical density of states. We highlight interesting properties of the energy relaxation rate not shared by other electronic properties, in particular its self-averaging character. We then present a detailed parametric and convergence study with the numerical parameters, including the system size, the number of bands and k points, and the physical approximations for the dielectric function and the exchange-correlation energy.

Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans–Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in Andrieux et al. (in New J Phys 11(4):043014, 2009). We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in Andrieux et al. (in New J Phys 11(4):043014, 2009), we recover the fluctuation–dissipation theorem in the linear response theory, short of Green–Kubo relations. We illustrate the general theory on a number of canonical models.

Theory of the electron-ion temperature relaxation rate spanning the hot solid metals and plasma phases.

We present a theory for the rate of energy exchange between electrons and ions-also known as the electron-ion coupling factor-in physical systems ranging from hot solid metals to plasmas, including liquid metals and warm dense matter. The paper provides the theoretical foundations of a recent work [J. Simoni and J. Daligault, Phys. Rev. Lett. 122, 205001 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.205001], where first-principles quantum molecular dynamics calculations based on this theory were presented for representative materials and conditions. We first derive a general expression for the electron-ion coupling factor that includes self-consistently the quantum mechanical and statistical nature of electrons, the thermal and disorder effects, and the correlations between particles. The electron-ion coupling is related to the friction coefficients felt by individual ions due to their nonadiabatic interactions with the electrons. Each coefficient satisfies a Kubo relation given by the time integral of the autocorrelation function of the interaction force of an ion with the electrons. Exact properties and different representations of the general expressions are discussed. We then show that our theory reduces to well-known models in limiting cases. In particular, we show that it simplifies to the standard electron-phonon coupling formula in the limit of hot solids with lattice and electronic temperatures much greater than the Debye temperature, and that it extends the electron-phonon coupling formula beyond the harmonic phonon approximation. For plasmas, we show that the theory readily reduces to the well-known Spitzer formula in the hot plasma limit, to the Fermi "golden rule" formula in the limit of weak electron-ion interactions, and to other models proposed to go beyond the latter approximation. We explain that the electron-ion coupling is particularly well adapted to average atom models, which offer an effective way to include nonideal interaction effects to the standard models and at a much reduced computational cost in comparison to first-principles quantum molecular dynamics simulations.

Ab initio molecular dynamics study of TiZrNbHfTa and VZrMoHfW liquid alloys

The structural and dynamical properties of liquid alloys are directly related to their solidification process, and consequently to the properties of as-cast solid alloys. Ab initio molecular dynamics is a powerful tool that can be used to study both of these factors. Thus, in this work, TiZrNbHfTa and VZrMoHfW liquid equiatomic alloys, exclusively consisting of transition metal (TM) elements, are studied using ab initio molecular dynamics via the CP2K software package; partial-pair correlation functions (PPCFs) are calculated, and their parameters are discussed. The calculated densities of these alloys are shown to be close to the ideal mixture densities. The analysis of the PPCFs is performed to predict the single-phase high-entropy alloy (HEA) formation. The obtained results agree with the semi-empirical phase formation rules. The diffusivities of the elements are calculated using the Green–Kubo relation and the mean square displacement (MSD) plot. The diffusivity is shown to decrease as the atomic mass and radius increase. The scattering process of different types of atoms in TiZrNbHfTa and VZrMoHfW are also discussed.

First-Principles Determination of Electron-Ion Couplings in the Warm Dense Matter Regime.

We present first-principles calculations of the rate of energy exchanges between electrons and ions in nonequilibrium warm dense plasmas, liquid metals, and hot solids, a fundamental property for which various models offer diverging predictions. To this end, a Kubo relation for the electron-ion coupling parameter is introduced, which includes self-consistently the quantum, thermal, nonlinear, and strong coupling effects that coexist in materials at the confluence of solids and plasmas. Most importantly, like other Kubo relations widely used for calculating electronic conductivities, the expression can be evaluated using quantum molecular dynamics simulations. Results are presented and compared to experimental and theoretical predictions for representative materials of various electronic complexity, including aluminum, copper, iron, and nickel.

Relaxation times for chiral transport phenomena and spin polarization in a strongly coupled plasma

We compute the dynamical relaxation times for chiral transport phenomena in strongly coupled regime using the AdS/CFT correspondence. These relaxation times can be a useful proxy for the dynamical time scale for achieving equilibrium spin-polarization of quasi-particles in the presence of magnetic field and fluid vorticity. We identify the Kubo relations for these relaxation times and clarify some previous issues regarding time-dependence of the Chiral Vortical Effect. We study the consequences of imposing time-reversal invariance on parity-odd thermal noise fluctuations that are related to chiral transport coefficients by the fluctuation-dissipation relation. We find that time-reversal invariance dictates the equality between some of the chiral transport coefficients as well as their relaxation times.

The Effect of Simulation Cell Size on the Diffusion Coefficient of an Ionic Surfactant Aggregate

Results of all-atom molecular dynamics simulation have been presented for salt-free aqueous solutions of sodium dodecyl sulfate at its fixed total concentration in a simulation cell containing one to four preliminarily formed quasi-stable ionic aggregates with equal aggregation numbers n = 32. The obtained molecular dynamics trajectories have been used to study the structural and transport properties of the micellar solution. The value of the counterion diffusion coefficient obtained using the Green–Kubo relation has turned out to be somewhat higher than the corresponding value calculated by the Einstein equation. The diffusion coefficients of the aggregates in the systems containing from two to four aggregates have appeared to be higher than the diffusion coefficient of a single aggregate in a cell. The mean force potential obtained for the interaction between the aggregates having aggregation number n = 32 as a function the distance between the aggregate centers of mass has a local minimum in the system containing four such aggregates.

