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The discrete velocity method (DVM) for rarefied flows and the unified methods (based on the DVM framework) for flows in all regimes have proven to be effective flow solvers over the past decades and have been successfully extended to other important physical fields. Both DVM and unified methods strive to physically model gas-gas interactions. However, for gas-surface interactions (GSI) at the wall boundary, they have only used the full accommodation boundary until now, which deviates from reality. Moreover, the accommodation degrees of momentum and thermal energy are often different. To overcome this limitation and extend the DVM and unified methods to more realistic boundary conditions, an algorithm for the Cercignani-Lampis (CL) boundary with different accommodation degrees of momentum and thermal energy is proposed and integrated into the DVM framework. A Gaussian distribution with slip velocity, including several undetermined parameters, is designed to describe the reflected distribution function. Once the accommodation coefficients (ACs) are known, these undetermined parameters are solved using the relationship between ACs and reflected macroscopic fluxes. In this paper, the Maxwell model and CLL model are adopted as benchmarks to evaluate the performance of the proposed CL model. Additionally, the current CL boundary for the DVM framework considers generality, accommodating both the recently developed efficient unstructured velocity space and the traditional Cartesian velocity space. Moreover, the proposed CL model allows for calculations of both monatomic gases and diatomic gases with internal degrees of freedom. Finally, by being integrated with the unified gas-kinetic scheme within the DVM framework, the performance of the present CL model is validated through a series of benchmark numerical tests across a wide range of Knudsen numbers.

A Multi-scale Boltzmann Equation (MBE) is found from the gas-kinetic theory and the direct modeling philosophy as a master equation for complex physical systems of neutral gases across all flow regimes, which locates between the continuum limit and the free-molecular limit, covering a vast range of applications such as hypersonic flows over aerospace crafts and delicate flows around MEMS. The most explicit characteristic of MBE is evolving the variable observation time in the expression, which distinguishes the MBE from the single-scale master or governing equation where a fixed scale is implied in the assumptions. The fundamental properties of MBE, such as the asymptotic property, are proved theoretically, while a concise numerical scheme is developed for MBE to demonstrate its validity by benchmark multi-scale problems.

The high-order gas-kinetic scheme(HGKS) with WENO spatial reconstruction method has been extensively validated through many numerical experiments, demonstrating its superior accuracy efficiency, and robustness. Compared with WENO class schemes, TENO class schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. In this paper, two kinds of fifth-order HGKS with TENO class schemes are designed. One involves replacing WENO5 scheme with the TENO5 scheme in the conventional WENO5-GKS. WENO and TENO schemes only provide the non-equilibrium state values at the cell interface. The slopes of the non-equilibrium state along with the equilibrium values and slopes, are obtained by additional linear reconstruction. Another kind of TENO5-D GKS is similar to WENO5-AO GKS. Following a strong scale-separation procedure, a tailored novel ENO-like stencil selection strategy is proposed such that the high-order accuracy is restored in smooth regions by selecting the candidate reconstruction on the large stencil while the ENO property is enforced near discontinuities by adopting the candidate reconstruction from smooth small stencils. The such TENO schemes are TENO-AA and TENO-D scheme. The HGKS scheme based on WENO-AO or TENO-D reconstruction take advantage of the large stencil to provide point values and slopes of the non-equilibrium state. By dynamically merging the reconstructed non-equilibrium slopes, extra reconstruction of the equilibrium state at the beginning of each time step can be avoided. The simplified schemes have better robustness and efficiency than the conventional WENO5-GKS or TENO5-GKS. TENO-D GKS is also as easy to develop as WENO-AO GKS to high-order finite volume method for unstructured mesh., Comment: arXiv admin note: text overlap with arXiv:2304.05572; text overlap with arXiv:1905.08489 by other authors

