Green-Kubo Formula Research Articles

Molecular dynamics simulation of annular flow boiling with the modified Lennard-Jones potential function

Molecular dynamics simulation of annular flow boiling in a nanochannel is numerically investigated. In this research, an annular flow model is developed to predict the superheated flow boiling heat transfer characteristics in a nanochannel. To characterize the forced annular boiling flow in a nanochannel, an external driving force $$ \overrightarrow {F}_{\text{ext}} $$ ranging from 1 to 12 PN (PN = pico newton) is applied along the flow direction to inlet fluid particles during the simulation. Based on an annular flow model analysis, it is found that saturation condition and superheat degree have great influences on the liquid–vapor interface. Also, the results show that due to the relatively strong influence of the surface tension in small channels, the interface between the liquid film and the vapor core is fairly smooth, and the mean velocity along the stream-wise direction does not change anymore. Also, it is found that the heat flux values depend on the boundary conditions. Finally, the Green–Kubo formula is used to calculate the thermal conductivity of liquid Argon. The simulations predict thermal conductivity of liquid Argon quite well.

Phonon Hall effect in ionic crystals in the presence of static magnetic field

We study phonon Hall effect (PHE) for ionic crystals in the presence of static magnetic field. Using Green-Kubo formula, we present an exact calculation of thermal conductivity tensor by considering both positive and negative frequency phonons. Numerical results are shown for some lattices such as hexagonal lattices, triangular lattices, and square lattices. We find that the PHE occurs on the nonmagnetic ionic crystal NaCl, although the magnitude is very small which is due to the tiny charge-to-mass ratio of the ions. The off-diagonal thermal conductivity is finite for nonzero magnetic field and changes sign for high value of magnetic field at high temperature. We also found that the off-diagonal thermal conductivity diverges as $\pm{1/T}$ at low temperature.

Symmetries in fluctuations far from equilibrium

Fluctuations arise universally in nature as a reflection of the discrete microscopic world at the macroscopic level. Despite their apparent noisy origin, fluctuations encode fundamental aspects of the physics of the system at hand, crucial to understand irreversibility and nonequilibrium behavior. To sustain a given fluctuation, a system traverses a precise optimal path in phase space. Here we show that by demanding invariance of optimal paths under symmetry transformations, new and general fluctuation relations valid arbitrarily far from equilibrium are unveiled. This opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations. We illustrate this concept studying symmetries of the current distribution out of equilibrium. In particular we derive an isometric fluctuation relation that links in a strikingly simple manner the probabilities of any pair of isometric current fluctuations. This relation, which results from the time-reversibility of the dynamics, includes as a particular instance the Gallavotti-Cohen fluctuation theorem in this context but adds a completely new perspective on the high level of symmetry imposed by time-reversibility on the statistics of nonequilibrium fluctuations. The new symmetry implies remarkable hierarchies of equations for the current cumulants and the nonlinear response coefficients, going far beyond Onsager's reciprocity relations and Green-Kubo formulas. We confirm the validity of the new symmetry relation in extensive numerical simulations, and suggest that the idea of symmetry in fluctuations as invariance of optimal paths has far-reaching consequences in diverse fields.

Determination of Shear Viscosity of Molecular Nitrogen (N2): Molecular Dynamic Hard Rotor Methodology and the Results

