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Special Issue "Understanding Molecular Dynamics via Stochastic Processes"

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (30 September 2017)

Special Issue Editors

Guest Editor
Prof. Dr. Giovanni Ciccotti

University of Roma “La Sapienza”, Piazzale A. Moro 2, 00185 Roma, Italy
Website | E-Mail
Interests: methods in molecular dynamics simulation of systems of statistical mechanical interest; equilibrium and non equilibrium molecular dynamics; rare events; computer simulation of complex molecular systems
Guest Editor
Prof. Dr. Mauro Ferrario

University of Modena and Reggio Emilia, Via G. Campi 213/A, 41100 Modena Italy
Website | E-Mail
Interests: molecular dynamics simulation of condensed matter systems; solvation in h-bonded liquids; friction at the nanoscale
Guest Editor
Prof. Dr. Christof Schuette

Freie Universität Berlin, ZIB, Takustraße 7, 14195 Berlin, Germany
Website | E-Mail
Interests: rare events statistics in molecular dynamics; coarse graining in molecular dynamics; conformation dynamics

Special Issue Information

Dear Colleagues,

Contrary to what the name seems to suggest, Molecular Dynamics (MD) is not only about generating a dynamical trajectory of a system of molecules, but also, and foremost, about understanding the statistical properties of this trajectory. As a result, to make sense of this solution is to extract its main statistical features, which must be done within the probabilistic framework of Statistical Mechanics. Pushing this viewpoint one step further, an MD trajectory can be thought of not so much as a small piece of the actual trajectory of a realistic system of molecules, but rather as a sampling device for the statistical mechanics properties of this system.

When speaking about equilibrium quantities like free energy, etc., the probabilistic interpretation of an MD trajectory as a sampling device has long be recognized, and so has its computational advantages. One is that the numerical accuracy of an MD trajectory should be analyzed in terms of the statistics it produces rather than by pathwise comparison with an actual solution, which simplifies matters. A second advantage is that the sampling performances can be improved by fiddling with MD dynamics as long as this does not affect the statistical properties one is aiming at. This is why, for example, Monte Carlo is used as a perfectly valid alternative to MD to compute equilibrium quantities: from the present perspective, the two approaches are not so different in spirit.

Contrary to a widespread opinion in the community of practitioners, there is no reason to restrict this type of approach to the computation of the standard equilibrium quantities of Statistical Mechanics: the same philosophy in which MD is fully integrated within a probabilistic perspective can be applied to understand dynamical properties such as correlation functions, transport coefficients, pathways and rates of rare events, etc. In this context, also, identifying the right statistical quantities first, then using MD or whichever modification thereof to sample them can prove valuable. However, developing the right probabilistic framework for the study of dynamical phenomena, such as rare events, is a formidable challenge. This part of Statistical Mechanics is still much less developed. It also requires more sophisticated tools from Stochastic Processes Theory, for dynamical properties are multiple- rather than single-time statistical properties of the systems, i.e., one must deal with a stochastic process rather than with random variables. As a result the probability distributions relevant to dynamical phenomena are more complicated objects, often not even readily available. However, establishing what these distributions are and how to use MD as a tool to sample them efficiently is the right way to go. The steady growth in computing power as well as the development of various computational tricks may permit to generate ever longer trajectories in ever bigger systems; however bare trajectories have very little use without the right probabilistic framework to under- stand their meaning and looking at them in their gory details may even be more confusing than helpful.

It is in this spirit that, in this Special Issue, we would like to collect papers focusing, with a pedagogical aim, on the importance of stochastic process modeling, to understand and put on solid basis classical statistical mechanics and MD simulations, showing that not brute force MD but intelligent use of probability will give us the possibility to solve the most challenging problems when modeling physical, chemical and biological processes on all space and time scales. To comply with this goal we have invited a group of highly qualified researchers working along these lines to produce some useful glimpses of the ongoing process.

Planned Papers*

  • Ron Elber, Juan M. Bello-Rivas, Piao Ma, Alfredo Cardenas and Arman Fathizadeh, Calculating isocommittor surfaces as optimal reaction coordinates with milestoning
  • Anastasia S. Georgiou, Juan M. Bello-Rivas, C. William Gear, Hau-Tieng Wu, Eliodoro Chiavazzo and Ioannis G. Kevrekidis, An exploration algorithm for stochastic simulators driven by energy gradients
  • Wei Zhang and Christof Schütte, Reliable approximation of long relaxation timescales in molecular dynamics.
  • Josh Fass, David A. Sivak, Gavin E. Crooks and John D. Chodera, Which integrator is best for biomolecular simulation? Comparing the efficiencies of Langevin integrators
  • Carsten Hartmann, Lorenz Richter, Christof Schütte and Wei Zhang, Variational characterisation of free energy: theory and algorithms
  • Mattias Sachs, Vincent Danos and Ben Leimkuhler, Variable-coefficient stochastic particle models and their stationary states.
  • Robert L Jack, Marcus Kaiser and Johannes Zimmer, Symmetries and geometrical properties of dynamical fluctuations in molecular dynamics"
  • Robert Skeel and Youhan Fang, Comparing Markov Chain Samplers for Molecular Simulation
  • Chloe Gao and David T. Limmer, Transport coefficients from path ensemble free energies
  • Carsten Hartmann, Susana Gomes and Grigorios A. Pavliotis, On the linear response of a system of infinitely many randomly perturbed oscillators

