A First and Second Law for Nonequilibrium Thermodynamics: Maximum Entropy Derivation of the Fluctuation-Dissipation Theorem and Entropy Production Functionals

A theory for non-equilibrium systems is derived from a maximum entropy approach similar in spirit to the equilibrium theory given by Gibbs. Requiring Hamilton's principle of stationary action to be satisfied on average during a trajectory, we add constraints on the transition probability distribution which lead to a path probability of the Onsager-Machlup form. Additional constraints derived from energy and momentum conservation laws then introduce heat exchange and external driving forces into the system, with Lagrange multipliers related to the temperature and pressure of an external thermostatic system. The result is a fully time-dependent, non-local description of a nonequilibrium ensemble. Detailed accounting of the energy exchange and the change in information entropy of the central system then provides a description of the entropy production which is not dependent on the specification or existence of a steady-state or on any definition of thermostatic variables for the central system. These results are connected to the literature by showing a method for path re-weighting, creation of arbitrary fluctuation theorems, and by providing a simple derivation of Jarzynski relations referencing a steady-state. In addition, we identify path free energy and entropy (caliber) functionals which generate a first law of nonequilibrium thermodynamics by relating changes in the driving forces to changes in path averages. Analogous to the Gibbs relations, the variations in the path averages yield fluctuation-dissipation theorems. The thermodynamic entropy production can also be stated in terms of the caliber functional, resulting in a simple proof of our microscopic form for the Clausius statement. We find that the maximum entropy route provides a clear derivation of the path free energy functional, path-integral, Langevin, Brownian, and Fokker-Planck statements of nonequilibrium processes.Comment: 35 page

Structures and orientation-dependent interaction forces of titania nanowires using molecular dynamics simulations

Engineering nano wires to develop new products and processes is highly topical due to their ability to provide highly enhanced physical, chemical, mechanical, thermal and electrical properties. In this work, using molecular dynamics simulations, we report fundamental information, about the structural and thermodynamic properties of individual anatase titania (TiO2) nanowires with cross-sectional diameters between 2 and 6 nm, and aspect ratio (Length: Diameter) of 6:1 at temperatures ranging from 300 to 3000 K. Estimates of the melting-transition temperature of the nanowires are between 2000 and 2500 K. The melting transition temperature predicted from the radial distribution functions (RDFs) shows strong agreement with those predicted from the total energy profiles. Overall, the transition temperature is in reasonable agreement with melting points predicted from experiments and simulations reported in the literature for spherical nanoparticles of similar sizes. Hence, the melting-transition temperature of TiO2 nanowires modelled here can be considered as shape independent. Furthermore, for the first time based on MD simulations, interaction forces between two nanowires are reported at ambient temperature (300 K) for different orientations: parallel, perpendicular, and end-to-end. It is observed that end-to-end orientations manifested the strongest attraction forces, while the parallel and perpendicular orientations, displayed weaker attractions. The results reported here could form a foundation in future multiscale modelling studies of the structured titania nanowire assemblies, depending on the inter-wire interaction forces

Perspective: Maximum caliber is a general variational principle for dynamical systems

We review here Maximum Caliber (Max Cal), a general variational principle for inferring distributions of paths in dynamical processes and networks. Max Cal is to dynamical trajectories what the principle of maximum entropy is to equilibrium states or stationary populations. In Max Cal, you maximize a path entropy over all possible pathways, subject to dynamical constraints, in order to predict relative path weights. Many well-known relationships of non-equilibrium statistical physics—such as the Green-Kubo fluctuation-dissipation relations, Onsager’s reciprocal relations, and Prigogine’s minimum entropy production—are limited to near-equilibrium processes. Max Cal is more general. While it can readily derive these results under those limits, Max Cal is also applicable far from equilibrium. We give examples of Max Cal as a method of inference about trajectory distributions from limited data, finding reaction coordinates in bio-molecular simulations, and modeling the complex dynamics of non-thermal systems such as gene regulatory networks or the collective firing of neurons. We also survey its basis in principle and some limitations

Inferring Microscopic Kinetic Rates from Stationary State Distributions

[Image: see text] We present a principled approach for estimating the matrix of microscopic transition probabilities among states of a Markov process, given only its stationary state population distribution and a single average global kinetic observable. We adapt Maximum Caliber, a variational principle in which the path entropy is maximized over the distribution of all possible trajectories, subject to basic kinetic constraints and some average dynamical observables. We illustrate the method by computing the solvation dynamics of water molecules from molecular dynamics trajectories