Determination of shear viscosity of molecular nitrogen (N(2)) by molecular dynamics (MD) in the high density range needs explicit incorporation of the rotational motion and therefore precise knowledge of angular dependence of N(2)-N(2) intermolecular potential. Newly designed Couette flow nonequilibrium molecular dynamic (NEMD) simulation procedure employs corrugated moving boundary, coupling the moving walls to translational and rotational motion exactly. Low density data on nitrogen viscosity show good agreement between MD data and experiment, confirming the radial dependence of the potential derived from quantum mechanical (QM) high precision calculations (coupled-cluster singles-and-doubles with a perturbative triples corrections [CCSD(T)]). Additionally, the angular dependence of the potential is verified using shear viscosity data for high density region, obtained from newly developed molecular dynamics (MD) simulations. It was demonstrated that the corrugated wall flow simulations provide results that are independent of the details of wall potential, fulfilling a basic requirement for application of MD simulations. These results are in good agreement with the equilibrium molecular dynamics (EMD) viscosity, derived from the Green-Kubo formula. Derived analytical dependence of the shear viscosity on the density and temperature shows that the MD data are in good agreement with experiment. Thus, MD simulations indicate that the CCSD(T) potential angular form is sufficiently precise for determination of the viscosity in a wide range of temperature and pressure.

Mass dependence of mutual diffusion coefficient: computer simulation study

Molecular dynamic computer simulations are performed to study the mass dependence of mutual diffusion coefficient in an isotopic, equimolar binary system. The particles of the system are assumed to be interacting via Lennard-Jones potential. The self- and mutual diffusion coefficients are calculated from the time dependence of the mean square displacement and from the velocity correlation function using the Green–Kubo formula. The study has been carried out at different densities and temperatures. Like the self-diffusion coefficient, the mutual diffusion coefficient is also found to be weakly dependent on the mass ratio. Our study also shows that the temperature of the system has a negligible effect on the mass dependence of the diffusion coefficients.

Molecular dynamics simulation of nanofluid’s effective thermal conductivity in high-shear-rate Couette flow

Effective thermal conductivity of Ar–Cu nanofluid in shear field is calculated by equilibrium molecular dynamics (EMD) simulation using Green–Kubo formula. The shear field is formed by imposing constant shear rate Couette flow with modified Lees–Edwards periodic boundary condition. The nanoparticle in the nanofluid in shear field rotates under the action of the velocity gradient. The rotation induces enhanced “microconvection” effect which is the main reason for the linear increase in the effective thermal conductivity of the shearing nanofluid with the shear rate increasing. The increase is more sharply with lower volume fraction of nanoparticle than with higher volume fraction, because the “microconvection” effect is weakened in the nanofluid with higher volume fraction of nanoparticle resulted by the slower nanoparticle rotation speed. The effective thermal conductivity obtained from the conventional correlation which is proposed for the flowing suspensions containing micro-sized particles are significantly lower than our numerical results. Moreover, the effect of nanoparticle volume fraction is more obvious in our numerical results. Therefore, the conventional correlation is not suitable when the sizes of the suspended particles are reduced to nanometers (nanofluid).

Mutual diffusion coefficients of heptane isomers in nitrogen: A molecular dynamics study

The accurate knowledge of transport properties of pure and mixture fluids is essential for the design of various chemical and mechanical systems that include fluxes of mass, momentum, and energy. In this study we determine the mutual diffusion coefficients of mixtures composed of heptane isomers and nitrogen using molecular dynamics (MD) simulations with fully atomistic intermolecular potential parameters, in conjunction with the Green-Kubo formula. The computed results were compared with the values obtained using the Chapman-Enskog (C-E) equation with Lennard-Jones (LJ) potential parameters derived from the correlations of state values: MD simulations predict a maximum difference of 6% among isomers while the C-E equation presents that of 3% in the mutual diffusion coefficients in the temperature range 500-1000 K. The comparison of two approaches implies that the corresponding state principle can be applied to the models, which are only weakly affected by the anisotropy of the interaction potentials and the large uncertainty will be included in its application for complex polyatomic molecules. The MD simulations successfully address the pure effects of molecular structure among isomers on mutual diffusion coefficients by revealing that the differences of the total mutual diffusion coefficients for the six mixtures are caused mainly by heptane isomers. The cross interaction potential parameters, collision diameter σ(12), and potential energy well depth ɛ(12) of heptane isomers and nitrogen mixtures were also computed from the mutual diffusion coefficients.

