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A comprehensive microscopic understanding of ambient liquid water is a major challenge for $ab$ $initio$ simulations as it simultaneously requires an accurate quantum mechanical description of the underlying potential energy surface (PES) as well as extensive sampling of configuration space. Due to the presence of light atoms (e.g., H or D), nuclear quantum fluctuations lead to observable changes in the structural properties of liquid water (e.g., isotope effects), and therefore provide yet another challenge for $ab$ $initio$ approaches. In this work, we demonstrate that the combination of dispersion-inclusive hybrid density functional theory (DFT), the Feynman discretized path-integral (PI) approach, and machine learning (ML) constitutes a versatile $ab$ $initio$ based framework that enables extensive sampling of both thermal and nuclear quantum fluctuations on a quite accurate underlying PES. In particular, we employ the recently developed deep potential molecular dynamics (DPMD) model---a neural-network representation of the $ab$ $initio$ PES---in conjunction with a PI approach based on the generalized Langevin equation (PIGLET) to investigate how isotope effects influence the structural properties of ambient liquid H$_2$O and D$_2$O. Through a detailed analysis of the interference differential cross sections as well as several radial and angular distribution functions, we demonstrate that this approach can furnish a semi-quantitative prediction of these subtle isotope effects., Comment: 19 pages, 5 figures, and 1 table

Fully anharmonic calculations of the dielectric response of insulators require costly $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ molecular dynamics simulations. Here, the authors show that this electronic response property can be described efficiently by a deep neural network that retains the accuracy of $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ molecular dynamics. The scheme is demonstrated with calculations of the infrared absorption spectrum of liquid water at standard conditions, and of the evolution of the spectrum of crystalline ice undergoing a pressure-induced structural transformation from molecular ice VII to ionic ice X.

The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. We introduce an order parameter to track the phase transition and resolve the corresponding phase diagram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension.

In recent years, promising deep learning based interatomic potential energy surface (PES) models have been proposed that can potentially allow us to perform molecular dynamics simulations for large scale systems with quantum accuracy. However, making these models truly reliable and practically useful is still a very non-trivial task. A key component in this task is the generation of datasets used in model training. In this paper, we introduce the Deep Potential GENerator (DP-GEN), an open-source software platform that implements the recently proposed "on-the-fly" learning procedure [Phys. Rev. Materials 3, 023804] and is capable of generating uniformly accurate deep learning based PES models in a way that minimizes human intervention and the computational cost for data generation and model training. DP-GEN automatically and iteratively performs three steps: exploration, labeling, and training. It supports various popular packages for these three steps: LAMMPS for exploration, Quantum Espresso, VASP, CP2K, etc. for labeling, and DeePMD-kit for training. It also allows automatic job submission and result collection on different types of machines, such as high performance clusters and cloud machines, and is adaptive to different job management tools, including Slurm, PBS, and LSF. As a concrete example, we illustrate the details of the process for generating a general-purpose PES model for Cu using DP-GEN., Comment: 29 pages, 6 figures

The instability of the two-dimensional Poiseuille flow in a long channel and the subsequent transition is studied using a thermodynamic approach. The idea is to view the transition process as an initial value problem with the initial condition being Poiseuille flow plus noise, which is considered as our ensemble. Using the mean energy of the velocity fluctuation and the skin friction coefficient as the macrostate variable, we analyze the transition process triggered by the initial noises with different amplitudes. A first order transition is observed at the critical Reynolds number $Re_* \sim 5772$ in the limit of zero noise. An action function, which relates the mean energy with the noise amplitude, is defined and computed. The action function depends only on the Reynolds number, and represents the cost for the noise to trigger a transition from the laminar flow. The correlation function of the spatial structure is analyzed.

We introduce a new family of trial wave-functions based on deep neural networks to solve the many-electron Schrodinger equation. The Pauli exclusion principle is dealt with explicitly to ensure that the trial wave-functions are physical. The optimal trial wave-function is obtained through variational Monte Carlo and the computational cost scales quadratically with the number of electrons. The algorithm does not make use of any prior knowledge such as atomic orbitals. Yet it is able to represent accurately the ground-states of the tested systems, including He, H2, Be, B, LiH, and a chain of 10 hydrogen atoms. This opens up new possibilities for solving large-scale many-electron Schrodinger equation.

