Abstract
The classical procedure devised by Irving and Kirkwood in 1950 and completed slightly later by Noll produces counterparts of the basic balance laws of standard continuum mechanics starting from an ordinary Hamiltonian description of the dynamics of a system of material points. Post-1980 molecular dynamics simulations of the time evolution of such systems use extended Hamiltonians such as those introduced by Andersen, Nosé, and Parrinello and Rahman. The additional terms present in these extensions affect the statistical properties of the system so as to capture certain target phenomenologies that would otherwise be beyond reach. We here propose a physically consistent application of the Irving-Kirkwood-Noll procedure to the extended Hamiltonian systems of material points. Our procedure produces balance equations at the continuum level featuring non-standard terms because the presence of auxiliary degrees of freedom gives rise to additional fluxes and sources that influence the thermodynamic and transport properties of the continuum model. Being aware of the additional contributions may prove crucial when designing multiscale computational schemes in which information is exchanged between the atomistic and continuum levels.
Highlights
To close the atomistic-to-continuum scale gap to the extent of associating a set of deterministic balances with a probabilistic account of the Newtonian evolution of a system of material points, we take the bottom-up path pioneered by Irving and Kirkwood[1] in a paper that appeared in 1950
The classical procedure devised by Irving and Kirkwood in 1950 and completed slightly later by Noll produces counterparts of the basic balance laws of standard continuum mechanics starting from an ordinary Hamiltonian description of the dynamics of a system of material points
Our procedure produces balance equations at the continuum level featuring non-standard terms because the presence of auxiliary degrees of freedom gives rise to additional fluxes and sources that influence the thermodynamic and transport properties of the continuum model
Summary
To close the atomistic-to-continuum scale gap to the extent of associating a set of deterministic balances with a probabilistic account of the Newtonian evolution of a system of material points, we take the bottom-up path pioneered by Irving and Kirkwood[1] in a paper that appeared in 1950. As is well-known, the IKN procedure tells us what statistical observables to associate with the continuum mechanical notions of mass, linear momentum, and energy density On evolving these observables ala Liouville with respect to a chosen Hamiltonian, three consequences of the Newtonian motion of a system of material points are derived. When setting up the IKN procedure for an extended Hamiltonian system, that is, to say, a particle system whose Hamiltonian H is nonstandard, two key observations are that (i) certain microscopic observables must be defined by combining ordinary physical variables with other possibly “unphysical” variables and that (ii) all macroscopic observables, whatever their microscopic antecedents, consist of ensemble averages weighted with respect to probability densities defined over the extended phase space This in itself guarantees a basic statistical-continuum consistency in the derived balances of macroscopic observables.
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