[*The above list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review]

Prof. Dr. Giovanni Ciccotti
Prof. Dr. Mauro Ferrario
Prof. Dr. Christof Schuette
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1500 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (8 papers)

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Research

Open AccessArticle Variational Characterization of Free Energy: Theory and Algorithms
Entropy 2017, 19(11), 626; doi:10.3390/e19110626
Received: 25 September 2017 / Revised: 7 November 2017 / Accepted: 15 November 2017 / Published: 20 November 2017
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Abstract
The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure
[...] Read more.
The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessFeature PaperArticle Transport Coefficients from Large Deviation Functions
Entropy 2017, 19(11), 571; doi:10.3390/e19110571
Received: 16 September 2017 / Revised: 18 October 2017 / Accepted: 19 October 2017 / Published: 25 October 2017
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Abstract
We describe a method for computing transport coefficients from the direct evaluation of large deviation functions. This method is general, relying on only equilibrium fluctuations, and is statistically efficient, employing trajectory based importance sampling. Equilibrium fluctuations of molecular currents are characterized by their
[...] Read more.
We describe a method for computing transport coefficients from the direct evaluation of large deviation functions. This method is general, relying on only equilibrium fluctuations, and is statistically efficient, employing trajectory based importance sampling. Equilibrium fluctuations of molecular currents are characterized by their large deviation functions, which are scaled cumulant generating functions analogous to the free energies. A diffusion Monte Carlo algorithm is used to evaluate the large deviation functions, from which arbitrary transport coefficients are derivable. We find significant statistical improvement over traditional Green–Kubo based calculations. The systematic and statistical errors of this method are analyzed in the context of specific transport coefficient calculations, including the shear viscosity, interfacial friction coefficient, and thermal conductivity. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessFeature PaperArticle Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics
Entropy 2017, 19(10), 562; doi:10.3390/e19100562
Received: 9 September 2017 / Revised: 18 October 2017 / Accepted: 19 October 2017 / Published: 22 October 2017
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Abstract
We describe some general results that constrain the dynamical fluctuations that can occur in non-equilibrium steady states, with a focus on molecular dynamics. That is, we consider Hamiltonian systems, coupled to external heat baths, and driven out of equilibrium by non-conservative forces. We
[...] Read more.
We describe some general results that constrain the dynamical fluctuations that can occur in non-equilibrium steady states, with a focus on molecular dynamics. That is, we consider Hamiltonian systems, coupled to external heat baths, and driven out of equilibrium by non-conservative forces. We focus on the probabilities of rare events (large deviations). First, we discuss a PT (parity-time) symmetry that appears in ensembles of trajectories where a current is constrained to have a large (non-typical) value. We analyse the heat flow in such ensembles, and compare it with non-equilibrium steady states. Second, we consider pathwise large deviations that are defined by considering many copies of a system. We show how the probability currents in such systems can be decomposed into orthogonal contributions that are related to convergence to equilibrium and to dissipation. We discuss the implications of these results for modelling non-equilibrium steady states. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessFeature PaperArticle Comparing Markov Chain Samplers for Molecular Simulation
Entropy 2017, 19(10), 561; doi:10.3390/e19100561
Received: 6 September 2017 / Revised: 17 October 2017 / Accepted: 19 October 2017 / Published: 21 October 2017
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Abstract
Markov chain Monte Carlo sampling propagators, including numerical integrators for stochastic dynamics, are central to the calculation of thermodynamic quantities and determination of structure for molecular systems. Efficiency is paramount, and to a great extent, this is determined by the integrated autocorrelation time
[...] Read more.
Markov chain Monte Carlo sampling propagators, including numerical integrators for stochastic dynamics, are central to the calculation of thermodynamic quantities and determination of structure for molecular systems. Efficiency is paramount, and to a great extent, this is determined by the integrated autocorrelation time (IAcT). This quantity varies depending on the observable that is being estimated. It is suggested that it is the maximum of the IAcT over all observables that is the relevant metric. Reviewed here is a method for estimating this quantity. For reversible propagators (which are those that satisfy detailed balance), the maximum IAcT is determined by the spectral gap in the forward transfer operator, but for irreversible propagators, the maximum IAcT can be far less than or greater than what might be inferred from the spectral gap. This is consistent with recent theoretical results (not to mention past practical experience) suggesting that irreversible propagators generally perform better if not much better than reversible ones. Typical irreversible propagators have a parameter controlling the mix of ballistic and diffusive movement. To gain insight into the effect of the damping parameter for Langevin dynamics, its optimal value is obtained here for a multidimensional quadratic potential energy function. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessFeature PaperArticle Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics
Entropy 2017, 19(7), 367; doi:10.3390/e19070367
Received: 25 April 2017 / Revised: 13 July 2017 / Accepted: 13 July 2017 / Published: 18 July 2017
Cited by 1 | PDF Full-text (515 KB) | HTML Full-text | XML Full-text
