Markov processes follow from the principle of Maximum Caliber

Markov models are widely used to describe processes of stochastic dynamics. Here, we show that Markov models are a natural consequence of the dynamical principle of Maximum Caliber. First, we show that when there are different possible dynamical trajectories in a time-homogeneous process, then the only type of process that maximizes the path entropy, for any given singlet statistics, is a sequence of identical, independently distributed (i.i.d.) random variables, which is the simplest Markov process. If the data is in the form of sequentially pairwise statistics, then maximizing the caliber dictates that the process is Markovian with a uniform initial distribution. Furthermore, if an initial non-uniform dynamical distribution is known, or multiple trajectories are conditioned on an initial state, then the Markov process is still the only one that maximizes the caliber. Second, given a model, MaxCal can be used to compute the parameters of that model. We show that this procedure is equivalent to the maximum-likelihood method of inference in the theory of statistics.Comment: 4 page

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