Hyperdynamics made simple: Accelerated molecular dynamics with the Bond-Boost method Kristen A. Fichthorn a,b,⇑ , Shafat Mubin b a Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, USA b Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA article info Article history: Received 1 October 2014 Received in revised form 3 December 2014 Accepted 4 December 2014 Available online xxxx Keywords: Molecular dynamics Rare events Transition-state theory Statistical mechanics abstract Hyperdynamics methods have significant potential for accelerating simulations of dynamical evolution in solids and solid-like materials, which is mediated by rare events. In this paper, we illustrate the main con- cepts associated with understanding hyperdynamics through a series of one-dimensional examples. These examples, which are mostly based on the Bond-Boost method, indicate the rigor of hyperdyamics methods and their potential to resolve key bottle-necks in simulating long-time evolution of materials. A significant advantage of hyperdynamics methods is their capability to resolve the small-barrier problem, which is ubiquitous in materials simulation. We present a simple boost potential with benefits for solving the small-barrier problem and discuss future challenges in hyperdynamics simulations. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction The time evolution of solids and solid-like materials often occurs via rare events, in which the system spends relatively long times localized in free-energy minima and occasionally exhibits ‘‘jumps’’ between neighboring minima. This situation can be difficult in molecular-dynamics (MD) simulations of materials because the time spent simulating localized motion in the free-energy minima can greatly exceed the ns–ls time scales typically covered in MD simulations. Thus, the capability of MD to render the time evolution of materials can be limited. To circumvent this difficulty, a variety of different methods have been proposed, ranging from kinetic Monte Carlo (KMC) [1,2] to accelerated MD [3–19]. Our group has worked to advance accelerated MD methods within the framework of hyperdynamics [3,4,7,9,11–15]. Below, we present the theory of hyperdynamics through a series of tutorial exercises. We discuss applications and future prospects for these methods. 2. Rates of rare events: transition-state theory An understanding of how to evaluate rate constants from atom- istic simulations is a prerequisite for properly conducting hyperdy- namics simulations. Below, we discuss how to evaluate rate constants within the framework of transition-state theory (TST) [20]. In TST, the rate of escape from a free-energy minimum A to a neighboring free-energy minimum B can be viewed in the con- text of equilibrium statistical thermodynamics as a canonical aver- age of the flux across the dividing surface between A and B. This can be expressed as k TST A!B ¼ 1 2 RR dx dpH A d y AB jv ? j expðbðK þ V ÞÞ RR dx dpH A expðbðK þ V ÞÞ ; ð1Þ where x and p are the 3N-dimensional configuration and momen- tum vectors, respectively, of the N-particle system, b ¼ 1=k B T ; K and V are the kinetic and potential energies, respectively, H A has a value of one if the system is in state A and is zero otherwise, d y AB is the delta function defining the location of the dividing hyper- surface and it is considered to reside within the domain of A; jv ? j is the velocity component orthogonal to the dividing hypersurface, and the factor of 1=2 limits the flux to trajectories that are exiting from A. Since the kinetic energy is independent of position and the velocities have the Maxwell–Boltzmann distribution, we can write Eq. (1) as k TST A!B ¼ 1 2 2 pbm   1=2 R dxH A d y AB expðbV Þ R dxH A expðbV Þ ; ð2Þ where m is the (effective) mass. Eq. (2) expresses the rate as the product of the probability of the system to reside at the dividing sur- face between A and B (the ratio of integrals) and the one-dimensional speed through the dividing surface, heading from A to B. http://dx.doi.org/10.1016/j.commatsci.2014.12.008 0927-0256/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author at: Department of Chemical Engineering, The Pennsyl- vania State University, University Park, PA 16802, USA. E-mail address: fichthorn@psu.edu (K.A. Fichthorn). Computational Materials Science xxx (2015) xxx–xxx Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci Please cite this article in press as: K.A. Fichthorn, S. Mubin, Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2014.12.008