Abstract
Many interesting rare events in molecular systems, like ligand association, protein folding or conformational changes, occur on timescales that often are not accessible by direct numerical simulation. Therefore, rare event approximation approaches like interface sampling, Markov state model building, or advanced reaction coordinate-based
[...] Read more.
Many interesting rare events in molecular systems, like ligand association, protein folding or conformational changes, occur on timescales that often are not accessible by direct numerical simulation. Therefore, rare event approximation approaches like interface sampling, Markov state model building, or advanced reaction coordinate-based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches. How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so we also allow for understanding deep connections between the different approaches. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessFeature PaperArticle An Exploration Algorithm for Stochastic Simulators Driven by Energy Gradients
Entropy 2017, 19(7), 294; doi:10.3390/e19070294
Received: 28 February 2017 / Revised: 24 April 2017 / Accepted: 8 May 2017 / Published: 22 June 2017
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Abstract
In recent work, we have illustrated the construction of an exploration geometry on free energy surfaces: the adaptive computer-assisted discovery of an approximate low-dimensional manifold on which the effective dynamics of the system evolves. Constructing such an exploration geometry involves geometry-biased sampling (through
[...] Read more.
In recent work, we have illustrated the construction of an exploration geometry on free energy surfaces: the adaptive computer-assisted discovery of an approximate low-dimensional manifold on which the effective dynamics of the system evolves. Constructing such an exploration geometry involves geometry-biased sampling (through both appropriately-initialized unbiased molecular dynamics and through restraining potentials) and, machine learning techniques to organize the intrinsic geometry of the data resulting from the sampling (in particular, diffusion maps, possibly enhanced through the appropriate Mahalanobis-type metric). In this contribution, we detail a method for exploring the conformational space of a stochastic gradient system whose effective free energy surface depends on a smaller number of degrees of freedom than the dimension of the phase space. Our approach comprises two steps. First, we study the local geometry of the free energy landscape using diffusion maps on samples computed through stochastic dynamics. This allows us to automatically identify the relevant coarse variables. Next, we use the information garnered in the previous step to construct a new set of initial conditions for subsequent trajectories. These initial conditions are computed so as to explore the accessible conformational space more efficiently than by continuing the previous, unbiased simulations. We showcase this method on a representative test system. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessArticle Ion Hopping and Constrained Li Diffusion Pathways in the Superionic State of Antifluorite Li2O
Entropy 2017, 19(5), 227; doi:10.3390/e19050227
Received: 21 March 2017 / Revised: 25 April 2017 / Accepted: 15 May 2017 / Published: 18 May 2017
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Abstract
Li2O belongs to the family of antifluorites that show superionic behavior at high temperatures. While some of the superionic characteristics of Li2O are well-known, the mechanistic details of ionic conduction processes are somewhat nebulous. In this work, we first
[...] Read more.
Li2O belongs to the family of antifluorites that show superionic behavior at high temperatures. While some of the superionic characteristics of Li2O are well-known, the mechanistic details of ionic conduction processes are somewhat nebulous. In this work, we first establish an onset of superionic conduction that is emblematic of a gradual disordering process among the Li ions at a characteristic temperature Tα (~1000 K) using reported neutron diffraction data and atomistic simulations. In the superionic state, the Li ions are observed to portray dynamic disorder by hopping between the tetrahedral lattice sites. We then show that string-like ionic diffusion pathways are established among the Li ions in the superionic state. The diffusivity of these dynamical string-like structures, which have a finite lifetime, shows a remarkable correlation to the bulk diffusivity of the system. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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Open AccessFeature PaperArticle Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning
Entropy 2017, 19(5), 219; doi:10.3390/e19050219
Received: 17 February 2017 / Revised: 24 April 2017 / Accepted: 8 May 2017 / Published: 11 May 2017
Cited by 1 | PDF Full-text (2983 KB) | HTML Full-text | XML Full-text
Abstract
Reaction coordinates are vital tools for qualitative and quantitative analysis of molecular processes. They provide a simple picture of reaction progress and essential input for calculations of free energies and rates. Iso-committor surfaces are considered the optimal reaction coordinate. We present an algorithm
[...] Read more.
Reaction coordinates are vital tools for qualitative and quantitative analysis of molecular processes. They provide a simple picture of reaction progress and essential input for calculations of free energies and rates. Iso-committor surfaces are considered the optimal reaction coordinate. We present an algorithm to compute efficiently a sequence of isocommittor surfaces. These surfaces are considered an optimal reaction coordinate. The algorithm analyzes Milestoning results to determine the committor function. It requires only the transition probabilities between the milestones, and not transition times. We discuss the following numerical examples: (i) a transition in the Mueller potential; (ii) a conformational change of a solvated peptide; and (iii) cholesterol aggregation in membranes. Full article
(This article belongs to the Special Issue Understanding Molecular Dynamics via Stochastic Processes)
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