Green-Kubo Formula Research Articles

Dynamic viscosity of water containing large-sized nanobubbles simulated with coarse-grained molecular dynamics model

Nanobubbles have received widespread interest due to prospective applications in various disciplines. A crucial approach to investigating nanobubbles is molecular dynamics simulation. However, investigating large-sized bulk nanobubbles requires expensive computer resources. By coarsening the water molecules, the computational complexity can be significantly reduced, making the investigation of large-sized nanobubbles possible. This article presents simulations and analysis of the microscopic characteristics of water containing large-sized nanobubbles using coarse-grained molecular dynamics, especially, the Green-Kubo formula is used to calculate the dynamic viscosity. We first observe the effects of temperature, pressure and diameter on the viscosity of nanobubble water. The results indicate that higher temperature leads to lower viscosity of nanobubble water while higher pressure leads to higher viscosity of nanobubble water. Additionally, it is observed that the viscosity of nanobubble water decreases gradually as the diameter of the nanobubble increases. It is a remarkable fact that the viscosity of nanobubble water is lower than that of water at the same temperature and pressure, which means the viscosity of water can be reduced with the addition of nanobubble. At last, a viscosity correction model for nanobubble water at room temperature is successfully developed.

Thermal Energy Transport through Nonbonded Native Contacts in Protein.

Within the protein interior, where we observe various types of interactions, nonuniform flow of thermal energy occurs along the polypeptide chain and through nonbonded native contacts, leading to inhomogeneous transport efficiencies from one site to another. The folded native protein serves not merely as thermal transfer medium but, more importantly, as sophisticated molecular nanomachines in cells. Therefore, we are particularly interested in what sort of "communication" is mediated through native contacts in the folded proteins and how such features are quantitatively depicted in terms of local transport coefficients of heat currents. To address the issue, we introduced a concept of inter-residue thermal conductivity and characterized the nonuniform thermal transport properties of a small globular protein, HP36, using equilibrium molecular dynamics simulation and the Green-Kubo formula. We observed that the thermal transport of the protein was dominated by that along the polypeptide chain, while the local thermal conductivity of nonbonded native contacts decreased in the order of H-bonding > π-stacking > electrostatic > hydrophobic contacts. Furthermore, we applied machine learning techniques to analyze the molecular mechanism of protein thermal transport. As a result, the contact distance, variance in contact distance, and H-bonding occurrence probability during MD simulations are found to be the top three important determinants for predicting local thermal transport coefficients.

Thermal conductivity calculation using homogeneous non-equilibrium molecular dynamics simulation with Allegro

In this study, we derive the heat flux formula for the Allegro model, one of machine-learning interatomic potentials using the equivariant deep neural network, to calculate lattice thermal conductivity using the homogeneous non-equilibrium molecular dynamics (HNEMD) method based on the Green-Kubo formula. Allegro can construct more advanced atomic descriptors than conventional ones, and can be applied to multicomponent and large-scale systems, providing a significant advantage in estimating the thermal conductivity of anharmonic materials, such as thermoelectric materials. In addition, the spectral heat current (SHC) method, recently developed for the HNEMD framework (HNEMD-SHC), allows the calculation of not only the total thermal conductivity but also its frequency components. The verification of the heat flux and the demonstration of HNEMD-SHC method are performed for the extremely anharmonic low-temperature phase of Ag2Se.

Green and Kubo forge the arrow of time

ABSTRACT Transport theory describes the response of a macroscopic current to a thermodynamic force, thus producing entropy and apparently violating time-reversal symmetry. In this note I report a pedagogical derivation of the Green–Kubo formula for transport coefficients, based on elementary equilibrium statistical mechanics and static response theory, that highlights the intrinsically dynamical nature of this formula and showcases the relation between the apparent breach of time-reversal symmetry and the non-commutativity of the low-frequency / low-wavevector limits of the conserved-density susceptibilities, from which the formula can be established.

