Nonequilibrium Problems Research Articles

Modeling vibrational relaxation rate using machine learning methods

The aim of the present study is to develop an efficient algorithm for simulating nonequilibrium gas-dynamic problems using the detailed state-to-state approach for vibrationalchemical kinetics. Optimization of the vibrational relaxation rate computation using machine learning algorithms is discussed. Since traditional calculation methods require a large number of operations, time and memory, it is proposed to predict the relaxation rates instead of explicit calculations. K-nearest neighbour and histogram based gradient boosting algorithms are applied. The algorithms were trained on datasets obtained using two classical models for the rate coefficients: the forced harmonic oscillator model and that of Schwartz-Slawsky-Herzfeld. Trained algorithms were used to solve the problem of spatially homogeneous relaxation of the O2-O mixture. Comparison of accuracy and calculation time by different methods is carried out. It is shown that the proposed algorithms allow one to predict the relaxation rates with good accuracy and to solve approximately the set of governing equations for the fluid-dynamic variables. Thus, we can recommend the use of machine learning methods in nonequilibrium gas dynamics coupled with detailed vibrational-chemical kinetics. The ways of further optimization of the considered methods are discussed.

A symmetrized and simplified Bernoulli trial collision scheme in direct simulation Monte Carlo

Here, a symmetrized and simplified Bernoulli trials (SSBT) scheme based on the probabilistic approach is introduced to provide less-restricted conditions in choosing selected pairs. Unlike the simplified Bernoulli trials (SBT) method, the SSBT scheme picks the second particle of a selected pair from a whole list of particles with equal probability; it prevents repetitive collisions by introducing a procedure to avoid duplicate colliding pairs. The efficiency of this newly introduced algorithm is investigated in benchmark problems such as a collision frequency test case, Fourier heat transfer, dissociation of simple gas, and hypersonic cylinder flow. Compared with SBT, no time counter (NTC), and nearest neighbor (NN) collision algorithms, the results show that the SSBT method predicts the solutions quite accurately. In the collision frequency test case and Fourier test case, we show that the SSBT scheme could work with few particles per cell (one or even less) if an appropriate space and time discretization is employed. The symmetrized algorithm of the SSBT scheme improves the quality of the selection process, which leads to a smaller sample size in the highly non-equilibrium problem of hypersonic cylinder flow to achieve the same convergence limit at that of the SBT and NN schemes. In addition, the SSBT scheme has inherently a lower separation of free paths in the stagnation point of the cylinder test case compared to the SBT scheme for the same grid test case. These features make SSBT a new, robust model that could be presented as an alternative to state-of-the-art models.

Bound-state characteristic temperature method and its applications

As the speed of a hypersonic vehicle increases, atomic emission lines in the shock-layer will be a main source of radiative heating. Therefore, it is very important to study the atomic excitation in the air plasma in the shock layer. For a thermal nonequilibrium air plasma, the equilibrium statistical theory is not applicable. Although full models (such as the collisional-radiative model) can be used to solve nonequilibrium problems with high accuracy, they are too expensive computationally and difficult to apply to engineering. In this work, we investigate the atomic excitation in air plasmas by the bound-state characteristic temperature (BCT) method. Some cases of equilibrium and nonequilibrium air plasmas associated with the well-known FIRE II flight experiment are considered. The calculated atomic energy level populations are in good agreement with those from the CR model, thereby showing that our calculation is reasonable and has a good accuracy. The computational efficiency is more than 2000 times higher than that from the CR model. If it is used in the flow field of a hypersonic vehicle, the computational cost can be greatly reduced.

Symmetries and phase diagrams with real-space mutual information neural estimation.

Real-space mutual information (RSMI) was shown to be an important quantity, formally and from a numerical standpoint, in finding coarse-grained descriptions of physical systems. It very generally quantifies spatial correlations and can give rise to constructive algorithms extracting relevant degrees of freedom. Efficient and reliable estimation or maximization of RSMI is, however, numerically challenging. A recent breakthrough in theoretical machine learning has been the introduction of variational lower bounds for mutual information, parametrized by neural networks. Here we describe in detail how these results can be combined with differentiable coarse-graining operations to develop a single unsupervised neural-network-based algorithm, the RSMI-NE, efficiently extracting the relevant degrees of freedom in the form of the operators of effective field theories, directly from real-space configurations. We study the information contained in the statistical ensemble of constructed coarse-graining transformations and its recovery from partial input data using a secondary machine learning analysis applied to this ensemble. In particular, we show how symmetries, also emergent, can be identified. We demonstrate the extraction of the phase diagram and the order parameters for equilibrium systems and consider also an example of a nonequilibrium problem.