We present a simple, yet general, end-to-end deep neural network representation of the potential energy surface for atomic and molecular systems. This methodology, which we call Deep Potential, is "first-principle" based, in the sense that no ad hoc approximations or empirical fitting functions are required. The neural network structure naturally respects the underlying symmetries of the systems. When tested on a wide variety of examples, Deep Potential is able to reproduce the original model, whether empirical or quantum mechanics based, within chemical accuracy. The computational cost of this new model is not substantially larger than that of empirical force fields. In addition, the method has promising scalability properties. This brings us one step closer to being able to carry out molecular simulations with accuracy comparable to that of quantum mechanics models and computational cost comparable to that of empirical potentials.

In general nonequilibrium steady states, directly replacing the canonical ensemble by the nonequilibrium invariant distribution yields a free energy function that is insufficient in characterizing the dynamical landscape. We address the problem by defining the free action, which is like a free energy on path space. Through a representative example, we demonstrate the conceptual and practical usefulness of the free action for quantifying the dynamics of nonequilibrium steady states, including those exhibiting phase transitions.

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, ${\it\lambda}$, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, ${\it\lambda}$, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, $t=O(1)$, and one where time tends to infinity at the rate $t=O(|\!\log {\it\lambda}|)$. The crucial result is that in the case where time is held constant, the ${\it\lambda}\rightarrow 0$ limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if ${\it\lambda}\rightarrow 0$ and $t\rightarrow \infty$, then contact line slippage is a leading-order singular effect.

The electronic structure of a smoothly deformed crystal is analyzed for the case when the effective Hamiltonian is a given function of the nuclei by considering the regime when the scale of the deformation is much larger than the lattice parameter. Wannier functions are defined by projecting the Wannier functions for the undeformed crystal to the space spanned by the wave functions of the deformed crystal. The exponential decay of such Wannier functions is proved for the case when the undeformed crystal is an insulator. The celebrated Cauchy–Born rule for crystal lattices is extended to the present situation for electronic structure analysis.

We consider the continuum limit for models of solids that arise in density functional theory and the QM-continuum approximation of such models. Two different versions of QM- continuum approximation are proposed, depending on the level at which the Cauchy-Born rule is used, one at the level of electron density and one at the level of energy. Consistency at the interface between the smooth and the non-smooth regions is analyzed. We show that if the Cauchy-Born rule is used at the level of electron density, then the resulting QM-continuum model is free of the so-called “ghost force” at the interface. We also present dynamic models that bridge naturally the Car-Parrinello method and the QM-continuum approximation.

The classic Hegselmann-Krause (HK) model for opinion dynam- ics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of all the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field limiting Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. The ordered phase features clusters whose width depends only on the noise level. We introduce an order parameter to track the phase transition and resolve the corresponding phase dia- gram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis also confirms previous simulations and predicts properties of the noisy HK model in higher dimension.

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier-Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec ≈ 2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

We developa general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we developan additional strategy for defect tracking. This method naturally decoup les the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented. r 2005 Elsevier Ltd. All rights reserved.

A method is presented for the study of rare events such as conformational changes arising in activated processes whose reaction coordinate is not known beforehand and for which the assumptions of transition state theory are invalid. The method samples the energy landscape adaptively and determines the isoprobability surfaces for the transition: by definition the trajectories initiated anywhere on one of these surfaces has equal probability to reach first one metastable set rather than the other. Upon weighting these surfaces by the equilibrium probability distribution, one obtains an effective transition pathway, i.e., a tube in configuration space inside which conformational changes occur with high probability, and the associated rate. The method is first validated on a simple two-dimensional example; then it is applied to a model of solid-solid transformation of a condensed system.

