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Microscopic and Macroscopic Stress with Gravitational and Rotational Forces

Many recent papers have questioned Irving and Kirkwood's atomistic expression for stress. In Irving and Kirkwood's approach both interatomic forces and atomic velocities contribute to stress. It is the velocity-dependent part that has been disputed. To help clarify this situation we investigate [1] a fluid in a gravitational field and [2] a steadily rotating solid. For both problems we choose conditions where the two stress contributions, potential and kinetic, are significant. The analytic force-balance solutions of both these problems agree very well with a smooth-particle interpretation of the atomistic Irving-Kirkwood stress tensor.


I. INTRODUCTION
In 2003 Zhou published his lengthy and detailed "New Look at the Atomic Level Virial Stress" in the Proceedings of the Royal Society of London [1]. He criticized the usual Irving-Kirkwood virial expression [2] for the pressure tensor P as a sum of potential and kinetic terms. The pressure tensor is the same thing as the comoving corotating momentum flux, and is also minus the stress tensor, σ ≡ −P . The detailed microscopic Irving-Kirkwood approach has been used for more than 50 years in the interpretation of atomistic molecular dynamics simulations. [3,4,5,6] Averaged over a homogeneous periodic volume V the Irving-Kirkwood expression for the pressure tensor gives: Here F ij is the force (for simplicity we assume a pairwise-additive potential) exerted on Particle i by Particle j, where the vector from j to i is r ij . Particle i, at location r i with mass m i and momentum p i , obeys Newton's equation of motion, Zhou stated that only the tensor force sum, (F r) ij ≡ F ij r ij , contributes to the stress, while the tensor momentum sum, (pp/m) i ≡ (p i p i /m i ), does not.
This idea -including the forces but not the momenta -is not quite so outlandish as it seems. In solids, where the longtime average of the particle location is a sensible quantity, the virial theorem can be written in a similar tensor form omitting the momenta: This form is derived in Section II.C of Reference 4. We use angular brackets here to indicate longtime averages. In situations including external forces the tensor force sum must also include either (F ext r) i or (F ext R) i . Subramaniyan and Sun [7] tested Zhou's ideas with molecular dynamics, heating a model atomistic solid subject to a variety of external boundary conditions on the particle coordinates. Their simulations showed that only the full Irving-Kirkwood pressure tensor, potential plus kinetic, was consistent with macroscopic thermodynamics. Liu and Qiu [8] recently provided a useful list of references supporting both sides of the question. In addition they suggest that the Zhou prescription is correct provided that external fields and rotation are not involved. Here we explore those latter two conditions separately and explicitly, showing that both [1] an external field (gravity) and [2] condensed-phase rotation can be analyzed properly with the Irving-Kirkwood pressure tensor, in a way compatible with macroscopic continuum mechanics. This suggests that the original Irving-Kirkwood approach is more generally useful than is Zhou's suggested modification of it.
In order to compute continuous differentiable field variables (density, velocity, energy, stress, heat flux, ... ) from atomistic molecular dynamics simulations, for comparison to corresponding fields generated by continuum mechanics solutions, we recommend the use of "smooth-particle" averages. These correspond to smearing individual particle properties over a spatial region of size h, the range of the smooth-particle weighting function, as is described in a recent text [9], summarized in Section II, and applied in Section III.
Because the derivation of the pressure tensor is familiar, and applies both at and away from equilibrium [4,5] we do not repeat that here. Instead, in Sections III and IV, we describe and study two specially instructive problems involving gravitational and rotational forces. We reserve our conclusions and closing remarks for Section V.