Atomistic configurational forces in crystalline fracture

Configurational atomistic forces contribute to the configurational mechanics (i.e. non-equilibrium) problem that determines the release of total potential energy of an atomistic system upon variation of the atomistic positions relative to the initial atomic configuration. These forces drive energetically favorable irreversible re-organizations of the material body, and thus characterize the tendency of crystalline defects to propagate. In this work, we provide new expressions for the atomistic configurational forces for two realistic interatomic potentials, i.e. the embedded atom potential (EAM) for metals, and second generation reactive bond order (REBO-II) potential for hydrocarbons. We present a range of numerical examples involving quasistatic fracture for both FCC metals and mono and bi-layer graphene at zero Kelvin that demonstrate the ability to predict defect nucleation and evolution using the proposed atomistic configurational mechanics approach. Furthermore, we provide the contributions for each potential including two-body stretching, three-body mixed-mode stretching-bending, and four-body mixed-mode stretching-bending-twisting terms that make towards defect nucleation and propagation.

Atomistic two-, three- and four-body potentials. Spatial and material settings

In molecular dynamics or molecular statics (MD/MS) multi-body potentials empirically capture the energetic interactions in atomistic systems enabling the computation of the corresponding atomistic forces as energetic conjugates to the atomistic positions. We distinguish here between spatial and material atomistic positions and consequently between the corresponding spatial and material atomistic forces. In quasi-statics, i.e. MS, the former, also denoted as deformational atomistic forces, contribute to the classical deformational mechanics (i.e., equilibrium) problem that seeks to minimise the total potential energy of an atomistic system with respect to the atomistic positions relative to the ambient space. The latter, also denoted as configurational atomistic forces, contribute to the configurational mechanics (i.e., non-equilibrium) problem that determines the release of total potential energy of an atomistic system upon variation of the atomistic positions relative to the ambient material, i.e., due to perturbations of the material (initial) atomistic configuration. The importance of material atomistic forces is that they drive energetically favourable re-organisations of the material atomistic configuration, thereby characterising the tendency of generic atomistic defects to propagate. In this contribution we focus on two-, three-, and four-body potentials, whereby we distinguish between novel stretch- and classical angle-based potentials for the two latter cases. Taken together, as the main contribution, we derive expressions for the corresponding spatial and, for the first time, material atomistic forces and highlight their striking formal similarity. The derivations are detailed but the final expression compact and well-suited for numerical implementation.

A general form of Macheret–Fridman classical impulsive dissociation model for nonequilibrium flows

The rate of dissociation behind a strong shock in thermochemical nonequilibrium depends on the vibrational excitation of the molecules, hence the rates become a function of translational-rotational and vibrational temperatures. The Macheret–Fridman (MF) model provides analytical expressions for nonequilibrium dissociation rates assuming the collision of molecules to be in the impulsive limit. However, the original form of the model was limited to the dissociation of homonuclear molecules. In this work, we present a general form of the Macheret–Fridman classical impulsive model by considering the dissociation of a heteronuclear molecule and present macroscopic rates applicable for modeling dissociation in computational fluid dynamics (CFD). The nonequilibrium dissociation rates from the MF-CFD model compared well with the available quasiclassical trajectory (QCT) data for some important reactions in the air. Additionally, we also present a comparison of the average vibrational energy removed in a dissociation reaction predicted by the MF-CFD model with QCT data for several reactions in air and propose some improvements to the model. The developed MF-CFD model was used to investigate various nonequilibrium flow problems and the results were compared with available experimental data. In general, the results from the MF-CFD model are promising and the model shows a possibility of becoming the standard tool for investigating nonequilibrium flows in CFD.