In this work, the Navier-Stokes (NS) solver is combined with the Direct simulation Monte Carlo (DSMC) solver in a direct way, under the wave-particle formulation [J. Comput. Phys. 401, 108977 (2020)]. Different from the classical domain decomposition method with buffer zone for overlap, in the proposed direct unified wave-particle (DUWP) method, the NS solver is coupled with DSMC solver on the level of algorithm. Automatically, in the rarefied flow regime, the DSMC solver leads the simulation, while the NS solver leads the continuum flow simulation. Thus advantages of accuracy and efficiency are both taken. At internal flow regimes, like the transition flow regime, the method is accurate as well because a kind of mesoscopic modeling is proposed in this work, which gives the DUWP method the multi-scale property. Specifically, as to the collision process, at $t < \tau$, it is supposed that only single collision happens, and the collision term of DSMC is just used. At $t > \tau$, it is derived that $1-\tau/\Delta t$ of particles should experience multiple collisions, which will be absorbed into the wave part and calculated by the NS solver. Then the DSMC and NS solver can be coupled in a direct and simple way, bringing about multi-scale property. The governing equation is derived and named as multi-scale Boltzmann equation. Different from the original wave-particle method, in the proposed DUWP method, the wave-particle formulation is no more restricted by the Boltzmann-BGK type model and the enormous research findings of DSMC and NS solvers can be utilized into much more complicated flows, like the thermochemical non-equilibrium flow. In this work, one-dimensional cases in monatomic argon gas are preliminarily tested, such as shock structures and Sod shock tubes.

The high-order gas-kinetic scheme (HGKS) features good robustness, high efficiency and satisfactory accuracy,the performaence of which can be further improved combined with WENO-AO (WENO with adaptive order) scheme for reconstruction. To reduce computational costs in the reconstruction procedure, this paper proposes to combine HGKS with a hybrid WENO-AO scheme. The hybrid WENO-AO scheme reconstructs target variables using upwind linear approximation directly if all extreme points of the reconstruction polynomials for these variables are outside the large stencil. Otherwise, the WENO-AO scheme is used. Unlike combining the hybrid WENO scheme with traditional Riemann solvers, the troubled cell indicator of the hybrid WENO-AO method is fully utilized in the spatial reconstruction process of HGKS. During normal and tangential reconstruction, the gas-kinetic scheme flux not only needs to reconstruct the conservative variables on the left and right interfaces but also to reconstruct the derivative terms of the conservative variables. By reducing the number of times that the WENO-AO scheme is used, the calculation cost is reduced. The high-order gas-kinetic scheme with the hybrid WENO-AO method retains original robustness and accuracy of the WENO5-AO GKS, while exhibits higher computational efficiency.

A computationally accurate and efficient numerical method under a unified framework is crucial to various multi-scale scientific and engineering problems. So far, many numerical methods have encountered various challenges in efficiently solving multi-scale non-equilibrium flows that cover a wide range of Knudsen numbers, especially the three-dimensional hypersonic flows. In this study, a conservative implicit scheme is further presented for three-dimensional steady flows of diatomic gases in all flow regimes, where both the implicit microscopic kinetic equations based on Rykov model and the corresponding implicit macroscopic governing equations are solved synchronously. Furthermore, a simplified multi-scale numerical flux inspired by the strategy of discrete unified gas kinetic scheme (DUGKS) is proposed to relieve the limitation of grid size and time step in all flow regimes. The flux is constructed through a backward Euler difference scheme with the consideration of collision effect in the physical reconstruction of the gas distribution function on the cell interface. Meanwhile, its asymptotic preserving property in the continuum limit is analyzed. In order to pursue high computational efficiency in three-dimensional flow simulations, the unstructured discrete velocity space (DVS) and MPI parallel in DVS are adopted to further speed up the calculation. Additionally, based on numerical experiments of Apollo 6 command module, an empirical generation criterion for three-dimensional unstructured DVS is proposed. Numerical results indicate that, the present method is about one to two orders of magnitude faster than the explicit conserved DUGKS method. The present method is proved to accurately and efficiently predict aerothermodynamic properties of hypersonic rarefied gas flows.

Orowan strengthening is a primary way of strengthening metallic materials.The effect of temperature on the Orowan strengthening mechanism is still debatable in the present day.In this study,the effect of temperature on the Orowan strengthening mechanism is systematically investigated,a thermal activation Orowan strengthening mechanism is developed,and the corresponding Orowan stress is deduced.The results indicate that the obstacle scales substantially affect the thermal activation-based dislocation that bypasses the precipitates processes.In small obstacle scales,the thermal activation contribution to Orowan stress cannot be ignored;while in large obstacle scales,the thermal activation contribution can be ignored even at high temperatures.In addition to temperature, the effects of shear modulus,strain rate, and dislocation density are investigated on Orowan stress.This work not only provides new insight into the Orowan strengthening mechanism but also aids in the development of new high-temperature structural materials.

