Single-site Potential Research Articles

Machine-learned coarse-grained potentials for particles with anisotropic shapes and interactions

Computational investigations of biological and soft-matter systems governed by strongly anisotropic interactions typically require resource-demanding methods such as atomistic simulations. However, these techniques frequently prove to be prohibitively expensive for accessing the long-time and large-length scales inherent to such systems. Conversely, coarse-grained models offer a computationally efficient alternative. Nonetheless, models of this type have seldom been developed to accurately represent anisotropic or directional interactions. In this work, we introduce a straightforward bottom-up, data-driven approach for constructing single-site coarse-grained potentials suitable for particles with arbitrary shapes and highly directional interactions. Our method for constructing these coarse-grained potentials relies on particle-centered descriptors of local structure that effectively encode dependencies on rotational degrees of freedom in the interactions. By using these descriptors as regressors in a linear model and employing a simple feature selection scheme, we construct single-site coarse-grained potentials for particles with anisotropic interactions, including surface-patterned particles and colloidal superballs in the presence of non-adsorbing polymers. We validate the efficacy of our models by accurately capturing the intricacies of the potential-energy surfaces from the underlying fine-grained models. Additionally, we demonstrate that this simple approach can accurately represent the contact function (shape) of non-spherical particles, which may be leveraged to construct continuous potentials suitable for large-scale simulations.

Derivation of Coupled KPZ Equations from Interacting Diffusions Driven by a Single-Site Potential

Derivation of Coupled KPZ Equations from Interacting Diffusions Driven by a Single-Site Potential

Wegner estimate and localisation for alloy-type operators with minimal support assumptions on the single site potential

Abstract We prove a Wegner estimate for alloy-type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set (a particular class of sets of positive measure). The proof exploits on one hand recently proven unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes and on the other hand the well developed machinery for proving Wegner estimates. We obtain a Wegner estimate with optimal volume dependence at all energies, and localization near the minimum of the spectrum, even for some non-stationary random potentials. We complement the result by showing that a lower bound on the potential by the characteristic function of a thick set is necessary for a Wegner estimate to hold. Hence, we have identified a sharp condition on the size for the support of random potentials that is sufficient and necessary for the validity of Wegner estimates.

Data-Driven Approach to Coarse-Graining Simple Liquids in Confinement.

We propose a data-driven framework for identifying coarse-grained (CG) Lennard-Jones (LJ) potential parameters in confined systems for simple liquids. Our approach involves the use of a Deep Neural Network (DNN) that is trained to approximate the solution of the Inverse Liquid State (ILST) problem for confined systems. The DNN model inherently incorporates essential physical characteristics specific to confined fluids, enabling an accurate prediction of inhomogeneity effects. By utilizing transfer learning, we predict single-site LJ potentials of simple multiatomic liquids confined in a slit-like channel, which effectively replicate both the fluid structure and molecular force of the target All-Atom (AA) system when the electrostatic interactions are not dominant. In addition, we showcase the synergy between the data-driven approach and the well-known Bottom-Up coarse-graining method utilizing Relative-Entropy (RE) Minimization. Through the sequential utilization of these two methods, the robustness of the iterative RE method is significantly augmented, leading to a remarkable enhancement in convergence.

On Bose–Einstein Condensation in One-Dimensional Noninteracting Bose Gases in the Presence of Soft Poisson Obstacles

We study Bose–Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on \(\mathbb R\) with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With a probability arbitrarily close to one when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the rth mean, \(r \ge 1\). Furthermore, in Poisson random potentials of any fixed strength a probability arbitrarily close to one for type-I g-BEC to occur is also obtained, but our upper bound for the number of macroscopically occupied one-particle states may be large in this case.

Bottom-up coarse-grain modeling of plasticity and nanoscale shear bands in α-RDX.

Computationally inexpensive particle-based coarse-grained (CG) models are essential for use in molecular dynamics (MD) simulations of mesoscopically slow cooperative phenomena, such as plastic deformations in solids. Molecular crystals possessing complex symmetry present enormous practical challenges for particle-based coarse-graining at molecularly resolved scales, when each molecule is in a single-site representation, and beyond. Presently, there is no published pairwise non-bonded single-site CG potential that is able to predict the space group and structure of a molecular crystal. In this paper, we present a successful coarse-graining at a molecular level from first principles of an energetic crystal, hexahydro-1,3,5-trinitro-s-triazine (RDX) in the alpha phase, using the force-matching-based multiscale coarse-graining (MSCG/FM) approach. The new MSCG/FM model, which implements an optimal pair decomposition of the crystal Helmholtz free energy potential in molecular center-of-mass coordinates, was obtained by force-matching atomistic MD simulations of liquid, amorphous, and crystalline states and in a wide range of pressures (up to 20GPa). The MSCG/FM potentials for different pressures underwent top-down optimization to fine-tune the mechanical and thermodynamic properties, followed by consolidation into a transferable density-dependent model referred to as RDX-TC-DD (RDX True-Crystal Density-Dependent). The RDX-TC-DD model predicts accurately the crystal structure of α-RDX at room conditions and reproduces the atomistic reference system under isothermal (300K) hydrostatic compression up to 20GPa, in particular, the Pbca symmetry of α-RDX in the elastic regime. The RDX-TC-DD model was then used to simulate the plastic response of uniaxially ([100]) compressed α-RDX resulting in nanoscale shear banding, a key mechanism for plastic deformation and defect-free detonation initiation proposed for many molecular crystalline explosives. Additionally, a comparative analysis of the effect of core-softening of the RDX-TC-DD potential and the degree of molecular rigidity in the all-atom treatment suggests a stress-induced short-range softening of the effective intermolecular interaction as a fundamental cause of plastic instability in α-RDX. The reported RDX-TC-DD model and overall workflow to develop it open up possibilities to perform high quality simulation studies of molecular energetic materials under thermal and mechanical stimuli, including extreme conditions.

