Understanding Molecular Dynamics via Stochastic Processes

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

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