Kinematic Basis of Emergent Energetics of Complex Dynamics

Stochastic kinematic description of a complex dynamics is shown to dictate an energetic and thermodynamic structure. An energy function $\varphi(x)$ emerges as the limit of the generalized, nonequilibrium free energy of a Markovian dynamics with vanishing fluctuations. In terms of the $\nabla\varphi$ and its orthogonal field $\gamma(x)\perp\nabla\varphi$, a general vector field $b(x)$ can be decomposed into $-D(x)\nabla\varphi+\gamma$, where $\nabla\cdot\big(\omega(x)\gamma(x)\big)=$ $-\nabla\omega D(x)\nabla\varphi$. The matrix $D(x)$ and scalar $\omega(x)$, two additional characteristics to the $b(x)$ alone, represent the local geometry and density of states intrinsic to the statistical motion in the state space at $x$. $\varphi(x)$ and $\omega(x)$ are interpreted as the emergent energy and degeneracy of the motion, with an energy balance equation $d\varphi(x(t))/dt=\gamma D^{-1}\gamma-bD^{-1}b$, reflecting the geometrical $\|D\nabla\varphi\|^2+\|\gamma\|^2=\|b\|^2$. The partition function employed in statistical mechanics and J. W. Gibbs' method of ensemble change naturally arise; a fluctuation-dissipation theorem is established via the two leading-order asymptotics of entropy production as $\epsilon\to 0$. The present theory provides a mathematical basis for P. W. Anderson's emergent behavior in the hierarchical structure of complexity science.Comment: 7 page

Current rectification by asymmetric molecules: An ab initio study

We study current rectification effect in an asymmetric molecule HOOC-C$_6$H$_4$-(CH$_2$)$_n$ sandwiched between two Aluminum electrodes using an {\sl ab initio} nonequilibrium Green function method. The conductance of the system decreases exponentially with the increasing number $n$ of CH$_2$. The phenomenon of current rectification is observed such that a very small current appears at negative bias and a sharp negative differential resistance at a critical positive bias when $n\ge 2$. The rectification effect arises from the asymmetric structure of the molecule and the molecule-electrode couplings. A significant rectification ratio of $\sim$38 can be achieved when $n=5$.Comment: to appear in J. Chem. Phy