Transport coefficient approach for characterizing nonequilibrium dynamics in soft matter

Nonequilibrium states in soft condensed matter require a systematic approach to characterize and model materials, enhancing predictability and applications. Among the tools, X-ray photon correlation spectroscopy (XPCS) provides exceptional temporal and spatial resolution to extract dynamic insight into the properties of the material. However, existing models might overlook intricate details. We introduce an approach for extracting the transport coefficient, denoted as [Formula: see text], from the XPCS studies. This coefficient is a fundamental parameter in nonequilibrium statistical mechanics and is crucial for characterizing transport processes within a system. Our method unifies the Green-Kubo formulas associated with various transport coefficients, including gradient flows, particle-particle interactions, friction matrices, and continuous noise. We achieve this by integrating the collective influence of random and systematic forces acting on the particles within the framework of a Markov chain. We initially validated this method using molecular dynamics simulations of a system subjected to changes in temperatures over time. Subsequently, we conducted further verification using experimental systems reported in the literature and known for their complex nonequilibrium characteristics. The results, including the derived [Formula: see text] and other relevant physical parameters, align with the previous observations and reveal detailed dynamical information in nonequilibrium states. This approach represents an advancement in XPCS analysis, addressing the growing demand to extract intricate nonequilibrium dynamics. Further, the methods presented are agnostic to the nature of the material system and can be potentially expanded to hard condensed matter systems.

Theoretical Study of Maxwell-Stefan Binary Diffusion Coefficients of n-alkanes-N2 at High Densities

ABSTRACT In this study, the Maxwell-Stefan binary diffusion coefficients ( D ij ) of C1-C6 n-alkanes-N2 are calculated using the Green-Kubo formula. The ratio of pD ij at different pressures (from 1 to 1000 bar) to that at 1 atm (reference condition) is studied. It is found that the ratio pD ij /(pD ij ) ref first decreases and then increases with increasing density, finally reaching 1.3 for n-C6H14-N2 at 1000 bar, 600 K. The gradual increase in molecular attraction at the beginning of the density increase results in a slower pressure increase compared to that of an ideal gas, contributing to the decreasing trend of pD ij /(pD ij ) ref . As the density continues to increase, the enhanced molecular repulsion and the molecular volume exclusion effect accelerate the pressure increase beyond that of an ideal gas, leading to an increasing trend in pD ij /(pD ij ) ref . Finally, the results obtained by the Green-Kubo method are compared with those obtained from empirical methods, and an equation for the refinement of D ij is formulated.

Prediction of phonon properties of cubic boron nitride with vacancy defects and isotopic disorders by using a neural network potential

Cubic boron nitride (c-BN) is a promising ultra-wide bandgap semiconductor for high-power electronic devices. Its thermal conductivity can be substantially modified by controlling the isotope abundance and by the quality of a single crystal. Consequently, an understanding of the phonon transport in c-BN crystals, with both vacancy defects and isotopic disorders at near-ambient temperatures, is of practical importance. In the present study, a neural network potential (NNP) for c-BN has been developed, which has facilitated the investigation of phonon properties under these circumstances. As a result, the phonon dispersion and the three- and four-phonon scattering rates that were predicted with this NNP were in close agreement with those obtained from density-functional theory (DFT) calculations. The thermal conductivities of the c-BN crystals were also investigated, with boron (B) vacancies ranging from 0.0% to 0.6%, by using equilibrium molecular dynamics simulations based on the Green-Kubo formula. These simulations accurately capture vacancy-induced phonon softening, localized vibration modes, and phonon localization effects. As has previously been experimentally prepared, four isotope-modified c-BN samples were selected for analyses in the evaluation of the impact of isotopic disorders. The calculated thermal conductivities aligned well with the DFT benchmarks. In addition, the present study was extended to include a c-BN crystal with a natural abundance of B atoms, which also contained B vacancies. Reasonable thermal conductivities and vibrational characteristics, within the temperature range of 250–500 K, were then obtained.