The rarefied flow and multi-scale flow are crucial for the aerodynamic design of spacecraft, ultra-low orbital vehicles and plumes. By introducing a discrete velocity space, the Boltzmann method, such as the discrete velocity method and unified methods, can capture complex and non-equilibrium velocity distribution functions (VDFs) and describe flow behaviors exactly. However, the extremely steep slope and high concentration of the gas VDFs in a local particle velocity space make it very difficult for the Boltzmann method with structured velocity space to describe high speed flow. Therefore, the adaptive velocity space (AVS) is required for the Boltzmann solvers to be practical in complex rarefied flow and multi-scale flow. This paper makes two improvements to the AVS approach, which is then incorporated into a general discrete velocity framework, such as the unified gas-kinetic scheme. Firstly, a global velocity mesh is used to prevent the interpolation of the VDFs at the physical interface during the calculation of the microscopic fluxes, maintaining the program's high level of parallelism. Secondly, rather than utilizing costly interpolation, the VDFs on a new velocity space were reconstruction using the ``consanguinity" relationship. In other words, a split child node's VDF is the same as its parent's VDF, and a merged parent's VDF is the average of its children's VDFs. Additionally, the discrete deviation of the equilibrium distribution functions is employed to maintain the proposed method's conservation. Moreover, an appropriate set of adaptive parameters is established to enhance the automation of the proposed method. Finally, a number of numerical tests are carried out to validate the proposed method.

The whole Al-Li phase diagram is predicted from first principles calculations and statistical mechanics including the effect of configurational and vibrational entropy. The formation enthalpy of different configurations at different temperatures was accurately predicted by means of cluster expansions that were fitted from first principles calculations. The vibrational entropic contribution of each configuration was determined from the bond length vs. bond stiffness relationships for each type of bond and the Gibbs free energy of the different phases was obtained as a function of temperature from Monte Carlo simulations. The predicted phase diagram was in excellent agreement with the currently accepted experimental one in terms of the stable (AlLi, Al2Li3, AlLi2, Al4Li9) and metastable (Al3Li) phases, of the phase boundaries between them and of the maximum stability temperature of line compounds. In addition, it provided accurate information about the gap between Al3Li and AlLi solvus lines. Finally, the influence of the vibrational entropy on the correct prediction of the phase diagram is discussed. Overall, the methodology shows that accurate phase diagrams of alloys of technological interest can be predicted from first principles calculations.

In this study, the multiple solutions of Nonlinear Coupled Constitutive Relation (NCCR) model are firstly observed and a way for identifying the physical solution is proposed. The NCCR model proposed by Myong is constructed from the generalized hydrodynamic equations of Eu, and aims to describe rarefied flows. The NCCR model is a complicated nonlinear system. Many assumptions have been used in the schemes for solving the NCCR equations. The corresponding numerical methods may be associated with unphysical solution and instability. At the same time, it is hard to analyze the physical accuracy and stability of NCCR model due to the uncertainties in the numerical discretization. In this study, a new numerical method for solving NCCR equations is proposed and used to analyze the properties of NCCR equations. More specifically, the nonlinear equations are converted into the solutions of an objective function of a single variable. Under this formulation, the multiple solutions of the NCCR system are identified and the criteria for picking up the physical solution are proposed. Therefore, a numerical scheme for solving NCCR equations is constructed. A series of flow problems in the near continuum and low transition regimes with a large variation of Mach numbers are conducted to validate the numerical performance of proposed method and the physical accuracy of NCCR model., Comment: 47 pages, 67 figures

The rarefied flow and multi-scale flow are crucial for the aerodynamic design of spacecraft, ultra-low orbital vehicles and plumes. By introducing a discrete velocity space, the discrete velocity method (DVM) and unified methods can capture complex and non-equilibrium distribution functions and describe flow behaviors exactly. The unified methods predict flows from continuum to rarefied regimes by adopting unified modeling, and they can be further applied to other multi-scale physics such as radiation heat transfer, phonon heat transfer and plasma. In the flow field, the concrete dynamic process needs to describe the gas-gas interaction and gas-surface interaction (GSI). However, in both DVM and unified methods, only a simple but not accurate GSI is used, which can be regarded as a Maxwell GSI with a fixed accommodation coefficient of 1 (full accommodation) at the present stage. To overcome the bottleneck in extending DVM and unified methods to the numerical experiment and investigate real multi-scale flow physics, this paper realizes precise GSI in the DVM framework by constructing the boundary conditions of a concrete Maxwell GSI with an adjustable accommodation coefficient. In the constructing process, the problems of macro-conservation and micro-consistency in the DVS at the boundary are well solved by reflected macroscopic flux and interpolation distribution function and interpolation error correction, respectively. Meanwhile, considering that the multi-scale flows in the background of aeronautics and aerospace are often at supersonic and hypersonic speeds, the unstructured velocity space (UVS) is essential. From the perspective of generality, the GSI is forced on UVS. Besides, by combined with the unified method (the unified gas-kinetic scheme in the paper), the effectiveness and validity of the present GSI on the DVM framework are verified by a series of simulations.