The least-action principle from the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise. The action is discretized and a preconditioned BFGS method is used to optimize the discrete action. Applications are presented for thermally activated reversal in the Ginzburg-Landau model in one and two dimensions, and for noise-induced excursion events in the Brusselator taken as an example of a nongradient system arising in chemistry. In the Ginzburg-Landau model, the reversal proceeds via interesting nucleation events, followed by propagation of domain walls. The issue of nucleation versus propagation is discussed, and the scaling for the number of nucleation events as a function of the reversal time and other material parameters is computed. Good agreement is found with the numerical results. In the Brusselator, whose deterministic dynamics has a single stable equilibrium state, the presence of noise is shown to induce large excursions by which the system cycles out of this equilibrium state. c � 2004 Wiley Periodicals, Inc.

We present a hybrid method for simulating epitaxial growth that combines kinetic Monte-Carlo (KMC) simulations with the Burton-Cabrera-Frank model for crystal growth. This involves partitioning the computational domain into KMC regions and regions where we time-step a discretized diffusion equation. Computational speed and accuracy are discussed. We find that the method is significantly faster than KMC while accounting for stochastic fluctuations in a comparable way.

The Gauss-Seidel projection method (GSPM) (Wang et al., J. Comp. Phys., vol.171, p.357-372, 2001) is a simple, efficient, and unconditionally stable method for micromagnetics simulations. We present an improvement of the method for small values of the damping parameter. With the new method, we are able to carry out fully resolved simulations of the magnetization reversal process in the presence of thermal noise.

Thermally activated switching and the energy landscape of submicron-sized magnetic elements are studied using the string method. For thin films, we found that switching proceeds by two generic scenarios: Domain-wall propagation and reconnection followed by edge domain switching, or vortex nucleation at the boundary followed by vortex propagation through the sample. The second scenario is preferred for thicker films whereas the first is preferred for thin (less than 20 nm) films. The energy landscape of such a system is nicely summarized on the plane spanned by the average magnetization in the in-plane directions. For three-dimensional samples, we found that switching proceeds by vortex propagation through the sample. The implication of the Landau–Lifshitz dynamics is also discussed.

A flat surface of stressed solid is unstable to small perturbations. This morphological instability is called Asaro–Tiller–Grinfield instability. Nonlinear evolution of this instability will result in the formation of cusp singularities. This instability can be described by a continuum model with surface diffusion driven by a stress-dependent chemical potential. The stress and strain in the solid are coupled with surface morphology and an elasticity problem must be solved numerically. We derive a nonlinear approximation equation governing the evolution of this instability in which the stress-dependent chemical potential is expressed explicitly as a function of the surface morphology. Linear instability analysis using our equation shows the same results as the well-known Asaro–Tiller–Grinfeld instability result. Compared with the exact solution of the elasticity problem for cycloid surface obtained by Chiu and Gao, our nonlinear approximation has a much wider range of applicability than linear approximatio...

In a ferromagnetic material, the dynamics of the relaxation process are affected by the presence of a strong shape or material anisotropy. In this article, we systematically explore this fact to derive the effective dynamical equation for a soft ferromagnetic thin film. We show that, as a consequence of the interplay between shape anisotropy and damping, the gyromagnetic term is effectively also a damping term for the in-plane components of the magnetization distribution. We validate our result through numerical simulation of the original Landau–Lifshitz equation and our effective equation.

A generalization of Kraichnan’s model of passive scalar advection is considered. Physically motivated regularizations of the model are considered which take into account both the effects of viscosity and molecular diffusion. The balance between these two effects on the inertial range behavior for the scalar is shown to be parameterized by a new turbulent Prandtl number. Three different regimes are identified in the parameter space depending on degrees of compressibility. In the strongly and weakly compressible regimes, the inertial range behavior of the scalar does not depend on the turbulent Prandtl number. In the regime of intermediate compressibility, the inertial range behavior does depend on the turbulent Prandtl number.

The continuum equations presented here model the growth of epitaxial films in terms of a local edge density ∼ ∇ h and surface concentration (number density) of adatoms. This model is more amenable to computations than existing models that feature discrete edges and solve continuum equations on each terrace; yet it offers a more detailed picture than continuum models that treat the surface height as the only dependent variable. This latter feature is especially important if one wishes to account for several species which may react on the surface of the film or at step edges to build complicated unit cells. The model is motivated by and compared with numerical solutions of rate equations which are derived from kinetic Monte-Carlo simulations. After introducing the model in a 1+1 dimensional setting, we extend it to a 2+1 dimensional setting assuming spatial derivatives become surface gradients. We also discuss extension for the case with multiple species.

