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We formulate a theory of non-equilibrium statistical thermodynamics for ensembles of atoms or molecules. The theory is an application of Jayne's maximum entropy principle, which allows the statistical treatment of systems away from equilibrium. In particular, neither temperature nor atomic fractions are required to be uniform but instead are allowed to take different values from particle to particle. In addition, following the Coleman-Noll method of continuum thermodynamics we derive a dissipation inequality expressed in terms of discrete thermodynamic fluxes and forces. This discrete dissipation inequality effectively sets the structure for discrete kinetic potentials that couple the microscopic field rates to the corresponding driving forces, thus resulting in a closed set of equations governing the evolution of the system. We complement the general theory with a variational meanfield theory that provides a basis for the formulation of computationally tractable approximations. We present several validation cases, concerned with equilibrium properties of alloys, heat conduction in silicon nanowires and hydrogen desorption from palladium thin films, that demonstrate the range and scope of the method and assess its fidelity and predictiveness. These validation cases are characterized by the need or desirability to account for atomiclevel properties while simultaneously entailing time scales much longer than * Corresponding author E-mail address: ortiz@caltech.edu (M. Ortiz). September 27, 2014 those accessible to direct molecular dynamics. The ability of simple meanfield models and discrete kinetic laws to reproduce equilibrium properties and long-term behavior of complex systems is remarkable. Preprint submitted to Journal of the Mechanics and Physics of Solids

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a b s t r a c tWe study dynamic nanovoid growth in copper single crystals under prescribed volumetric strain rates ranging from moderate ( _ ¼ 10 5 s À1 ) to high ( _ ¼ 10 10 s À1 ). We gain access to lower strain rates by accounting for thermal vibrations in an entropic sense within the framework of maximum-entropy non-equilibrium statistical mechanics. We additionally account for heat conduction by means of empirical atomic-level kinetic laws. The resulting mean trajectories of the atoms are smooth and can be integrated implicitly using large time steps, greatly in excess of those required by molecular dynamics. We also gain access to large computational cells by means of spatial coarse-graining using the quasicontinuum method. On this basis, we identify a transition, somewhere between 10 7 and 10 8 s À1 , between two regimes: a quasistatic regime characterized by nearly isothermal behavior and low dislocation velocities; and a dynamic regime characterized by nearly adiabatic conditions and high dislocation velocities. We also elucidate the precise mechanisms underlying dislocation emission from the nanovoids during cavitation. We additionally investigate the sensitivity of the results of the analysis to the choice of interatomic potential by comparing two EAM-type potentials.

SUMMARYThe present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplace's problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co-ordinates and the resulting integrals can be numerically evaluated in a simple and e cient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulÿl the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation.

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