In this paper, a unified gas-kinetic scheme (UGKS) with simplified multi-scale numerical flux is proposed for the thermodynamic non-equilibrium flow simulation involving the excitation of molecular vibrational degrees of freedom in all flow regimes. The present UGKS keep the basic conservation laws of the macroscopic flow variables and the microscopic gas distribution function in a discretized space. In order to improve the efficiency of the UGKS, a simplify multi-scale numerical flux is directly constructed from the characteristic difference solution of the kinetic model equation. In addition, a new BGK-type kinetic model for diatomic gases is proposed to describe the high-temperature thermodynamic non-equilibrium effect, which is a phenomenological relaxation model with the continuous distribution modes of rotational and vibrational energies. In present model, the equilibrium distribution functions is constructed by using a multi-dimensional Hermitian expansion around the Maxwellian distribution to achieve the correct Prandtl number and proper relaxation rate of heat fluxes. Furthermore, the application of the unstructured discrete velocity space (DVS) and a simple integration error correction reduce the number of velocity mesh significantly and make the present method be a efficient tool for simulations of flows in all flow regimes. The new scheme are examined in a series of cases, such as Sod's shock tube, high non-equilibrium shock structure, hypersonic flow around a circular cylinder with Knudsen (Kn) number Kn = 0.01, and the rarefied hypersonic flow over a flat plate with a sharp leading edge. The present UGKS results agree well with the benchmark data of DSMC and the other validated methods.

In this paper, the nonlinear squeeze-film damping (SFD) involving rarefied gas effect in the micro-electro-mechanical-systems (MEMS) is investigated. Considering the motion of structures (beam, cantilever, and membrane) in MEMS, the dynamic response of structure will be influenced largely by the squeeze-film damping. In the traditional model, a viscous damping assumption that damping force is linear with moving velocity is used. As the nonlinear damping phenomenon is observed for a micro-structure oscillating with a high-velocity, this assumption is invalid and will generates error result for predicting the response of micro-structure. In addition, due to the small size of device and the low pressure of encapsulation, the gas in MEMS usually is rarefied gas. Therefore, to correctly predict the damping force, the rarefied gas effect must be considered. To study the nonlinear SFD phenomenon involving the rarefied gas effect, a kinetic method, namely discrete unified gas kinetic scheme (DUGKS), is introduced. And based on DUGKS, two solving methods, a traditional decoupled method (Eulerian scheme) and a coupled framework (arbitrary Lagrangian-Eulerian scheme), are adopted. With these two methods, two basic motion forms, linear (perpendicular) and tilting motions of a rigid micro-beam, are studied with forced and free oscillations.

The multi-scale flow mechanism is crucial for force and heat loaded on near-space and reentry vehicles, the control of spacecraft by thrusters, the propelling and cooling of MEMS, etc. Since the continuum flow and rarefied flow often exist simultaneously, the modeling and prediction for such a multi-scale flow field is very complicated. One important and efficient way of predicting the multi-scale flow is constructing numerical methods by adapting the multi-scale properties of the temporal integral solutions (or equivalent characteristic line solutions) for the model equations in the gas-kinetic theory. The model equations can be classified into FP-type and BGK-type, and the numerical methods can have deterministic form or stochastic particle form. Since these numerical methods are strictly based on model equations, they are also restricted by the model equations. The difficulty and complexity in constructing model equation that has complete asymptotic preserving property for gas mixture with non-equilibrium internal energy will prevent the further extension of these methods. Therefore, this paper addresses the question whether a multi-scale numerical method can be established not on the model equations, but directly adapts the relaxation rates of macroscopic variables such as stress and heat flux. Since the algorithm of multi-scale particle method based on BGK-type equation is clear and simple, its collision step is modified in this work, where the macroscopic variables firstly evolve according to their relaxation rates, and then the molecules that participate in inter-molecular collisions get their after-collision velocities from the after-evolution macroscopic variables. Therefore, this direct relaxation process does not depend on the model equations anymore. Finally, the validity and accuracy of the particle method with direct relaxation process are examined with five test cases., Comment: 39 pages, 46 figures