The statistics of the multivalued solutions of the forced Riemann equation, ut+uux=f, is considered. An exact equation for the signed probability density function of these solutions and their gradient ξ=ux is derived, and some properties of this equation are analyzed. It is shown in particular that the tails of the signed probability density function generally decay as |ξ|−3 for large |ξ|. Further considerations give bounds on the cumulative probability density function for the velocity gradient of the solution of Burgers equation.

We compare two models describing the dynamics of phase separation of incompressible mixtures: A local model proposed by de Gennes, and a nonlocal model proposed by E and Palffy-Muhoray. We find that in the interfacial regime, the local model gives rise to interfacial motion via the Mullins-Sekerka law for moderate quenches, and surface diffusion for deep quenches. The interface dynamics is slowed down sharply as the quench depth is increased. The nonlocal model, on the other hand, has an additional convective mechanism which gives rise to motion by Hele-Shaw dynamics. This additional mechanism is insensitive to the quench depth. While both models explain qualitatively the observed pinning phenomenon for deeply quenched off-critical mixtures, only the nonlocal model predicts the correct dependence of the coarsening rate on the quench depth.

We characterize the tails of the probability distribution functions for the solution of Burgers’ equation with Gaussian initial data and its derivatives ∂kv(x,t)/∂xk, k=0,1,2,... . The tails are ‘‘stretched exponentials’’ of the form P(θ)∝exp[−(Re)−ptqθr], where Re is the Reynolds number. The exponents p, q, and r depend on the initial spectrum as well as on the order of differentiation, k. These exact results are compared with those obtained using the mapping closure technique.

We study the dynamics of vortices in time-dependent Ginzburg-Landau theories in the asymptotic limit when the vortex core size is much smaller than the inter-vortex distance. We derive reduced systems of ODEs governing the evolution of these vortices. We then extend these to study the dynamics of vortices in extremely type-II superconductors. Dynamics of vortex lines is also considered. For the simple Ginzburg-Landau equation without the magnetic field, we find that the vortices are stationary in the usual diffusive scaling, and obey remarkably simple dynamic laws when time is speeded up by a logarithmic factor. For columnar vortices in superconductors, we find a similar dynamic law with a potential which is screened by the current. For curved vortex lines in curerconductors, we find that the vortex lines move in the direction of the normal with a speed proportional to the curvature. Comparisons are made with the previous results of John Neu.

Two‐dimensional Boussinesq convection is studied numerically using two different methods: a filtered pseudospectral method and a high‐order accurate eno scheme. The issue whether finite time singularity occurs for initially smooth flows is investigated. In contrast to the findings of Pumir and Siggia who reported finite time collapse of the bubble cap, the present numerical results suggest that the strain rate corresponding to the intensification of the density gradient across the front saturates at the bubble cap. Consequently, the thickness of the bubble decreases exponentially. On the other hand, the bubble experiences much stronger straining and intensification of gradients at its side. As the bubble rises, a secondary front also forms from its tail. Together with the primary front, they constitute a pair of tightly bound plus and minus double vortex sheet structure which is highly unstable and vulnerable to viscous dissipation.

We study the two dimensional Kolmogorov flows in the limit as the forcing frequency goes to infinity. Direct numerical simulation indicates that the low frequency energy spectrum evolves to a universal kappa (exp -4) decay law. We derive effective equations governing the behavior of the large scale flow quantities. We then present numerical evidence that with smooth initial data, the solution to the effective equation develops a kappa (exp -4) type singularity at a finite time. This gives a convenient explanation for the kappa (exp -4) decay law exhibited by the original Kolmogorov flows.

Analytical solution of Maxwell's equations is obtained for general linear optical materials: lossy and optically active crystals. Explicit expressions are obtained for the dispersion relation and the propagating eigenmodes. In general, four rather than two distinct modes are present. The results are useful in describing light propagation in optically complex media.