Accurate estimation of transport coefficients using time correlation functions without fitting any analytic forms in equilibrium simulations

Transport coefficients, such as the diffusion coefficient and shear viscosity, are important material properties that are calculated in computer simulations. In this study, the criterion for the accurate estimation of viscosity, as an example of transport coefficients, is determined by using the Green-Kubo formula without fitting any analytic forms for the stress decay function. The related algorithm is given by the estimation of the viscosities of polyethylene oxide solutions by using a molecular dynamics simulation for testing. The algorithm can be used in the simulations of complex systems with a long tail of correlations typically found in macromolecular and biological simulation systems.

General validity of the second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state and further show that if the system is highly degenerate, then the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and hence generally applies to nonequilibrium systems driven by stochastic forces. These theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that our heat transfer model yields better predictions than the Green-Kubo formula for systems far from the equilibrium.

Construction of machine-learning interatomic potential under heat flux regularization and its application to power spectrum analysis for silver chalcogenides

We propose a data-driven approach for constructing machine-learning interatomic potentials (MLIPs) trained under a regularization with the aim of avoiding nonphysical heat flux that arises from the many-body effect of MLIPs and is sensitive to their initial weight parameters. Specifically, we introduce a regularization term for the heat flux into the cost function of MLIPs to be minimized. Since the treatment of heat flux using MLIPs with regularization can be decomposed into elemental contributions or conducted in frequency space, this approach is expected to be useful for investigating the origin of thermal conductivity obtained from the Green-Kubo formula. However, the strength of regularization needs to be appropriately set because it may reduce not only the nonphysical part but also the intrinsic heat flux one. To this end, we investigated the conditions for constructing MLIPs that can reproduce the power spectra of heat flux associated with the empirical interatomic potential of Ag2Se, which consists of pairwise functions and do not contain a nonphysical heat flux. The power spectrum is useful because it helps to verify the effect of regularization and also contains the information on thermal conductivity and phonon. The appropriate strength could be estimated from the variation of the magnitude of regularization term as well as root mean square errors for total potential energy, atomic force, and virial stress with respect to the strengths, without reference spectrum data. As an application example, we explored the differences in power spectra between superionic and nonsuperionic conducting phases based on the heat flux regularization to MLIPs trained with the first-principles calculation data of Ag2S. Furthermore, our results demonstrate that training with the regularization improves the robustness of MLIPs as well as the reduction of the nonphysical heat flux.

Diffusion coefficient scaling of a free Brownian particle with velocity-dependent damping.

An analytical expression for the diffusion coefficient D of a free Brownian particle with velocity-dependent damping γ(v) is derived from the Green-Kubo formula. A special case of damping that decreases monotonically with velocity is considered. At high temperature T, the diffusion coefficient is found to exhibit two scaling types: (i) for a power-law decrease of damping with the particle's kinetic energy, γ(v)∝1/v^{2α}, it scales as D∝T^{α+1}; (ii) for a Gaussian function γ(v), it diverges at temperatures above a critical value T_{c} and behaves as D∝1/sqrt[T_{c}-T] at T slightly below T_{c}. At T>T_{c}, the particle trajectory contains long flight events, which are not observed at T<T_{c} in case (ii) and at all temperatures in case (i).

Self-Assembly and the Properties of Micro-Mesoporous Carbon.

This study introduces a new approach for constructing atomistic models of nanoporous carbon by randomly distributing carbon atoms and pore volumes in a periodic box and then using empirical and ab initio molecular simulation tools to find the suitable energy-minimum structures. The models, consisting of 5000, 8000, 12000, and 64000 atoms, each at mass densities of 0.5, 0.75, and 1 g/cm3, were analyzed to determine their structural characteristics and relaxed pore size distribution. Surface analysis of the pore region revealed that sp atoms exist predominantly on surfaces and act as active sites for oxygen adsorption. We also investigated the electronic and vibrational properties of the models, and localized states near the Fermi level were found to be primarily situated at sp carbon atoms through which electrical conduction may occur. Additionally, the thermal conductivity was calculated using heat flux correlations and the Green-Kubo formula, and its dependence on pore geometry and connectivity was analyzed. The behavior of the mechanical elasticity moduli (Shear, Bulk, and Young's moduli) of nanoporous carbons at the densities of interest was discussed.

