A non-equilibrium multiscale simulation paradigm Shaofan Li * , Ni Sheng, Xiaohu Liu Department of Civil and Environmental Engineering, University of California, 783 Davis Hall, Berkeley, CA 94720, USA Received 23 October 2007; in final form 30 November 2007 Available online 8 December 2007 Abstract A con-current multiscale non-equilibrium molecular dynamics is proposed. The notion of multiscale canonical element ensemble is put forth, which enables us to employ the coarse grain field as a heat bath, and it is accomplished by using a distributed Nos ´e–Hoover ther- mostat network. In doing so, it guarantees that the non-equilibrium molecular dynamics returns to a canonical equilibrium state when external disturbances vanish, which may have not been the case for the non-equilibrium molecular dynamics in the literature. We have shown that the non-equilibrium distribution function is canonical. Ó 2007 Elsevier B.V. All rights reserved. 1. Introduction The non-equilibrium thermal–mechanical process at small scales, for instance nanoscale heat transfer, in which the length and/or time scales span from the molecular to the continuum, is a subject of increasing importance to energy conversion, biotechnology, microelectronics, biochemical detection, and material synthesis and failure analysis. The capacity to simulate thermal–mechanical couplings in non-equilibrium states at small scales are vital to the understanding of transport mechanisms of energy conversion and to the advancement of reliability of micro and nano-electronics. The conventional molecular dynamics (MD) is a dynam- ics of the micro-canonical ensemble, and it is unable to pro- vide statistical characters to the physics problem that it simulates. To extrapolate statistical thermodynamics infor- mation from molecular dynamics simulations, the simula- tions of equilibrium ensemble molecular dynamics (EEMD) are required at a fixed temperature or fixed pres- sure or specified chemical potentials. Various EEMDs have been implemented by using different thermostats, e.g. [1–5]. However, the EEMD is unable to simulate problems with spatial or temporal temperature gradients. Since early 1980s, the non-equilibrium molecular dynamics (NEMD) has become a major simulation tool for simulations of non-equilibrium processes. The NEMD has been used to compute transport coefficients [6–8] and to simulate viscous flows [9–11] and plastic deformations [12]. In the literatures, there are mainly three types of NEMDs: (1) Prescribed-flow-driven NEMD (2) The synthetic NEMD, and (3) Boundary-flux-driven NEMD The representatives of the first type NEMDs are the well- known DOLLS and SLLOD algorithms, e.g. [9–11,13–16], in which the MD system is driven out of equilibrium by pre- scribed constant flow field. To the best of the authors’ knowledge, this type of NEMDs are only used in special cases such as simulation of the Couette flows for extrapolat- ing the viscous coefficient. In the second type of NEMDs [7,14,17–19], an artificial external field is prescribed to drive the system out of equilibrium, however, the artificial exter- nal field is judicially chosen such that it renders the phase– space flux divergence-free, or it enforces the so-called adia- batic incompressibility condition (AIC) [14]. The synthetic NEMD is related to the Green–Kubo linear response theory, and it has been mainly used to extrapolate the transport coefficients of non-equilibrium states from an 0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.11.099 * Corresponding author. Fax: +1 510 642 8928. E-mail address: shaofanli@gmail.com (S. Li). www.elsevier.com/locate/cplett Available online at www.sciencedirect.com Chemical Physics Letters 451 (2008) 293–300