In this study, the kinetic inviscid flux (KIF) is improved and an implicit strategy is coupled. The recently proposed KIF is a kind of inviscid flux, whose microscopic mechanism makes it good at solving shock waves, with advantages against the shock instability phenomenon. When developing the implicit KIF, a phenomenon is noticed that the kinetic flux vector splitting (KFVS) part in boundary layers not only reduces the accuracy, but seriously reduces the Courant-Friedrichs-Lewy (CFL) number as well. As a result, in this paper, a new weight is proposed about how to combine the KFVS method well with the totally thermalized transport (TTT) method. Besides admitting the using of larger CFL numbers, this new weight brings about more accurate numerical results like pressure, friction coefficient and heat flux when solving shock waves, boundary layers and complex supersonic/hypersonic flows. In order to examine the validity, accuracy and efficiency of the present method, six numerical test cases, covering the whole speed regime, are conducted, including the hypersonic viscous flow past a cylinder, the hypersonic double-cone flow, the hypersonic double-ellipsoid flow, the laminar shock boundary layer interaction, the supersonic flow around a ramp segment and the lid-driven cavity flow. The advantages of this scheme and corresponding mechanisms are to be discussed in detail., Comment: 29 pages, 30 figures

The goal of this paper is to develop a high order numerical method based on Kinetic Inviscid Flux (KIF) method and Flux Reconstruction (FR) framework. The KIF aims to find a balance between the excellent merits of Gas-Kinetic Scheme (GKS) and the lower computational costs. The idea of KIF can be viewed as an inviscid-viscous splitting version of the gas-kinetic scheme, and Shu and Ohwada have made the fundamental contribution. The combination of Totally Thermalized Transport (TTT) scheme and Kinetic Flux Vector Splitting (KFVS) method are achieved in KIF. Using a coefficient which is related to time step $\delta t$ and averaged collision time $\tau$, KIF can adjust the weights of TTT and KFVS flux in the simulation adaptively. By doing the inviscid-viscous splitting, KIF is very suitable and easy to integrate into the existing framework. The well understood FR framework is used widely for the advantages of robustness, economical costs and compactness. The combination of KIF and FR is originated by three motivations. The first purpose is to develop a high order method based on the gas kinetic theory. The second reason is to keep the advantages of GKS. The last aim is that the designed method should be more efficient. In present work, we use the KIF method to replace the Riemann flux solver applied in the interfaces of elements. The common solution at the interface is computed according to the gas kinetic theory, which makes the combination of KIF and FR scheme more reasonable and available. The accuracy and performance of present method are validated by several numerical cases. The Taylor-Green vortex problem has been used to verify its potential to simulate turbulent flows., Comment: 33 Pages, 21 Figures

A Kinetic Inviscid Flux (KIF) is proposed for simulating incompressible and compressible flows. It is constructed based on the direct modeling of multi-scale flow behaviors, which is used in the Gas-Kinetic Scheme (GKS), the Unified Gas-Kinetic Scheme (UGKS), the Discrete Unified Gas-Kinetic Scheme (DUGKS), etc.. In KIF, the discontinuities (such as the shock wave) that can not be well resolved by mesh cells are mainly solved by the Kinetic Flux Vector Splitting (KFVS) method representing the free transport mechanism (or micro-scale mechanism), while in other flow regions that are smooth, the flow behavior is solved mainly by the central-scheme-like Totally Thermalized Transport (TTT). The weights of KFVS and TTT in KIF is automatically determined by those in the theory of direct modeling. Two ways of choosing the weights in KIF are proposed, which are actually the weights adopted in the UGKS and the DUGKS, respectively. By using the test cases of Sod shock tube, rarefaction wave, the boundary layer of flat plate, the cavity flow and hypersonic flow over circular cylinder, the validity and accuracy of the present method are examined. The KIF does not suffer from the carbuncle phenomenon, and does not introduce extra numerical viscosity in smooth regions. Especially, in the case of hypersonic cylinder, it gives a quite sharp and clear density and temperature contours. The KIF can be viewed as an inviscid-viscous splitting version of the GKS. By the doing this splitting, it is easy to be used in the traditional CFD frameworks. It can also be classified as a new type in the numerical schemes based on the kinetic theory that are represented by the works in Ref. \cite{SunA} and Ref. \cite{ohwada2018a}, except the weights are determined by the weights of direct modeling., Comment: 31 pages, 39 figures