Two approaches for the efficient rational approximation of the Fermi-Dirac function are discussed: one uses the contour integral representation and conformal mapping and the other is based on a version of the multipole representation of the Fermi-Dirac function that uses only simple poles. Both representations have logarithmic computational complexity. They are of great interest for electronic structure calculations., 16 pages, 8 figures, dedicated to Professor Andy Majda on the occasion of his 60th birthday

We propose a multipole representation of the Fermi-Dirac function and the Fermi operator, and use this representation to develop algorithms for electronic structure analysis of metallic systems. The new algorithm is quite simple and efficient. Its computational cost scales logarithmically with $\beta\Delta\eps$ where $\beta$ is the inverse temperature, and $\Delta \eps$ is the width of the spectrum of the discretized Hamiltonian matrix., Comment: 10 pages, 3 figures, 3 tables

We present a linear-scaling method for electronic structure computations in the context of Kohn-Sham density functional theory (DFT). The method is based on a subspace iteration, and takes advantage of the nonorthogonal formulation of the Kohn-Sham functional, and the improved localization properties of nonorthogonal wave functions. A one-dimensional linear problem is presented as a benchmark for the analysis of linear-scaling algorithms for Kohn-Sham DFT. Using this one-dimensional model, we study the convergence properties of the localized subspace-iteration algorithm presented. We demonstrate the efficiency of the algorithm for practical applications by performing fully three-dimensional computations of the electronic density of alkane chains.

Through detailed comparisons between Embedded Atom Method (EAM) and first-principles calculations for Al, we find that EAM tends to fail when there are large electron density gradients present. We attribute the observed failures to the violation of the uniform density approximation (UDA) underlying EAM. To remedy the insufficiency of UDA, we propose a gradient-corrected EAM model which introduces gradient corrections to the embedding function in terms of exchange-correlation and kinetic energies. Based on the perturbation theory of "quasiatoms" and density functional theory, the new embedding function captures the essential physics missing in UDA, and paves the way for developing more transferable EAM potentials. With Voter-Chen EAM potential as an example, we show that the gradient corrections can significantly improve the transferability of the potential., Comment: 8 pages, 6 figures

This paper presents a systematic approach for finding efficient boundary conditions for molecular dynamics simulations of crystalline solids. These boundary conditions effectively eliminate phonon reflection at the boundary and at the same time allow the thermal energy from the bath to be introduced to the system. Our starting point is the Mori-Zwanzig formalism [R. Zwanzig, J. Chem. Phys. 32, 1173 (1960); in Systems Far from Equilibrium, edited by L. Garrido (Interscience, New York, 1980); H. Mori, Prog. Theor. Phys. 33, 423 (1965)] for eliminating the thermal bath, but we take the crucial next step that goes beyond this formalism in order to obtain memory kernels that decay faster. An equivalent variational formulation allows us to find the optimal approximate boundary conditions, after specifying the spatial-temporal domain of dependence for the positions of the boundary atoms. Application to a one-dimensional chain, a two-dimensional Lennard-Jones system, and a three-dimensional model of $\ensuremath{\alpha}$-iron with embedded atom potential is presented to demonstrate the effectiveness of this approach.

The accuracy of the quasicontinuum method is studied by reformulating the summation rules in terms of reconstruction schemes for the local atomic environment of the representative atoms. The necessary and sufficient condition for uniform first-order accuracy and, consequently, the elimination of the ``ghost force'' is formulated in terms of the reconstruction schemes. The quasi-nonlocal approach is discussed as a special case of this condition. Examples of reconstruction schemes that satisfy this condition are presented. Transition between atom-based and element-based summation rules are studied.

We present an elementary and systematic discussion on the derivation of continuum theories from atomistic models for studying the elastic deformation of plates, sheets, and rods. The derivation is based on various generalizations of the classical Cauchy-Born rule. In particular, we discuss a so-called local Cauchy-Born rule which is very general and particularly easy to use. As an application, we use the atomistically derived continuum models to study the elastic deformation of carbon nanotubes.