Kinetic evolution and correlation of fluctuations in an expanding quark gluon plasma

Evolution of spatially anisotropic perturbation created in the system formed after Relativistic Heavy Ion Collisions has been studied. The microscopic evolution of the fluctuations has been examined within the ambit of Boltzmann Transport Equation (BTE) in a hydrodynamically expanding background. The expansion of the background composed of quark gluon plasma (QGP) is treated within the framework of relativistic hydrodynamics. Spatial anisotropic fluctuations with different geometries have been evolved through Boltzmann equation. It is observed that the trace of such fluctuation survives the evolution. Within the relaxation time approximation, analytical results have been obtained for the evolution of these anisotropies. Explicit relations between fluctuations and transport coefficients have been derived. The mixing of various Fourier (or k) modes of the perturbations during the evolution of the system has been explicitly demonstrated. This study is very useful in understanding the presumption that the measured anisotropies in the data from heavy ion collisions at relativistic energies imitate the initial state effects. The evolution of correlation function for the perturbation in pressure has been studied and shows that the initial correlation between two neighbouring points in real space evolves to a constant value at later time which gives rise to Dirac delta function for the correlation function in Fourier space. The power spectrum of the fluctuation in thermodynamic quantities (like temperature estimated in this work) can be connected to the fluctuation in transverse momentum of the thermal hadrons measured experimentally. The bulk viscous coefficient of the QGP has been estimated by using correlations of pressure fluctuation with the help of Green–Kubo relation. Angular power spectrum of the anisotropies has been estimated in the appendix.

Multi-charge-state molecular dynamics and self-diffusion coefficient in the warm dense matter regime

We present a multi-ion molecular dynamics (MIMD) simulation and apply it to calculating the self-diffusion coefficients of ions with different charge-states in the warm dense matter (WDM) regime. First, the method is used for the self-consistent calculation of electron structures of different charge-state ions in the ion sphere, with the ion-sphere radii being determined by the plasma density and the ion charges. The ionic fraction is then obtained by solving the Saha equation, taking account of interactions among different charge-state ions in the system, and ion–ion pair potentials are computed using the modified Gordon–Kim method in the framework of temperature-dependent density functional theory on the basis of the electron structures. Finally, MIMD is used to calculate ionic self-diffusion coefficients from the velocity correlation function according to the Green–Kubo relation. A comparison with the results of the average-atom model shows that different statistical processes will influence the ionic diffusion coefficient in the WDM regime.

Viscosity of a sheared correlated (near-critical) model fluid in confinement

Second-order phase transitions are characterized by a divergence of the spatial correlation length of the order parameter fluctuations. For confined systems, this is known to lead to remarkable equilibrium physical phenomena, including finite-size effects and critical Casimir forces. We explore here some non-equilibrium aspects of these effects in the stationary state resulting from the action of external forces: by analyzing a model of a correlated fluid under shear, spatially confined by two parallel plates, we study the resulting viscosity within the setting of (Gaussian) Landau–Ginzburg theory. Specifically, we introduce a model in which the hydrodynamic velocity field (obeying the Stokes equation) is coupled to an order parameter with dissipative dynamics. The well-known Green–Kubo relation for bulk systems is generalized for confined systems. This is shown to result in a non-local Stokes equation for the fluid flow, due to the correlated fluctuations. The resulting effective shear viscosity shows universal as well as non-universal contributions, which we study in detail. In particular, the deviation from the bulk behavior is universal, depending on the ratio of the correlation length and the film thickness L. In addition, at the critical point the viscosity is proportional to , where is a dynamic length scale. These findings are expected to be experimentally observable, especially for systems where the bulk viscosity is affected by critical fluctuations.

Thermal conductivity calculation of nano-suspensions using Green–Kubo relations with reduced artificial correlations

The presence of artificial correlations associated with Green–Kubo (GK) thermal conductivity calculations is investigated. The thermal conductivity of nano-suspensions is calculated by equilibrium molecular dynamics (EMD) simulations using GK relations. Calculations are first performed for a single alumina (Al2O3) nanoparticle dispersed in a water medium. For a particle size of 1 nm and volume fraction of 9%, results show enhancements as high as 235%, which is much higher than the Maxwell model predictions. When calculations are done with multiple suspended particles, no such anomalous enhancement is observed. This is because the vibrations in alumina crystal can act as low frequency perturbations, which can travel long distances through the surrounding water medium, characterized by higher vibration frequencies. As a result of the periodic boundaries, they re-enter the system resulting in a circular resonance of thermal fluctuations between the alumina particle and its own image, eventually leading to artificial correlations in the heat current autocorrelation function (HCACF), which when integrated yields abnormally high thermal conductivities. Adding more particles presents ‘obstacles’ with which the fluctuations interact and get dissipated, before they get fed back to the periodic image. A systematic study of the temporal evolution of HCACF indicates that the magnitude and oscillations of artificial correlations decrease substantially with increase in the number of suspended nanoparticles.

Determining transport coefficients for a microscopic simulation of a hadron gas

Quark-Gluon plasmas produced in relativistic heavy-ion collisions quickly expand and cool, entering a phase consisting of multiple interacting hadronic resonances just below the QCD deconfinement temperature, $T\sim 155$ MeV. Numerical microscopic simulations have emerged as the principal method for modeling the behavior of the hadronic stage of heavy-ion collisions, but the transport properties that characterize these simulations are not well understood. Methods are presented here for extracting the shear viscosity, and two transport parameters that emerge in Israel-Stewart hydrodynamics. The analysis is based on studying how the stress-energy tensor responds to velocity gradients. Results agree with expectations based on Kubo relations.