A Quantified Model-Competition (QMC) mechanism for multi-scale flows is extracted from the integral (analytical) solution of the Boltzmann-BGK model equation. In the QMC mechanism, the weight of the rarefied model and the weight of the continuum (aerodynamic/hydrodynamic) model are quantified. Then, a Simplified Unified Wave-Particle method (SUWP) is constructed based the on the QMC mechanism. In the SUWP, the stochastic particle method and the continuum Navier-Stokes method are combined together. Their weights are determined by the QMC mechanism quantitatively in every discrete cells of the computational domain. The validity and accuracy of the present numerical method are examined using a series of test cases including the high non-equilibrium shock wave structure case, the unsteady Sod shock-tube case with a wide range of Kn number, the hypersonic flow around the circular cylinder from the free-molecular regime to the near continuum regime, and the viscous boundary layer case. In the construction process of the present method, an anti-dissipation effect in the continuum mechanism is also discussed., Comment: 39 pages, 15 figures

In this paper, authors focus effort on improving the conventional discrete velocity method (DVM) into a multiscale scheme in finite volume framework for gas flow in all flow regimes. Unlike the typical multiscale kinetic methods unified gas-kinetic scheme (UGKS) and discrete unified gas-kinetic scheme (DUGKS), which concentrate on the evolution of the distribution function at the cell interface, in the present scheme the flux for macroscopic variables is split into the equilibrium part and the nonequilibrium part, and the nonequilibrium flux is calculated by integrating the discrete distribution function at the cell center, which overcomes the excess numerical dissipation of the conventional DVM in the continuum flow regime. Afterwards, the macroscopic variables are finally updated by simply integrating the discrete distribution function at the cell center, or by a blend of the increments based on the macroscopic and the microscopic systems, and the multiscale property is achieved. Several test cases, involving unsteady, steady, high speed, low speed gas flows in all flow regimes, have been performed, demonstrating the good performance of the multiscale DVM from free molecule to continuum Navier-Stokes solutions and the multiscale property of the scheme is proved., Comment: Added more discussions about the relevent works and added the corresponding references

Discrete unified gas-kinetic scheme (DUGKS) is a multi-scale numerical method for flows from continuum limit to free molecular limit, and is especially suitable for the simulation of multi-scale flows, benefiting from its multi-scale property. To reduce integration error of the DUGKS and ensure the conservation property of the collision term in isothermal flow simulations, a Conserved-DUGKS (C-DUGKS) is proposed. On the other hand, both DUGKS and C-DUGKS adopt Cartesian-type discrete velocity space, in which Gaussian and Newton-Cotes numerical quadrature are used for calculating the macroscopic physical variables in low speed and high speed flows, respectively. While, the Cartesian-type discrete velocity space leads to huge computational cost and memory demand. In this paper, the isothermal C-DUGKS is extended to the non-isothermal case by adopting coupled mass and inertial energy distribution functions. Moreover, since the unstructured mesh, such as the triangular mesh in two dimensional case, is more flexible than the structured Cartesian mesh, it is introduced to the discrete velocity space of C-DUGKS, such that more discrete velocity points can be arranged in the velocity regions that enclose large number of molecules, and only a few discrete velocity points need to be arranged in the velocity regions with small amount of molecules in it. By using the unstructured discrete velocity space, the computational efficiency of C-DUGKS is significantly increased. A series of numerical tests in a wide range of Knudsen numbers, such as the Couette flow, lid-driven cavity flow, two-dimensional rarefied Riemann problem and the supersonic cylinder flows, are carried out to examine the validity and efficiency of the present method.

