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On the dynamics of reaction coordinates in classical, time-dependent, many-body processes Meyer, Hugues ; ; in Journal of Chemical Physics (2019), 150(17), 174118 Complex microscopic many-body processes are often interpreted in terms of so-called “reaction coordinates,” i.e., in terms of the evolution of a small set of coarse-grained observables. A rigorous method ... [more ▼] Complex microscopic many-body processes are often interpreted in terms of so-called “reaction coordinates,” i.e., in terms of the evolution of a small set of coarse-grained observables. A rigorous method to produce the equation of motion of such observables is to use projection operator techniques, which split the dynamics of the observables into a main contribution and a marginal one. The basis of any derivation in this framework is the classical Heisenberg equation for an observable. If the Hamiltonian of the underlying microscopic dynamics and the observable under study do not explicitly depend on time, this equation is obtained by a straightforward derivation. However, the problem is more complicated if one considers Hamiltonians which depend on time explicitly as, e.g., in systems under external driving, or if the observable of interest has an explicit dependence on time. We use an analogy to fluid dynamics to derive the classical Heisenberg picture and then apply a projection operator formalism to derive the nonstationary generalized Langevin equation for a coarse-grained variable. We show, in particular, that the results presented for time-independent Hamiltonians and observables in the study by Meyer, Voigtmann, and Schilling, J. Chem. Phys. 147, 214110 (2017) can be generalized to the time-dependent case. [less ▲] Detailed reference viewed: 109 (4 UL)On the non-stationary generalized Langevin Equation Meyer, Hugues ; ; in Journal of Chemical Physics (2017), 147(21), 214110 In molecular dynamics simulations and single molecule experiments, observables are usually measured along dynamic trajectories and then averaged over an ensemble (“bundle”) of trajectories. Under ... [more ▼] In molecular dynamics simulations and single molecule experiments, observables are usually measured along dynamic trajectories and then averaged over an ensemble (“bundle”) of trajectories. Under stationary conditions, the time-evolution of such averages is described by the generalized Langevin equation. By contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable has. We employ the formalism of time-dependent projection operator techniques to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. The equation is similar in structure to the generalized Langevin equation but exhibits a time-dependent memory kernel as well as a fluctuating force that implicitly depends on the initial conditions of the process. We also derive a relation between this memory kernel and the autocorrelation function of the fluctuating force that has a structure similar to a fluctuation-dissipation relation. In addition, we show how the choice of the projection operator allows us to relate the Taylor expansion of the memory kernel to data that are accessible in MD simulations and experiments, thus allowing us to construct the equation of motion. As a numerical example, the procedure is applied to Brownian motion initialized in non-equilibrium conditions and is shown to be consistent with direct measurements from simulations. [less ▲] Detailed reference viewed: 103 (2 UL)Sequencing chess ; Schilling, Tanja ; in Europhysics Letters (2016), 116(10009), We analyze the structure of the state space of chess by means of transition path sampling Monte Carlo simulations. Based on the typical number of moves required to transpose a given configuration of chess ... [more ▼] We analyze the structure of the state space of chess by means of transition path sampling Monte Carlo simulations. Based on the typical number of moves required to transpose a given configuration of chess pieces into another, we conclude that the state space consists of several pockets between which transitions are rare. Skilled players explore an even smaller subset of positions that populate some of these pockets only very sparsely. These results suggest that the usual measures to estimate both the size of the state space and the size of the tree of legal moves are not unique indicators of the complexity of the game, but that considerations regarding the connectedness of states are equally important. [less ▲] Detailed reference viewed: 117 (4 UL)Dissipation by a crystallization process Dorosz, Sven ; ; Schilling, Tanja in Europhysics Letters (2016), 113(10004), We discuss crystallization as a non-equilibrium process. In a system of hard spheres under compression at a constant rate, we quantify the amount of heat that is dissipated during the crystallization ... [more ▼] We discuss crystallization as a non-equilibrium process. In a system of hard spheres under compression at a constant rate, we quantify the amount of heat that is dissipated during the crystallization process. We interpret the dissipation as arising from the resistance of the system against phase transformation. An intrinsic compression rate is identified that separates a quasistatic regime from one of rapidly driven crystallization. In the latter regime the system crystallizes more easily, because new relaxation channels are opened, at the cost of forming a higher fraction of non-equilibrium crystal structures. We rationalize the change in the crystallization mechanism by analogy with shear thinning, in terms of a kinetic competition between near-equilibrium relaxation and external driving. [less ▲] Detailed reference viewed: 133 (3 UL) |
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