Multi-scale Analysis of Random Alloy Models with Summable Site Potentials of Infinite Range

We propose a reformulation of the probabilistic component of the Multi-Scale Analysis adapted to the alloy-type Anderson models with a non-negligible dependence at large distances due to an infinite range of the media-particle interaction. Despite a considerable wealth of results accumulated in the spectral theory of random operators with short-range potentials, much less is known about the alloy models with a slow (e.g., inverse polynomial) decay of interaction. Applied to the alloys with a power-law interaction, the new approach gives rise to stronger localization results than in prior works under an optimal assumption (summability) on the single-site potential.

Metastability in a continuous mean-field model at low temperature and strong interaction

We consider a system of N∈N mean-field interacting stochastic differential equations that are driven by Brownian noise and a single-site potential of the form z↦z4∕4−z2∕2. The strength of the noise is measured by a small parameter ε>0 (which we interpret as the temperature), and we suppose that the strength of the interaction is given by J>0. Choosing the empirical mean (P:RN→R, Px=1∕N∑ixi) as the macroscopic order parameter for the system, we show that the resulting macroscopic Hamiltonian has two global minima, one at −mε⋆<0 and one at mε⋆>0. Following this observation, we are interested in the average transition time of the system to P−1(mε⋆), when the initial configuration is drawn according to a probability measure (the so-called last-exit distribution), which is supported around the hyperplane P−1(−mε⋆). Under the assumption of strong interaction, J>1, the main result is a formula for this transition time, which is reminiscent of the celebrated Eyring–Kramers formula (see Bovier et al. (2004)) up to a multiplicative error term that tends to 1 as N→∞ and ε↓0. The proof is based on the potential-theoretic approach to metastability.In the last chapter we add some estimates on the metastable transition time in the high-temperature regime, where ε=1, and for a large class of single-site potentials.

Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows

We show that the time evolution of a quantum wavepacket in a periodic potential converges in a combined high-frequency/Boltzmann-Grad limit, up to second order in the coupling constant, to terms that are compatible with the linear Boltzmann equation. This complements results of Eng and Erd\"os for low-density random potentials, where convergence to the linear Boltzmann equation is proved in all orders. We conjecture, however, that the linear Boltzmann equation fails in the periodic setting for terms of order four and higher. Our proof uses Floquet-Bloch theory, multi-variable theta series and equidistribution theorems for homogeneous flows. Compared with other scaling limits traditionally considered in homogenisation theory, the Boltzmann-Grad limit requires control of the quantum dynamics for longer times, which are inversely proportional to the total scattering cross section of the single-site potential.

Morphological analysis of chiral rod clusters from a coarse-grained single-site chiral potential.

We present a coarse-grained single-site potential for simulating chiral interactions, with adjustable strength, handedness, and preferred twist angle. As an application, we perform basin-hopping global optimisation to predict the favoured geometries for clusters of chiral rods. The morphology phase diagram based upon these predictions has four distinct families, including previously reported structures for potentials that introduce chirality based on shape, such as membranes and helices. The transition between these two configurations reproduces some key features of experimental results for fd bacteriophage. The potential is computationally inexpensive, intuitive, and versatile; we expect it will be useful for large scale simulations of chiral molecules. For chiral particles confined in a cylindrical container we reproduce the behaviour observed for fusilli pasta in a jar. Hence this chiropole potential has the capability to provide insight into structures on both macroscopic and molecular length scales.

Lifshitz tails for Schrödinger operators with random breather potential

We prove a Lifshitz tail bound on the integrated density of states of random breather Schrödinger operators. The potential is composed of translated single-site potentials. The single-site potential is an indicator function of the set tA where t is from the unit interval and A is a measurable set contained in the unit cell. The challenges of this model are that, since A is not assumed to be star-shaped, the dependence of the potential on the parameter t is not monotone. It is also non-linear and not differentiable.

Eigenfunction statistics for Anderson model with Hölder continuous single site potential

We consider random Schr\"{o}dinger operators on $\ell^2(\mathbb{Z}^d)$ when the distribution of single site potentials is $\alpha$-H\"{o}lder continuous ($0<\alpha\leq 1$). In localized regime we study the distribution of eigenfunctions simultaneously in space and energy. In a certain scaling limit we prove limits point are Poisson.