A conservative discrete velocity method (DVM) is developed for the ellipsoidal Fokker-Planck (ES-FP) equation in prediction of non-equilibrium neutral gas flows in this paper. The ES-FP collision operator is solved in discrete velocity space in a concise and quick finite difference framework. The conservation problem of discrete ES-FP collision operator is solved by multiplying each term in it by extra conservative coefficients whose values are very closed to unity. Their differences to unity are in the same order of the numerical error in approximating the ES-FP operator in discrete velocity space. All the macroscopic conservative variables (mass, momentum and energy) are conserved in the present modified discrete ES-FP collision operator. Since the conservation property in discrete element of physical space is very important for numerical scheme when discontinuity and large gradient exist in flow field, a finite volume framework is adopted for the transport term of ES-FP equation. For $nD$-$3V$ ($n<3$) cases, a $nD$-quasi $nV$ reduction is specially proposed for ES-FP equation and the corresponding FP-DVM method, which can greatly reduce the computational cost. The validity and accuracy of both ES-FP equation and FP-DVM method are examined using a series of $0D$-$3V$ homogenous relaxation cases and $1D$-$3V$ shock structure cases with different $Mach$ numbers, in which $1D$-$3V$ cases are reduced to $1D$-quasi $1V$ cases. Both the predictions of $0D$-$3V$ and $1D$-$3V$ cases match well with the benchmark results such as analytical Boltzmann solution, direct full-Boltzmann numerical solution and DSMC result. Especially, the FP-DVM predictions match well with the DSMC results in the $Mach$ 8.0 shock structure case, which is in high non-equilibrium, and is a challenge case of the model Boltzmann equation and the corresponding numerical methods.

We propose a variational approach to study renormalized phonons in momentum conserving nonlinear lattices with either symmetric or asymmetric potentials. To investigate the influence of pressure to phonon properties, we derive an inequality which provides both the lower and upper bound of the Gibbs free energy as the associated variational principle. This inequality is a direct extension to the Gibbs-Bogoliubov inequality. Taking the symmetry effect into account, the reference system for the variational approach is chosen to be harmonic with an asymmetric quadratic potential which contains variational parameters. We demonstrate the power of this approach by applying it to one dimensional nonlinear lattices with a symmetric or asymmetric Fermi-Pasta- Ulam type potential. For a system with a symmetric potential and zero pressure, we recover existing results. For other systems which beyond the scope of existing theories, including those having the symmetric potential and pressure, and those having the asymmetric potential with or without pressure, we also obtain accurate sound velocity., Comment: 10 pages, 10 figures, comments are welcome

Transport and diffusion of heat in one dimensional (1D) nonlinear systems which {\it conserve momentum} is typically thought to proceed anomalously. Notable exceptions, however, exist of which the rotator model is a prominent case. Therefore, the quest arises to identify the origin of manifest anomalous transport in those low dimensional systems. Here, we develop the theory for both, momentum/heat diffusion and its corresponding momentum/heat transport. We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux. This result in turn relates, via the integrated momentum flux correlation, to an effective viscosity, or equivalently, to the underlying momentum diffusivity. We put forward the intriguing hypothesis that a fluid-like momentum dynamics with a {\it finite viscosity} causes {\it normal} heat transport; its corollary being that superdiffusive momentum diffusion with an intrinsic {\it diverging viscosity} in turn yields {\it anomalous} heat transport. This very hypothesis is corroborated over wide extended time scales by use of precise molecular dynamics simulations. The numerical validation of the hypothesis involves three distinct archetype classes of nonlinear pair-interaction potentials: (i) a globally bounded pair interaction (the noted rotator model), (ii) unbounded interactions acting at large distances (the Fermi-Pasta-Ulam $\beta$ model, or the rotator model amended with harmonic pair interactions) and (iii), a pair interaction potential being unbounded at short distances while displaying an asymptotic free part (Lennard-Jones model)., Comment: New J. Phys. (in press)

Guided by a stylized experiment we develop a self-consistent anharmonic phonon concept for nonlinear lattices which allows for explicit "visualization." The idea uses a small external driving force which excites the front particles in a nonlinear lattice slab and subsequently one monitors the excited wave evolution using molecular dynamics simulations. This allows for a simultaneous, direct determination of the existence of the phonon mean free path with its corresponding anharmonic phonon wavenumber as a function of temperature. The concept for the mean free path is very distinct from known prior approaches: the latter evaluate the mean free path only indirectly, via using both, a scale for the phonon relaxation time and yet another one for the phonon velocity. Notably, the concept here is neither limited to small lattice nonlinearities nor to small frequencies. The scheme is tested for three strongly nonlinear lattices of timely current interest which either exhibit normal or anomalous heat transport.

