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

Bidirectional outflows as evidence of magnetic reconnection leading to a solar microflare

Magnetic reconnection is a rapid energy release process that is believed to be responsible for flares on the Sun and stars. Nevertheless, such flare-related reconnection is mostly detected to occur in the corona, while there have been few studies concerning the reconnection in the chromosphere or photosphere. Here we present both spectroscopic and imaging observations of magnetic reconnection in the chromosphere leading to a microflare. During the flare peak time, chromospheric line profiles show significant blueshifted/redshifted components on the two sides of the flaring site, corresponding to upflows and downflows with velocities of $\pm$(70--80) km s$^{-1}$, comparable with the local Alfv\'{e}n speed as expected by the reconnection in the chromosphere. The three-dimensional nonlinear force-free field configuration further discloses twisted field lines (a flux rope) at a low altitude, cospatial with the dark threads in He I 10830 \r{A} images. The instability of the flux rope may initiate the flare-related reconnection. These observations provide clear evidence of magnetic reconnection in the chromosphere and show the similar mechanisms of a microflare to those of major flares.Comment: 16 pages, 5 figures, accepted for publication in ApJ