II. SMOOTH-PARTICLE AVERAGES OF ATOMISTIC PROPERTIES
Irving and Kirkwood chose to localize particle properties at the particle locations using delta functions. Though this is convenient for formal analyses, and even natural for mass and momentum, a smoothed or smeared-out particle contribution to potential energy and to fluxes often simplifies comparisons with continuum mechanics. The smeared approach can provide field variables with two continuous spatial derivatives, as we show below.
Because "action at a distance" makes the exact location of momentum and energy fluxes ambiguous we choose to smear out particle contributions within a spatial region somewhat larger in extent than the spacing between particles. We use a local weight function with a range h, w(r, h) to convert particle properties to continuum field properties. Consider, for example, the density ρ and the velocity v in a fluid or solid composed of particles with individual masses and velocities {m i , v i }. In the smooth-particle approach [9,10] field variables, such as the density and velocity at the point r, are defined as h-dependent (range-dependent) sums of nearby particle contributions: The sums include all particles within a distance h of point r. A good feature of this approach is that these definitions of density and velocity satisfy the continuity equation,ρ/ρ ≡ −∇ · v, exactly. Here, as is usual, the dot indicates a comoving time derivative following the motion.
Lucy was one of the inventors of the smooth-particle approach [10]. For convenience we use his form for the weighting function in all of our smooth-particle sums, This form has two continuous derivatives everywhere. The normalizing prefactor C D depends on the dimensionality D, C is chosen so that the spatial integral of the weight function is unity: Lucy's polynomial form is the simplest normalized weight function with a maximum value at the origin and two continuous derivatives everywhere. In the following section, where we consider the mechanical equilibrium of a two-dimensional fluid in a one-dimensional gravitational field, we compute average values of the pressure tensor using the one-dimensional form of Lucy's weight function.

III. GRAVITATIONAL EQUILIBRATION
Gravitational equilibration is a problem in which both the potential and kinetic contributions to stress can play a rôle. Where a constant gravitational acceleration acts downward in y, the simple force balance equation for mechanical equilibrium is, The stationary density profile, ρ(y), can be found provided that the dependence of pressure P on the density ρ is known. As a simple example problem, chosen to highlight the kinetic and potential contributions to the virial, we choose to study the molecular dynamics of an atomistic system which closely approximates the isothermal fluid equation of state This equation of state closely corresponds to the virial equation of state for two-dimensional particles of unit mass at unit temperature interacting with a "Cusp" potential chosen to have a spatial integral of unity: We use this cusp interaction for the interparticle forces because the model closely corresponds to the simple and useful thermodynamic equation of state given above. We choose the range of the Cusp pair potential h = 3, so that the deviation of the potential part of the pressure tensor from that macroscopic equation of state is of order one percent.
For periodic two-dimensional systems the virial-theorem expression for the potential part of the pressure tensor can be expressed in terms of sums over all N(N − 1)/2 pairs of interacting particles [4,5]. For a hydrostatic fluid, where P Φ xx and P Φ yy are each equal to the potential part P Φ of the hydrostatic pressure P , we have: For a completely random distribution of particles in the volume V the potential part of the pressure is then given by a force integral. The integral can be related to the integral of the pair potential using integration by parts. With our particular choice of pair potential φ, with an integral of unity, and particle mass, unity, the resulting hydrostatic pressure is simply half the square of the density: A snapshot from an isokinetic (constant kinetic temperature) simulation appears in Figure 1.
For convenience we have chosen a situation in which the potential and kinetic parts of the pressure are equally important. At unit temperature (kT = 1) and a density of 2 We choose the gravitational acceleration g so that the "weight" of a column of unit width and containing n y particles is equal to the maximum pressure, 4, at the maximum density, ρ(y = 0) = 2. In this case the mechanical equilibrium force-balance density and pressure profiles are: (ρ + 1)(dρ/dy) = −ρg −→ ρ − 2 + ln(ρ/2) = −gy ; P (y) = P Φ (y) + P K (y) = g ∞ y ρ(y)dy .
We test these analytic results against a molecular dynamics simulation carried out isothermally [4,5,6] at a constant temperature of unity. At and below the bottom y = 0 of the column we place 6n x boundary particles in an area of 3n x (corresponding to the maximum density, 2). See Figure 1. We also include a short-ranged repulsive force, which is applied to those few moving particles which occasionally penetrate the boundary at y = 0.
With periodic boundaries in x and a repulsive boundary at y = 0 a 9216-particle simulation gives the typical configuration we showed in Figure 1. The corresponding kinetic and potential pressure profiles, averaged vertically with Lucy's one-dimensional weight function, are compared to the analytic force-balance profile in Figure 2. Evidently the agreement is quite good, and would be qualitatively in error were the kinetic contribution to the pressure tensor omitted.
-50 < x < +50 nearest-neighbor Hooke's-Law interaction, with the force constant κ, characteristic length d, and particle mass m all set equal to unity. To make contact with continuum mechanics we write the stress tensor in terms of the displacement vector u and elastic constants λ and η: where I is the unit tensor, with I xx = I yy ≡ 1 and I xy = I yx = 0. For the nearest-neighbor Hooke's-law crystal the Lamé constants are known, The radial displacement in a rotating disk of radius R, u(r), as well as the corresponding stress tensor σ are well-known results of linear elastic theory [11]. A derivation for our two-dimensional situation is sketched in the Appendix. The results are: The stress components satisfy the radial force-balance equation for a plane-polar-coordinate volume element rdrdθ rotating at the angular frequency ω: +ρr = −ρrω 2 = (dσ rr /dr) + (σ rr − σ θθ )/r .
In the comoving and corotating frame, where stress is the negative of the momentum flux, rotation provides a centrifugal force per unit mass varying as ω 2 .
To compare these results from linear elasticity to molecular dynamics simulations, consider the stationary rotation of a Hooke's-Law lattice. Figure 3 shows two nominally stationary states of a 2335-particle solid with an angular velocity of ω = 0.01. The cold crystal is shown at the left. The kinetic temperature of the warm crystal shown on the right is kT = 0.01. The 2335-particle crystal is nearly circular. It is the smallest with 36 particles equidistant from the origin (at √ 637 ≃ 25.239). Both these rotational problems were initialized by thermostating the radial momenta [4,5] while rescaling the angular momenta to generate thermally-equilibrated steadily-rotating solid disks. During the first half of each run two separate rescaling, or "Gaussian", thermostats were applied, so as to keep the radial temperature and the angular velocity constant. The Hooke's-law nature of the particle interactions guarantees that our model crystals will not melt. But as temperature rises the deformation can become quite large, so that linear elastic theory no longer applies. Figure 5 is a typical view of a rotating specimen at a rotation rate of ω = 0.01 and a kinetic temperature (relative to rigid-body rotation) kT = 0.02.
The simplest route to the polar-coordinate stress tensor is, first, to calculate the kinetic and potential parts of each particle's pressure tensor in Cartesian coordinates: In keeping with the Irving-Kirkwood picture, the potential contributions to the pressure tensor are divided evenly between pairs {i, j} of interacting particles. The polar-coordinate representation for each particle's pressure tensor follows from the Cartesian representation by a simple rotation, which can be written as a pair of matrix multiplications:

V. CONCLUSION
Both the gravitational and the rotational problems show excellent correspondence between conventional continuum mechanics and atomistic mechanics provided that both the kinetic and potential parts of the pressure tensor are included in the analysis. Although for stationary solids the solely potential form for the virial theorem is correct, the number and type of problems which can be studied numerically is greatly enhanced by including Irving and Kirkwood's ideas coupled with the smooth-particle averaging introduced by Lucy and Monaghan in 1977. For well-defined local properties, both at, and especially away from equilibrium, it is essential that these properties be measured in a coordinate frame that moves with the material. It is no accident that the fundamental equations of continuum mechanics take their simplest form in the comoving frame. In particular, the pressure (or stress) and temperature tensors, as well as the heat flux, only make sense in this frame.
Stress and pressure cannot depend upon the chosen coordinate system. Hence we must choose the "comoving" "corotating" "Lagrangian" frame. In that frame the pressure tensor is simply the momentum flux, and has both potential and kinetic contributions, as shown clearly in the two problems solved here.

VI. APPENDIX
The stationary rotation, at angular velocity ω, of an elastic disk of radius R with equal Lamé constants λ = η = 3/16 obeys the force-balance equation in the comoving frame, 0 = +ρω 2 r + (∂σ rr /∂r) + (σ rr − σ θθ )/r . This solution can be used to generate the maximum tensile stress in the disk as well as the stress and strain profiles.