Consider anomalous energy spread in solid phases, i.e., $MSD= \int (x -{< x >}_E)^2 \rho_E(x,t)dx \propto t^{\beta}$, as induced by a small initial excess energy perturbation distribution $\rho_{E}(x,t=0)$ away from equilibrium. The associated total thermal equilibrium heat flux autocorrelation function $C_{JJ}(t)$ is shown to obey rigorously the intriguing relation, $d^2 MSD/dt^2 = 2C_{JJ}(t)/(k_BT^2c)$, where $c$ is the specific volumetric heat capacity. Its integral assumes a time-local Helfand-moment relation; i.e. $ dMSD/dt|_{t=t_s} = 2/(k_BT^2c)\int_0^{t_s} C_{JJ}(s)ds$, where the chosen cut-off time $t_s$ is determined by the maximal signal velocity for heat transfer. Given the premise that the averaged nonequilibrium heat flux is governed by an anomalous heat conductivity, energy diffusion scaling necessarily determines a corresponding anomalous thermal conductivity scaling behavior., Comment: This paper has been withdrawn by the authors. This arXiv article is now the improved and amended version of arXiv:1103.2835v4

We study full counting statistics for classical heat transport through anharmonic/nonlinear molecular junctions formed by interacting oscillators. Analytical result of the steady state heat flux for an overdamped anharmonic junction with arbitrary temperature bias is obtained. It is found that the thermal conductance can be expressed in terms of temperature dependent effective force constant. The role of anharmonicity is identified. We also give the general formula for the second cumulant of heat in steady state, as well as the average geometric heat flux when two temperatures are modulated adiabatically. We present an anharmonic example for which all cumulants for heat can be obtained exactly. For a bounded single oscillator model with mass we found that the cumulants are independent of the nonlinear potential.

Thermal transport is an important energy transfer process in nature. Phonon is the major energy carrier for heat in semiconductor and dielectric materials. In analogy to Ohm's law for electrical conductivity, Fourier's law is a fundamental rule of heat transfer in solids. It states that the thermal conductivity is independent of sample scale and geometry. Although Fourier's law has received great success in describing macroscopic thermal transport in the past two hundreds years, its validity in low dimensional systems is still an open question. Here we give a brief review of the recent developments in experimental, theoretical and numerical studies of heat transport in low dimensional systems, include lattice models, nanowires, nanotubes and graphenes. We will demonstrate that the phonon transports in low dimensional systems super-diffusively, which leads to a size dependent thermal conductivity. In other words, Fourier's law is breakdown in low dimensional structures.

We study classical heat conduction in a dissipative open system composed of interacting oscillators. By exactly solving a twisted Fokker-Planck equation which describes the full counting statistics of heat flux flowing through the system, we identify the geometric-phase-like effect and examine its impact on the classical heat transport. Particularly, we find that the nonlinear interaction as well as the closely related temperature-dependence of system-parameters are crucial in manifesting the geometric-phase contribution of heat flux. Finally, we propose an electronic experiment based on RC circuits to verify our theoretical predictions., Comment: 4+ pages, 3 figures, supplement included, accepted by PRL

Consider anomalous energy spread in solid phases, i.e., $MSD= \int (x -{\langle x \rangle}_E)^2 \rho_E(x,t)dx \propto t^{\beta}$, as induced by a small initial excess energy perturbation distribution $\rho_{E}(x,t=0)$ away from equilibrium. The associated total thermal equilibrium heat flux autocorrelation function $C_{JJ}(t)$ is shown to obey rigorously the intriguing relation, $d^2 MSD/dt^2 = 2C_{JJ}(t)/(k_BT^2c)$, where $c$ is the specific volumetric heat capacity. Its integral assumes a time-local Helfand-moment relation; i.e. $ dMSD/dt|_{t=t_s} = 2/(k_BT^2c)\int_0^{t_s} C_{JJ}(s)ds$, where the chosen cut-off time $t_s$ is determined by the maximal signal velocity for heat transfer. Given the premise that the averaged nonequilibrium heat flux is governed by an anomalous heat conductivity, energy diffusion scaling determines a corresponding anomalous thermal conductivity scaling behaviour.