A fast synthetic iterative scheme for the stationary phonon Boltzmann transport equation

The heat transfer in solid materials at the micro- and nano-scale can be described by the mesoscopic phonon Boltzmann transport equation (BTE), rather than the macroscopic Fourier’s heat conduction equation that works only in the diffusive regime. The implicit discrete ordinate method (DOM) is efficient to find steady-state solutions to the BTE for highly non-equilibrium heat transfer problems, but converges extremely slowly in the near-diffusive regime. In this paper, a fast synthetic iterative scheme is developed to expedite convergence for the implicit DOM. The key innovative point of the present scheme is the introduction of a macroscopic diffusion-type equation for the temperature variation, which is exactly derived from the phonon BTE and valid for all Knudsen numbers. The synthetic equation, which is asymptomatically preserving to Fourier’s heat conduction equation in the diffusive regime, contains a term related to Fourier’s law and a term determined by the second-order moment of the distribution function that reflects the non-Fourier heat transfer. The mesoscopic kinetic equation and macroscopic diffusion-type equations are tightly coupled, because the macroscopic equation provides the temperature for the BTE, while the BTE provides a high-order moment to the diffusion-type equation. This synthetic iterative scheme strengthens the coupling of phonons with different wave vectors in the phase space to facilitate fast convergence from the diffusive to ballistic regimes. Typical numerical tests in one-, two-, and three-dimensional problems demonstrate that our scheme can describe the multiscale heat transfer problems accurately and efficiently. For all test cases, the present convergence is one to three orders of magnitude faster than the traditional implicit DOM in the near-diffusive regime.

A study of Darcy–Bénard regular and chaotic convection using a new local thermal non-equilibrium formulation

The onset of Darcy–Bénard regular and chaotic convection in a porous medium is studied by considering phase-lag effects that naturally arise in the thermal non-equilibrium heat transfer problem between the fluid and solid phases. A new type of heat equation is derived for both the phases. Using a double Fourier series and a novel decomposition, an extended Vadasz–Lorenz model with three phase-lag effects is derived. New parameters arise due to the phase-lag effects between local acceleration, convective acceleration, and thermal diffusion. The principle of exchange of stabilities is found to be valid and the subcritical instability is discounted. The new perspective supports the finding of an analytical expression for the critical Darcy–Rayleigh numbers representing, respectively, the onset of regular and chaotic convection. The understanding of the transition from the local thermal non-equilibrium situation to the local thermal equilibrium one is also best explained through the new perspective. In its present elegant form, the extended Vadasz–Lorenz system with three phase-lag effects is analyzed using the largest Lyapunov exponent and the bifurcation diagram. It is found that the lag effects not only give rise to a quantitative difference in the above two metrics concerning chaos, but also present a qualitative difference as well in the form of the very nature of chaos.

Numerical simulation of hypersonic reaction flows with nonlinear coupled constitutive relations

As all known, there exists extremely thermal and chemical non-equilibrium phenomenon in hypersonic flows. Much effort has been put into the development of computational models for the prediction of such non-equilibrium flow. In order to obtain the real physical solution of the challenging non-equilibrium problems, Eu proposed a set of generalized hydrodynamic equations (GHE) from the viewpoint of generalized hydrodynamics. Based on Eu's GHE equations, Myong developed an efficient computational model, named as the nonlinear coupled constitutive relations (NCCR). The present study is based on the 3-D FVM NCCR model in conjunction with 7-species chemical reaction model to consider the reacting gas effect for hypersonic reentry vehicles. For further examining the performance of the proposed computational model, the typical engineering cases of Radio Attenuation Measurements (RAM)-C II vehicle are adopted. The flow contours, distribution profiles along stagnation line and surface properties are compared between nonlinear coupled constitutive model and those of the Navier-Stokes equation. The accuracy and physical consistency of the NCCR model are critically examined and validated for hypersonic reaction flow in the continuum region. For the near continuum region, the NCCR model can predict non-equilibrium phenomenon more accurately than NS equation through the comparison of shock stand-off distance and electron number density. The comparisons between NCCR model with one component perfect gas and multi-species reacting gas effect are also made to show the importance of chemical reaction source term.

A detailed multiscale study of rotational–translational relaxation process of diatomic molecules

