Variational Characterization of Free Energy: Theory and Algorithms

Abstract

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 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.

Data assimilation: The Schrödinger perspective

“…We also recall the definition of the evidence (10). The quantity F = − log β is called the free energy in statistical physics (Hartmann et al 2017). An appropriate transition kernel q 1 ( z 1 |z 1 ), satisfying (14), is required in order to complete the transition from π 0 to π 1 following scenario (A) from definition 2.4.…”

“…It is often assumed in optimal control or rare event simulations arising from statistical mechanics that in (2.1) is a point measure, that is, the starting point of the simulation is known exactly. See, for example, Hartmann, Richter, Schütte and Zhang (2017). This corresponds to (2.3) with .…”

“…See also Hartmann et al. (2017) for an in-depth discussion of variational formulations and their numerical implementation in the context of rare event simulations for which it is generally assumed that in (2.3), that is, the ensemble size is when viewed within the context of this paper.…”

Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices

“…LD efficiently explore around a mode of a target distribution using the gradient information without being trapped by local minima thanks to added Gaussian noise. Many previous studies theoretically and numerically proved LD's superior performance [2][3][4][5]. Since non-reversible dynamics generally improves mixing performance [6,7], research on introducing non-reversible dynamics to LD for better sampling performance is attracting attention [8].…”

Molecular Dynamics vs. Stochastic Processes: Are We Heading Anywhere?

“…Given the growing importance of advanced statistical techniques and stochastic modeling to understand and develop a more solid basis for computational statistical mechanics and, in particular, MD simulation, we need a novel effort to overcome the traditional way of approaching this problem. The present Special Issue is a first attempt to collect some of these new theory-related research efforts in the framework of MD, specially addressing algorithms [ 1 , 2 , 3 , 4 ], theoretical methods [ 5 , 6 , 7 ] and rigorous mathematical formulations [ 8 , 9 ]. The issue also presents some further applications [ 10 , 11 ] of molecular dynamics which can be of use while widening the perspective.…”