We derive a set of equations for the dynamics of evolving fluid membranes, such as cell membranes, in the presence of bulk fluids. We model the membrane as a surface endowed with a director field, which describes the local average orientation of the molecules on the membrane. A model for the elastic energy of a surface endowed with a director field is derived using liquid crystal theory. This elastic energy reduces to the well-known Helfrich energy in the limit when the directors are constrained to be normal to the surface. We then derive the full dynamic equations for the membrane that incorporate both the elastic and viscous effects, with and without the presence of bulk fluids. We also consider the effect of local spontaneous curvature, arising from the presence of membrane proteins. Overall, the systems of equations allow us to carry out stable, accurate, and robust numerical modeling for the dynamics of the membranes.

We consider phase separation in multicomponent systems in the limit as the free volume goes to zero. Excluded volume effects are taken into account via a pressure field which implicitly relates currents and forces. The pressure, obtainable from a divergence condition, may give rise to circulating currents. A simple description of the dynamics of phase separation at late times follows. @S1063-651X~97!50504-6#

In this paper we discuss continuum models of phase separation in polymer solutions, with emphasis on the thermodynamic foundation of these models. We demand that these models obey a free energy dissipation relation, which in the present context plays the role of the second law of thermodynamics, since the system is isothermal. First, we derive a modified two-fluid model for viscoelastic phase separation from nonequilibrium thermodynamics. Then we study the special case when only diffusion is present, and hydrodynamic effects are neglected. Numerical results demonstrate that our models show better stability properties and at the same time reproduce the expected physical phenomena such as volume shrinking and phase inversion. Our findings suggest that these important phenomena are caused by a diffusional asymmetry of the constituent molecules.

We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates, in the space of imaginary-time trajectories, the minimal action path (MAP) between two minima of the imaginary-time action. From the MAP, the saddle-point (``bounce'') action associated with the exponential barrier penetration probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the zero-temperature escape rates for the metastable zero-voltage states in the current-biased Josephson tunnel junction model. In the regime close to the critical bias current, direct comparison of the numerical and analytical results yields good agreement. Our calculations indicate that for the nanojunctions encountered in many experiments today, the (absolute) escape rates should be measurable at bias current much below the critical current., 12 pages including 10 figures

Complex interdisciplinary phenomena, such as drug design, crackpropagation, heterogeneous catalysis, turbulent combustion and many others, raise a growing demand of simulational methods capable of handling the simultaneous interaction of multiple space and time scales. Computational schemes aimed at such type of complex applications often involve multiple levels of physical and mathematical description, and are consequently referred to as to multiphysics methods [1, 2, 3]. The opportunity for multiphysics methods arises whenever single-level methods, say molecular dynamics and partial differential equations of continuum mechanics, expand their range of scales to the point where overlap becomes possible. In order to realize this multiphysics potential specific efforts must be directed towards the development of robust and efficient interfaces dealing with “hand-shaking” regions where the exchange of information between the different schemes takes place. Two-level schemes combing atomistic and continuum methods for crack propagation in solids or strong shock fronts in rarefied gases have made their appearance in the early 90s. More recently, threelevel schemes for crack dynamics, combining finite-element treatment of continuum mechanics far away from the crack with molecular dynamics treatment of atomic motion in the near-crack region and a quantum mechanical description of bond-snapping in the crack tip have been demonstrated.

We present a systematic introduction to the basic concepts and techniques for determining transition pathways and transition rates in systems with multiple metastable states. After discussing the classical transition state theory and its limitations, we derive a new set of equations for the optimal dividing surfaces. We then discuss transition path sampling, which is the most general technique currently available for determining transition regions and rates. This is followed by a discussion on minimal energy path for systems with smooth energy landscapes. For systems with rough energy landscapes, our presentation is centered around the notion of reaction coordinates. We discuss the two related notions of free energies associated with a reaction coordinate, and show that at least in the high friction limit, there does exist an optimal reaction coordinate that gives asymptotically the correct prediction for the transition rates. Variational principles associated with the optimal reaction coordinates are exploited under the assumption that the transition paths are restricted to tubes, and this provides a theoretical justification for the finite temperature string method. Blue moon sampling techniques, metadynamics and a new form of accelerated dynamics are also discussed.