Shear viscosity to entropy density ratio in BUU transport model

Shear viscosity (eta) is a basic transport coefficient of the medium. In this work, we calculate shear viscosity to entropy density ratio (eta/S) of an equilibrated system in intermediate energy heavy ion collisions within the framework of the Boltzmann-Uehling-Uhlenbeck model (BUU) model. After the equilibration of Au + Au system at central collision in a fixed volume is reached, temperature, pressure and energy density are extracted by the phase space information and then eta/S is calculated using the Green-Kubo formulas. The results show that eta/S drops with the incident energy and its value is not so drastically different from the RHIC results.

Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps

We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation. Using a technique by which Taylor–Green–Kubo formulae are evaluated in terms of generalized Takagi functions, we derive exact, fully analytical expressions for the diffusion coefficients. Typically, for simple maps these quantities are fractal functions of control parameters. However, our family of four maps exhibits both fractal and linear behaviour. We explain these different structures by looking at the topology of the Markov partitions and the ergodic properties of the maps.

Asymptotic analysis of the Green-Kubo formula

[Received on 30 November 2009; revised on 10 May 2010; accepted on 13 May 2010] A detailed study of various distinguished limits of the Green–Kubo formula for the self-diffusion coefficient is presented in this paper. First, an alternative representation of the Green–Kubo formula in terms of the solution of a Poisson equation is derived when the microscopic dynamics is Markovian. Then the techniques developed in Golden & Papanicolaou (1983, Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys. , 90, 473–491) and Avellaneda & Majda (1991, An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows. Commun. Math. Phys. , 138, 339–391) are used to obtain a Stieltjes integral representation formula for the symmetric and antisymmetric parts of the diffusion tensor. The effect of irreversible microscopic dynamics on the diffusion coefficient is analysed and various asymptotic limits of physical interest are studied. Several examples are presented that confirm the findings of our theory.

The Green–Kubo formula for general Markov processes with a continuous time parameter

For general Markov processes, the Green–Kubo formula is shown to be valid under a mild condition. A class of stochastic evolution equations on a separable Hilbert space and three typical infinite systems of locally interacting diffusions on (irreversible in most cases) are shown to satisfy the Green–Kubo formula, and the Einstein relations for these stochastic evolution equations are shown explicitly as a corollary.

Approach to a Stationary State in an External Field

We study relaxation towards a stationary out-of-equilibrium state by analyzing a one-dimensional stochastic process followed by a particle accelerated by an external field and propagating through a thermal bath. The effect of collisions is described within one-dimensional formulation of Boltzmann’s kinetic theory. We present analytical solutions for the Maxwell gas and for the very hard particle model. The exponentially fast relaxation of the velocity distribution towards the stationary form is demonstrated. In the reference frame moving with constant drift velocity the hydrodynamic diffusive mode is shown to govern the distribution in the position space. We show that the exact value of the diffusion coefficient for any value of the field is correctly predicted by the Green-Kubo autocorrelation formula generalized to the stationary state.

How to improve the accuracy of equilibrium molecular dynamics for computation of thermal conductivity?

Equilibrium molecular dynamics (EMD) simulations through Green–Kubo formula (GKF) have been widely used in the study of thermal conductivity of various materials. However, there exist controversial simulation results which have huge discrepancies with experimental ones in literatures. In this Letter, we demonstrate that the fluctuation in calculated thermal conductivity is due to the uncertainty in determination of the truncation time, which is related to the ensemble and size dependent phonon relaxation time. We thus propose a new scheme in the direct integration of heat current autocorrelation function (HCACF) and a nonzero correction in the double-exponential-fitting of HCACF to describe correctly the contribution to thermal conductivity from low frequency phonons. By using crystalline Silicon (Si) and Germanium (Ge) as examples, we demonstrate that our method can give rise to the values of thermal conductivity in an excellent agreement with experimental ones.