Conditional Wegner Estimate for the Standard Random Breather Potential

We prove a conditional Wegner estimate for Schr\"odinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in previous papers, in particular in recent years. We consider the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. While our methods work for a substantially larger class of random breather potentials, we discuss in this particular paper only the standard model in order to make the arguments and ideas easily accessible.

Electric field effect on the donor levels bound to the X-valleys in AlAs/GaAs quantum wells

We study the Si donor states bound to X-valleys in a GaAs/AlAs/GaAs layer system in the presence of the electric field using a simple Koster–Slater impurity model. The multilayer structure is modeled with the spds⁎ tight binding approximation and the impurity is described by a single site potential. By analyzing the local density of states we have obtained the values of the binding energies of impurity levels bound to “parallel” Xz and “perpendicular” Xx,y valleys (with respect to the direction of the electric field) as function of barrier width and impurity position in the barrier for various values of the electric field. For the case of zero electric field we have obtained very similar results to the k·p calculations in Carneiro and Weber [1].

Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials

We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation, a Wegner estimate, which is polynomial in the volume of the box and linear in the size of the energy interval, holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis.

Modified logarithmic Sobolev inequalities for canonical ensembles

In this paper, we prove modified logarithmic Sobolev inequalities for canonical ensembles with superquadratic single-site potential. These inequalities were introduced by Bobkov and Ledoux, and are closely related to concentration of measure and transport-entropy inequalities. Our method is an adaptation of the iterated two-scale approach that was developed by Menz and Otto to prove the usual logarithmic Sobolev inequality in this context. As a consequence, we obtain convergence in Wasserstein distance W p for Kawasaki dynamics on the Ginzburg−Landau’s model.

Existence, uniqueness and anisotropic-decay-caused Lifshitz tails of the integrated density of surface states for random surface models

Abstract The current paper is devoted to the study of existence, uniqueness and Lifshitz tails of the integrated density of surface states (IDSS) for Schrödinger operators with alloy type random surface potentials. We prove the existence and uniqueness of the IDSS for negative energies, which is defined as the thermodynamic limit of the normalized eigenvalue counting functions of localized operators on strips with sections being special cuboids. Under the additional assumption that the single-site impurity potential decays anisotropically, we also prove that the IDSS for negative energies exhibits Lifshitz tails near the bottom of the almost sure spectrum in the following three regimes: the quantum regime, the quantum-classical/classical-quantum regime and the classical regime. We point out that the quantum-classical/classical-quantum regime is new for random surface models.

Decay of Correlations in 1D Lattice Systems of Continuous Spins and Long-Range Interaction

We consider a one-dimensional lattice system of unbounded and continuous spins. The Hamiltonian consists of a perturbed strictly-convex single-site potential and a product term with longe-range interaction. We show that if the interactions have an algebraic decay of order \(2+\alpha \), \(\alpha >0\), then the correlations also decay algebraically of order \(2+ \tilde{\alpha }\) for some \(\alpha > \tilde{\alpha }> 0\). For the argument we generalize a method due to Zegarlinski from finite-range to infinite-range interaction to get a preliminary decay of correlations, which is improved to the correct order by a recursive scheme based on Lebowitz inequalities. Because the decay of correlations yields the uniqueness of the Gibbs measure, the main result of this article yields that the one-phase region of a continuous spin system is at least as large as for the Ising model. This shows that there is no-phase transition in one-dimensional systems of unbounded and continuous spins as long as the interaction decays algebraically of order \(2+\alpha \), \(\alpha >0\).

Representability and Transferability of Kirkwood-Buff Iterative Boltzmann Inversion Models for Multicomponent Aqueous Systems.

We discuss the application of the Kirkwood-Buff iterative Boltzmann inversion (KB-IBI) method for molecular coarse-graining (Ganguly et al. J. Chem. Theory Comput. 2012, 8, 1802) to multicomponent aqueous mixtures. Using a fixed set of effective single-site solvent-solvent potentials previously derived for binary urea-water systems, solute-solvent and solute-solute KB-IBI coarse-grained (CG) potentials have been derived for benzene in urea-water mixtures. Preferential solvation and salting-in coefficients of benzene are reproduced in quantitative agreement with the atomistic force field model. The transferability of the CG models is discussed, and it is shown that free energies of formation of hydrophobic benzene clusters obtained from simulations with the CG model are in good agreement with results obtained from all-atom simulations. The state-point representability of the CG models is discussed with respect to reproducing thermodynamic quantities such as pressure, isothermal compressibility, and preferential solvation. Combined use of KB-IBI and pressure corrections in deriving single-site CG models for pure-water, binary mixtures of urea and water, and ternary mixtures of benzene in urea-water at infinite benzene dilution provides an improved scheme to representing the atomistic pressure and the preferential solvation between the solution components. It is also found that the application of KB-IBI leads to a faster and improved convergence of the pressure and potential energy compared to the IBI method.

Minami’s Estimate: Beyond Rank One Perturbation and Monotonicity

In this note, we prove Minami’s estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami’s estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).