On the binary diffusion coefficients of n-alkanes in He/N2

The binary diffusion coefficients of fuel molecules in bath gasses are key parameters for accurate predictions of flame properties such as extinction strain rates. In combustion simulations, they are often calculated using the Hirschfelder-Bird-Spotz (HBS) equation, which requires the availability of effective Lennard-Jones parameters, i.e. collision diameters and well depths, which can be obtained from the intermolecular potentials of two interacting species by applying an orientation-averaging rule. Besides the HBS equation, the Green-Kubo formula can be used to obtain binary diffusion coefficients, using velocity autocorrelation and cross-correlation functions from molecular dynamics simulations. In this work, we propose a new orientation-averaging rule, called VolW-σ-ε, based on the characteristic of intermolecular potentials, and evaluate different approaches to computing the diffusion coefficients of n-alkanes in He/N2 mixtures. Several orientation-averaging rules are evaluated and compared with experimental data of n-alkanes in He and N2 and the results of molecular dynamics simulations. Finally, using the binary diffusion coefficients of n-dodecane in N2 evaluated by the VolW-σ-ε method, we reproduce previously reported experimental extinction strain rates reported for counterflow n-dodecane/N2 versus O2 jets.

Formation of Amorphous Carbon Multi‐Walled Nanotubes from Random Initial Configurations

Amorphous carbon nanotubes (a-CNT) with up to four walls and sizes ranging from 200 to 3200 atoms have been simulated, starting from initial random configurations and using the Gaussian Approximation Potential [Phys. Rev. B 95, 094203 (2017)]. The important variables (like density, height, and diameter) required to successfully simulate a-CNTs, were predicted with a machine learning random forest technique. The models were validated $\textit{ex post facto}$ using density functional codes. The a-CNT models ranged from 0.55 nm - 2 nm wide with an average inter-wall spacing of 0.31 nm. The topological defects in a-CNTs were discussed and new defect configurations were observed. The electronic density of states and localization in these phases were discussed and delocalized electrons in the $\pi$ subspace were identified as an important factor for inter-layer cohesion. Spatial projection of the electronic conductivity favors axial transport along connecting hexagons, while non-hexagonal parts of the network either hinder or bifurcate the electronic transport. A vibrational density of states was calculated and is potentially an experimentally testable fingerprint of the material and the appearance of a low-frequency radial breathing mode was discussed. The thermal conductivity at 300 K was calculated using the Green-Kubo formula.

Time-translational symmetry in statistical dynamics dictates Einstein relation, Green-Kubo formula, and their generalizations.

A stochastic dynamics has a natural decomposition into a drift capturing mean rate of change and a martingale increment capturing randomness. They are two statistically uncorrelated, but not necessarily independent, components contributing to the overall fluctuations of the dynamics, representing the uncertainties in the past and in the future. We show that fluctuation-dissipation relations of the two aforementioned components, such as the Einstein relation and the Green-Kubo formula, can be formulated for any stochastic process with a steady state, without additional supposition of the process being Markovian, reversible, or linear. Further, by considering the adjoint process defined by the time reversal at the steady state, we show that reversibility in equilibrium leads to an additional symmetry in the covariance between system's state and drift. Potential directions of further generalizing our results to processes without steady states is briefly discussed.

Quantum local-equilibrium approach to dissipative hydrodynamics.

