An important aspect of multi-scale modelling of materials is to link continuum concepts, such as fields, to the underlying discrete microscopic behaviour in a seamless manner. With the growing importance of atomistic calculations to understand material behaviour, reconciling continuum and discrete concepts is necessary to interpret molecular and quantum-mechanical simulations. In this work, we provide a quantum-mechanical framework to a distinctly continuum quantity: mechanical stress. While the concept of the global macroscopic stress tensor in quantum mechanics has been well established, there still exist open issues when it comes to a spatially varying local quantum stress tensor. We attempt to shed some light on this topic by establishing a general quantum-mechanical operator-based approach to continuity equations and from those, introduce a local quantum-mechanical stress tensor. Further, we elucidate the analogies that exist between the (classical) molecular-dynamics-based stress definition and the quantum stress. Our derivations appear to suggest that the local quantum-mechanical stress may not be an observable in quantum mechanics and therefore traces the non-uniqueness of the atomistic stress tensor to the gauge arbitrariness of the quantum-mechanical state function. Lastly, the virial stress theorem (of empirical molecular dynamics) is re-derived in a transparent manner that elucidates the analogy between quantum-mechanical global stresses.
An important aspect of multi-scale modelling of materials is to link continuum concepts, such as fields, to the underlying discrete microscopic behaviour in a seamless manner. With the growing importance of atomistic calculations to understand material behaviour, reconciling continuum and discrete concepts is necessary to interpret molecular and quantum-mechanical simulations. In this work, we provide a quantum-mechanical framework to a distinctly continuum quantity: mechanical stress. While the concept of the global macroscopic stress tensor in quantum mechanics has been well established, there still exist open issues when it comes to a spatially varying local quantum stress tensor. We attempt to shed some light on this topic by establishing a general quantum-mechanical operator-based approach to continuity equations and from those, introduce a local quantum-mechanical stress tensor. Further, we elucidate the analogies that exist between the (classical) molecular-dynamics-based stress definition and the quantum stress. Our derivations appear to suggest that the local quantum-mechanical stress may not be an observable in quantum mechanics and therefore traces the non-uniqueness of the atomistic stress tensor to the gauge arbitrariness of the quantum-mechanical state function. Lastly, the virial stress theorem (of empirical molecular dynamics) is re-derived in a transparent manner that elucidates the analogy between quantum-mechanical global stresses.
The concept of a stress tensor is fundamental in classical continuum mechanics, encapsulating internal forces per unit area that a material can sustain. In quantum many‑body systems, however, the notion of pressure and stress has traditionally been limited to scalar or local quantities derived from the equation of state or from density functional approximations. In this work we introduce a fully tensorial representation of electronic internal forces that arises naturally from the quantum mechanical description of electrons in condensed matter systems. We define the Electron Quantum Stress Tensor (EQST) as a second‑rank tensor operator constructed from the kinetic and interaction contributions to the many‑electron Hamiltonian. By deriving explicit expressions for the EQST in terms of the electron density, its gradients, and the exchange–correlation potential, we establish a direct link between the microscopic wavefunction and macroscopic mechanical observables. The EQST framework provides a new lens for interpreting phenomena in two‑dimensional electron gases, quantum Hall fluids, and other strongly correlated electronic materials, offering a potential route toward uncovering novel mechanical properties of electrons that could have implications for future Nobel‑level discoveries.
Within the density-functional theory of Hohenberg, Kohn and Sham, a comprehensive stress tensor is defined for a many-electron polyatomic system. This consists of contributions from kinetic energy, Coulomb energy and exchange correlation energy. The covariant derivative of the stress tensor yields the 'local' force density in three-dimensional space, which vanishes for static stationary states due to the balance between classical and quantum force densities. Some interesting implications of the zero force density are discussed. In particular, this turns out to be a special case of the Euler equation or the Navier-Stokes equation in fluid dynamics. The fluid-dynamical interpretation of the stress tensor is discussed and the viscosity coefficients are derived. The present work is likely to provide a basis for 'classical' interpretations of many molecular and solid-state phenomena.
The equation of motion technique is used to derive quantum operators corresponding to the microscopic stress tensor and pressure. The technique is demonstrated for a one-dimensional system. Stress tensor and related operators are obtained for a nonrelativistic, many-body, Coulomb system. The stress tensor operators occur as a momentum flux density in continuity equations which are the quantum analog to Newton's second law (F = ma). The derivation also produces operators with obvious classical analogs such as linear and angular momentum densities in a context which makes their physical role evident.
I show how the pressure in Fermi and Bose systems, identified in standard discussions of quantum statistical mechanics by the use of thermodynamic analogies, can be derived directly in terms of the flux of momentum across a surface by using the quantum mechanical stress tensor. In this approach, which is analogous to classical kinetic theory, the pressure is naturally defined locally. The approach leads to a simple interpretation of the pressure in terms of the momentum flow encoded in the wave functions. The stress-tensor and thermodynamic approaches are related by an interesting application of boundary perturbation theory for quantum systems. I investigate the properties of quasi-continuous systems, the relations for Fermi and Bose pressures, shape-dependent effects and anisotropies, and the treatment of particles in external fields, and note several interesting problems for graduate courses in statistical mechanics.
In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.
Introduction: Quantum computing has the potential to transform scientific computing by providing efficient algorithms for solving large-scale numerical problems that arise in engineering and materials science. However, integrating quantum algorithms into multiscale mechanics remains largely unexplored due to the complexity of coupling atomistic and continuum models and the nonlinear nature of molecular dynamics. In this work, we develop a unified hybrid classical–quantum framework for molecular dynamics and multiscale materials modeling based on the Variational Quantum Linear Solver (VQLS). Materials and methods: The proposed framework reformulates three representative mechanics problems as quantum-compatible linear systems: static atomistic–continuum coupling through reduced stiffness equations, dynamic atomistic–continuum coupling using a boundary-reduced Berry–Childs–Ostrander–Wang formulation, and nonlinear molecular dynamics through Carleman linearization. These systems are solved using shot-free quantum-circuit simulations within a common VQLS framework. Results: For the static problem with 1024 free displacement degrees of freedom, the VQLS solution achieves R 2 = 0.99784 and a relative L 2 error of 2.70 % . For the dynamic problem, the five-qubit restarted systems produce maximum final-time displacement and velocity differences of 1.39 × 10 − 4 nm and 6.74 × 10 − 5 nm/ps, respectively. For the nonlinear problem, comparison of the six-qubit VQLS trajectory with a high-accuracy integration of the full Lennard–Jones equations over 150 time steps gives maximum displacement and velocity differences of 4.09 × 10 − 4 nm and 2.87 × 10 − 3 nm/ps, respectively. The reported velocities are reconstructed from the displacement histories using a common backward-difference rule with the prescribed zero initial velocity. The numerical comparisons quantify agreement with classical reference trajectories and distinguish selected modeling, temporal-discretization, and variational-solution errors. Conclusions: By providing a unified quantum linear-system formulation for representative multiscale mechanics problems, this deterministic, shot-free, simulator-based study establishes a methodological proof of concept for integrating VQLS formulations with computational materials models, rather than demonstrating near-term hardware performance or scalable quantum computational advantage.