This article continues our cycle devoted to comprehensive investigation of the diatomic molecule collision process. In this paper, we focus particularly on the in-depth study of the rotational–translational (R–T) energy exchange process and Borgnakke–Larsen (BL) energy exchange model used in the direct simulation Monte Carlo method. The present study, which was performed on several levels of description (molecular, microscopic, and macroscopic), is based mainly on the highly detailed dataset (around 1011 configurations) of binary N2–N2 collisions, obtained via the classical trajectory calculation (CTC) method. This dataset, along with the explicit mathematical representation of the Borgnakke–Larsen model derived in the present paper, allowed us to obtain new results regarding the R–T energy exchange process: (1) we present an ab initio method to derive physically accurate expressions for inelastic collision probability pr in the BL model directly from CTC data; (2) we present a new two-parametric model for pr and compared it to the previously known models, including the recent nonequilibrium-direction-dependent model of Zhang et al. [“Nonequilibrium-direction-dependent rotational energy model for use in continuum and stochastic molecular simulation,” AIAA J. 52(3), 604 (2014)]; (3) it showed that apart from the well-known dependence of the rotational relaxation rate on “direction to equilibrium” (ratio between translational and rotational temperatures), on molecular scale, rotationally over-excited molecule pairs demonstrate almost zero energy transfer to the translational energy mode (even in the case of very significant discrepancies between translational and rotational energies); (4) it was also shown that the Borgnakke–Larsen approach itself may require reassessment since it fails to give a proper description of distribution of post-collision energies. Throughout this paper, we also tried to put together and analyze the existing works studying the rotational relaxation process and estimating the rotational collision number Zrot by performing reviews and assessment of (1) numerical approaches to simulate non-equilibrium problems, (2) models for inelastic collision probabilities pr, (3) approaches to estimate Zrot, and (4) intermolecular potentials used for molecular dynamics and CTC simulations. The corresponding conclusions are given in this paper.

Target Search of Active Agents Crossing High Energy Barriers.

Target search by active agents in rugged energy landscapes has remained a challenge because standard enhanced sampling methods do not apply to irreversible dynamics. We overcome this nonequilibrium rare-event problem by developing an algorithm generalizing transition-path sampling to active Brownian dynamics. This method is exemplified and benchmarked for a paradigmatic two-dimensional potential with a high barrier. We find that even in such a simple landscape the structure and kinetics of the ensemble of transition paths changes drastically in the presence of activity. Indeed, active Brownian particles reach the target more frequently than passive Brownian particles, following longer and counterintuitive search patterns.

Quenched solitons and shock waves in Bose-Einstein condensates

The Bose-Einstein condensate (BEC) formed by ultracold atomic gases provides an ideal platform for studying various quantum phenomena. In this platform, researchers have explored in depth the important equilibrium and steady phenomena including superfluidity, vortices, and solitons, and recently started to study nonequilibrium problems. In a classical system, nonequilibrium problems, such as explosion, usually occur together with shock waves, which is presented when the explosion speed is larger than the local sound speed. For BEC systems which possess quantum properties, how to produce and understand the shock waves becomes a hot research topic. In this work, we systematically discuss the possibility of quantum shock wave and its essential mechanism in a one-dimensional BEC initially containing dark solitons through quenching interactions. When the system is quenched to the limit of non-interaction, we analytically obtain the post-quench dynamics of initially immobile dark solitons, and find the existence of shock wave, which can be explained through the quantum interference effect. When the system is quenched to finite interaction, we find similar phenomena through numerically solving the Gross-Pitaevskii equation, and analyze different situations. When the system is quenched to a finite weaker interaction, the situation is similar to a non-interaction case; when the system is quenched to a stronger interaction, the shock wave is accompanied by the splitting of the initial soliton, and the two objects can synchronously change; specifically when the quenched ratio of strength is an integer squared, the shock wave disappears, and the soliton is split perfectly. We further explore the properties of the shock wave including its amplitude and speed, and obtain the full scenario as the quenched interaction varies. This work provides theoretical guidance for realizing and measuring shock wave in experiment.

Rotational Forms of Large Eddy Simulation Turbulence Models: Modeling and Mathematical Theory

In this paper the authors present a derivation of a back-scatter rotational Large Eddy Simulation model, which is the extension of the Baldwin & Lomax model to non-equilibrium problems. The model is particularly designed to mathematically describe a fluid filling a domain with solid walls and consequently the differential operators appearing in the smoothing terms are degenerate at the boundary. After the derivation of the model, the authors prove some of the mathematical properties coming from the weighted energy estimates, which allow to prove existence and uniqueness of a class of regular weak solutions.

Quantum Advantage in the Charging Process of Sachdev-Ye-Kitaev Batteries.

