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See detailSystematic derivation of Generalized Langevin Equations for coarse-graining and bridge-scaling procedures
Meyer, Hugues UL

Scientific Conference (2020, July)

In many branches of physics, one must often deal with processes involving a huge number of degrees of freedom. Instead of describing the dynamics of each individual of them, one rather wants to ... [more ▼]

In many branches of physics, one must often deal with processes involving a huge number of degrees of freedom. Instead of describing the dynamics of each individual of them, one rather wants to characterize the process of interest via a small set of observ- ables that capture its main features of the process. Even if the microscopic dynamics can be resolved using Newton’s equations of motion, it quickly becomes a computation- ally very expensive calculation to make. It is however much more convenient to come up with a self-consistent equation of motion for the ’global’ observable of interest itself in order to reduce the complexity of the problem. The development of the Mori-Zwanzig formalism in the 1960’s allowed to systematically derive such equations for arbitrary observables in stationary processes. This framework, derived from first principles by means of projection operator techniques, proves the structure of what is now known as the Generalized Langevin Equation, i.e. a stochastic equation of motion which a priori exhibits memory effects in the form on non-localities in time. We propose to extend the formalism and its corollaries to a broad class of out-of- equilibrium processes. We show that the structure of the Generalized Langevin Equa- tion is overall robust but must be adapted to account for the non-stationary dynamics [1,2]. The function that controls memory effects the stochastic term are related through a relation that can be associated to fluctuation-dissipation theorems. This formalism is very convenient to study two-time auto-correlation functions for which we can write a self-consistent differential equation as well. We finally show a new method to evaluate the memory function from numerical or experimental data [3]. [less ▲]

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See detailOn the dynamics of reaction coordinates in classical, time-dependent, many-body processes
Meyer, Hugues UL; Voigtmann, Thomas; Schilling, Tanja

in The 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 ▲]

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See detailMemory Effects in the Fermi–Pasta–Ulam Model
Amati, Graziano; Meyer, Hugues UL; Schilling, Tanja

in Journal of Statistical Physics (2019), 174(1), 219-257

We study the intermediate scattering function (ISF) of the strongly-nonlinear Fermi–Pasta–Ulam Model at thermal equilibrium, using both numerical and analytical methods. From the molecular dynamics ... [more ▼]

We study the intermediate scattering function (ISF) of the strongly-nonlinear Fermi–Pasta–Ulam Model at thermal equilibrium, using both numerical and analytical methods. From the molecular dynamics simulations we distinguish two limit regimes, as the system behaves as an ideal gas at high temperature and as a harmonic chain for low excitations. At intermediate temperatures the ISF relaxes to equilibrium in a nontrivial fashion. We then calculate analytically the Taylor coefficients of the ISF to arbitrarily high orders (the specific, simple shape of the two-body interaction allows us to derive an iterative scheme for these). The results of the recursion are in good agreement with the numerical ones. Via an estimate of the complete series expansion of the scattering function, we can reconstruct within a certain temperature range its coarse-grained dynamics. This is governed by a memory-dependent Generalized Langevin Equation (GLE), which can be derived via projection operator techniques. Moreover, by analyzing the first series coefficients of the ISF, we can extract a parameter associated to the strength of the memory effects in the dynamics. [less ▲]

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See detailOn the non-stationary generalized Langevin Equation
Meyer, Hugues UL; Voigtmann, Thomas; Schilling, Tanja

in The 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 ▲]

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See detailConnectivity percolation in suspensions of attractive square-well spherocylinders
Dixit, Mohit UL; Meyer, Hugues UL; Schilling, Tanja UL

in Physical Review. E : Statistical, Nonlinear, and Soft Matter Physics (2016), (93), 012116

We have studied the connectivity percolation transition in suspensions of attractive square-well spherocylinders by means ofMonte Carlo simulation and connectedness percolation theory. In the 1980s the ... [more ▼]

We have studied the connectivity percolation transition in suspensions of attractive square-well spherocylinders by means ofMonte Carlo simulation and connectedness percolation theory. In the 1980s the percolation threshold of slender fibers has been predicted to scale as the fibers’ inverse aspect ratio [Phys. Rev. B 30, 3933 (1984)]. The main finding of our study is that the attractive spherocylinder system reaches this inverse scaling regime at much lower aspect ratios than found in suspensions of hard spherocylinders. We explain this difference by showing that third virial corrections of the pair connectedness functions, which are responsible for the deviation from the scaling regime, are less important for attractive potentials than for hard particles. [less ▲]

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See detailPercolation in suspensions of polydisperse hard rods: Quasi universality and finite-size effects
Meyer, Hugues UL; Van der Schoot, Paul; Schilling, Tanja UL

in Journal of Chemical Physics (2015), 143(4), 044901

We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the ... [more ▼]

We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter, and the connectedness criterion, and we invoke bimodal, Gaussian, and Weibull distributions for these. The main finding from our simulations is that the percolation threshold shows quasi universal behaviour, i.e., to a good approximation, it depends only on certain cumulants of the full size and connectivity distribution. Our connectedness percolation theory hinges on a Lee-Parsons type of closure recently put forward that improves upon the often-used second virial approximation [T. Schilling, M. Miller, and P. van der Schoot, e-print arXiv:1505.07660 (2015)]. The theory predicts exact universality. Theory and simulation agree quantitatively for aspect ratios in excess of 20, if we include the connectivity range in our definition of the aspect ratio of the particles. We further discuss the mechanism of cluster growth that, remarkably, differs between systems that are polydisperse in length and in width, and exhibits non-universal aspects. [less ▲]

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