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

Criticality in Translation-Invariant Parafermion Chains

In this work we numerically study critical phases in translation-invariant $\mathbb{Z}_N$ parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $\mathbb{Z}_N$ spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translation invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual $\mathbb{Z}_N$ clock models. For $3\leq N\leq 6$ we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a $N=3$ chain with both nearest and next-nearest neighbor hopping and six critical phases with central charges being $4/5$, 1 or 2 are found. We find continuous phase transitions between $c=1$ and $c=2$ phases, while the phase transition between $c=4/5$ and $c=1$ is conjectured to be of Kosterlitz-Thouless type.Comment: published versio

Simultaneously encoding movement and sEMG-based stiffness for robotic skill learning

Transferring human stiffness regulation strategies to robots enables them to effectively and efficiently acquire adaptive impedance control policies to deal with uncertainties during the accomplishment of physical contact tasks in an unstructured environment. In this work, we develop such a physical human-robot interaction (pHRI) system which allows robots to learn variable impedance skills from human demonstrations. Specifically, the biological signals, i.e., surface electromyography (sEMG) are utilized for the extraction of human arm stiffness features during the task demonstration. The estimated human arm stiffness is then mapped into a robot impedance controller. The dynamics of both movement and stiffness are simultaneously modeled by using a model combining the hidden semi-Markov model (HSMM) and the Gaussian mixture regression (GMR). More importantly, the correlation between the movement information and the stiffness information is encoded in a systematic manner. This approach enables capturing uncertainties over time and space and allows the robot to satisfy both position and stiffness requirements in a task with modulation of the impedance controller. The experimental study validated the proposed approach