Evaluation of heat current formulations for equilibrium molecular dynamics calculations of thermal conductivity

Equilibrium molecular dynamics combined with the Green-Kubo formula can be used to calculate the thermal conductivity of materials such as germanium and carbon. The foundation of this calculation is extracting the heat current from the results and implementing it into the Green-Kubo formula. This work considers all formulations from the literature that calculate the heat current for the Tersoff potential, the interatomic potential most applicable to semiconductor materials. The formulations for the heat current are described, and results for germanium and carbon are presented. The formulations are compared with respect to how well they capture the physics of the Tersoff potential and how well the calculated value of the thermal conductivity reflects the experimentally measured value.

Approach to a stationary state in an external field

We study relaxation towards a stationary out of equilibrium state by analizing a one-dimensional stochastic process followed by a particle accelerated by an external field and propagating through a thermal bath. The effect of collisions is described within Botlzmann's kinetic theory. We present analytical solutions for the Maxwell gas and for the very hard particle model. The exponentially fast relaxation of the velocity distribution toward the stationary form is demonstrated. In the reference frame moving with constant drift velocity the hydrodynamic diffusive mode is shown to govern the distribution in the position space. We show that the exact value of the diffusion coefficient for any value of the field is correctly predicted by Green-Kubo autocorrelation formula generalized to the stationary state.

Generalized Green-Kubo Formula for a Dissipative Quantum System

A generalized Green-Kubo formula is derived for a quantum dissipative system of driven Brownian particle, in which the coupling between the system and the environment is linear. The structure is essentially the same as that for the generalized Green-Kubo formula for driven granular particles. It is demonstrated that the correction to the conventional Green-Kubo formula is zero for a free Brownian particle.

Mori–Zwanzig formalism as a practical computational tool

An operational procedure is presented to compute explicitly the different terms in the generalized Langevin equation (GLE) for a few relevant variables obtained within Mori-Zwanzig formalism. The procedure amounts to introducing an artificial controlled parameter which can be tuned in such a way that the so-called projected dynamics becomes explicit and the GLE reduces to a Markovian equation. The projected dynamics can be realised in practice by introducing constraints, and it is shown that the Green-Kubo formulae computed with these dynamics do not suffer from the plateau problem. The methodology is illustrated in the example of star polymer molecules in a melt using their center of mass as relevant variables. Through this example, we show that not only the effective potentials, but also the friction forces and the noise play a very important role in the dynamics.

Correlations and the Ring-Kinetic Equation in Dense Sheared Granular Flows

A formal way of deriving fluctuation-correlation relations in dense sheared granular media, starting with the Enskog approximation for the collision integral in the Chapman-Enskog theory, is discussed. The correlation correction to the viscosity is obtained using the ring-kinetic equation, in terms of the correlations in the hydrodynamic modes of the linearised Enskog equation. It is shown that the Green-Kubo formula for the shear viscosity emerges from the two-body correlation function obtained from the ring-kinetic equation.

Boltzmann equation and hydrodynamic fluctuations

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.

On the definition of discrete hydrodynamic variables

The Green-Kubo formula for discrete hydrodynamic variables involves information about not only the fluid transport coefficients but also about discrete versions of the differential operators that govern the evolution of the discrete variables. This gives an intimate connection between discretization procedures in fluid dynamics and coarse-graining procedures used to obtain hydrodynamic behavior of molecular fluids. We observed that a natural definition of discrete hydrodynamic variables in terms of Voronoi cells leads to a Green-Kubo formula which is divergent, rendering the full coarse-graining strategy useless. In order to understand this subtle issue, in the present paper we consider the coarse graining of noninteracting Brownian particles. The discrete hydrodynamic variable for this problem is the number of particles within Voronoi cells. Thanks to the simplicity of the model we spot the origin of the singular behavior of the correlation functions. We offer an alternative definition, based on the concept of a Delaunay cell that behaves properly, suggesting the use of the Delaunay construction for the coarse graining of molecular fluids at the discrete hydrodynamic level.