The macroscopic hydrodynamic equationsare derived for many-body systems in the local-equilibrium approach, using the Schrödinger picture of quantum mechanics. In this approach, statistical operators are defined in terms of microscopic densities associated with the fundamentally conserved quantities and other slow modes possibly emerging from continuous symmetry breaking, as well as macrofields conjugated to these densities. Functional identities can be deduced, allowing us to identify the reversible and dissipative parts of the mean current densities, to obtain general equationsfor the time evolution of the conjugate macrofields, and to establish the relationship to projection-operator methods. The entropy production is shown to be nonnegative by applying the Peierls-Bogoliubov inequality to a quantum integral fluctuation theorem. Using the expansion in the gradients of the conjugate macrofields, the transport coefficients are given by Green-Kubo formulas and the entropy production rate can be expressed in terms of quantum Einstein-Helfand formulas, implying its nonnegativity in agreement with the second law of thermodynamics. The results apply to multicomponent fluids and can be extended to condensed matter phases with broken continuous symmetries.

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.

Green–Kubo formula for electrical conductivity of a driven $$0$$–$$\pi$$ qubit

Green–Kubo formula for electrical conductivity of a driven $$0$$–$$\pi$$ qubit

Green-Kubo formula for electrical conductivity of a driven $0$ - $\pi$ qubit

Одна из наиболее важных концепций квантовых вычислений основана на использовании состояний конденсата куперовских пар в джозефсоновском контакте, и в связи с этими состояниями в настоящее время большой интерес представляет такая конфигурация, как $0$ - $\pi$ кубит. Линейный отклик этой системы на внешнее воздействие найден путем решения уравнения Лиувилля для функции распределения в фазовом пространстве. Как следствие, получается "закон Ома", т. е. выражение для электропроводности системы через новые корреляционные функции. Этот общий результат проверен на примере $0$ - $\pi$ кубита с параметрами, которые использовались в недавних экспериментах.

Dynamics and lifetimes of resonant phonons in the quasiparticle and nonquasiparticle regimes

The dynamics and lifetimes of a phonon with model resonant interactions from the quasiparticle to nonquasiparticle regime were investigated employing Green's function and the Green-Kubo method. In the weak-coupling case, the dynamics of the resonant phonon are analogous to a damped harmonic oscillator and its lifetime $\ensuremath{\tau}$ is in accordance with the standard phonon transport theory $\ensuremath{\tau}=\frac{1}{2\mathrm{\ensuremath{\Gamma}}}$ ($\mathrm{\ensuremath{\Gamma}}$ being the imaginary part of phonon self-energy). In the strong-coupling nonquasiparticle regime, however, the resonant phonon ``propagates'' in a complex form of wave packets and the actual phonon lifetime ${\ensuremath{\tau}}_{\mathrm{GK}}$ as determined from the Green-Kubo formula significantly deviates from the standard $\ensuremath{\tau}=\frac{1}{2\mathrm{\ensuremath{\Gamma}}}$ relation. Taking the four-phonon resonant ${\mathrm{AgCrSe}}_{2}$ and three-phonon resonant PbSe model systems as examples, the phonon nonquasiparticle dynamics and their lifetimes in real materials are further investigated by first-principle calculations. Substantial discrepancies between ${\ensuremath{\tau}}_{\mathrm{GK}}$ and $\ensuremath{\tau}$ are found for the strongly resonant phonons at high-symmetrical points in both materials. Meanwhile, the lifetime ${\ensuremath{\tau}}_{\mathrm{GK}}$ of the phonons that are not subjected to resonant phonon interactions almost recovers to the conventional theoretical result $\ensuremath{\tau}$ despite the non-Lorentzian spectral feature of these phonons. It is suggested that $\ensuremath{\omega}{\ensuremath{\tau}}_{\mathrm{GK}}$, instead of the conventional $\ensuremath{\omega}\ensuremath{\tau}$, can be applied as a criterion to distinguish phonon quasiparticles and nonquasiparticles in addition to the phonon spectral functions.