The exactly solvable Sachdev-Ye-Kitaev (SYK) model has recently received considerable attention in both condensed matter and high energy physics because it describes quantum matter without quasiparticles, while being at the same time the holographic dual of a quantum black hole. In this Letter, we examine SYK-based charging protocols of quantum batteries with N quantum cells. Extensive numerical calculations based on exact diagonalization for N up to 16 strongly suggest that the optimal charging power of our SYK quantum batteries displays a superextensive scaling with N that stems from genuine quantum mechanical effects. While the complexity of the nonequilibrium SYK problem involved in the charging dynamics prevents us from an analytical proof, we believe that this Letter offers the first (to the best of our knowledge) strong numerical evidence of a quantum advantage occurring due to the maximally entangling underlying quantum dynamics.

Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and non-equilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.

Ionic Layering and Overcharging in Electrical Double Layers in a Poisson-Boltzmann Model.

Electrical double layers (EDLs) play a significant role in a broad range of physical phenomena related to colloidal stability, diffuse-charge dynamics, electrokinetics, and energy storage applications. Recently, it has been suggested that for large ion sizes or multivalent electrolytes, ions can arrange in a layered structure inside the EDLs. However, the widely used Poisson-Boltzmann models for EDLs are unable to capture the details of ion concentration oscillations and the effect of electrolyte valence on such oscillations. Here, by treating a pair of ions as hard spheres below the distance of closest approach and as point charges otherwise, we are able to predict ionic layering without any additional parameters or boundary conditions while still being compatible with the Poisson-Boltzmann framework. Depending on the combination of ion valence, size, and concentration, our model reveals a structured EDL with spatially oscillating ion concentrations. We report the dependence of critical ion concentration, i.e., the ion concentration above which the oscillations are observed, on the counter-ion valence and the ion size. More importantly, our model displays quantitative agreement with the results of computationally intensive models of the EDL. Finally, we analyze the nonequilibrium problem of EDL charging and demonstrate that ionic layering increases the total charge storage capacity and the charging timescale.

Characterizing the Non-equilibrium Dynamics of Field-Driven Correlated Quantum Systems

Recent experimental advances in ultrafast phenomena have triggered renewed interest in the dynamics of correlated quantum systems away from equilibrium. We review nonequilibrium dynamical mean-field theory studies of both the transient and the steady states of a DC field-driven correlated quantum system. In particular, we focus on the nonequilibrium behavior and how it relates to the fluctuation-dissipation theorem. The fluctuation-dissipation theorem emerges as an indicator for how the system thermalizes and for how it reaches a steady state. When the system thermalizes in an infinite temperature steady state it can pass through a succession of quasi-thermal states that approximately obey the fluctuation-dissipation theorem. We also discuss the Wigner distribution and what its evolution tells us about the nonequilibrium many-body problem.

Breaking the Entanglement Barrier: Tensor Network Simulation of Quantum Transport.

The recognition that large classes of quantum many-body systems have limited entanglement in the ground and low-lying excited states led to dramatic advances in their numerical simulation via so-called tensor networks. However, global dynamics elevates many particles into excited states, and can lead to macroscopic entanglement and the failure of tensor networks. Here, we show that for quantum transport-one of the most important cases of this failure-the fundamental issue is the canonical basis in which the scenario is cast: When particles flow through an interface, they scatter, generating a "bit" of entanglement between spatial regions with each event. The frequency basis naturally captures that-in the long-time limit and in the absence of inelastic scattering-particles tend to flow from a state with one frequency to a state of identical frequency. Recognizing this natural structure yields a striking-potentially exponential in some cases-increase in simulation efficiency, greatly extending the attainable spatial and time scales, and broadening the scope of tensor network simulation to hitherto inaccessible classes of nonequilibrium many-body problems.

PENERAPAN MODIFIED VOGEL'S APPROXIMATION METHOD (MVAM) UNTUK MEMINIMUMKAN BIAYA TRANSPORTASI (STUDI KASUS: PABRIK TAHU TAUFIK)

This research aims to determine the tofu product distribution transportation model at Taufik Tofu Factory. This type of research is applied research with a quantitative approach. The method used to solve transportation problems in this study is the Modified Vogel’s Approximation Method (MVAM) and optimized using the Modified Distribution (MODI) method. The results of this study indicate that the application of MVAM at the Taufik Tofu Factory provides a more minimum solution for calculating transportation costs, both in equilibrium and non-equilibrium problems. MVAM provides the same transportation costs as the optimal solution for the simplex method, which is equal to Rp1.164.911,00 for equilibrium problems and Rp2.176.838,00 for unequal problems. These results have been tested for optimism using MODI and better than the real cost calculations issued by the factory.