Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations

Phase transitions and effects of external noise on many body systems are one of the main topics in physics. In mean field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of L\'evy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with L\'evy noise have attracted much attention and behaviors of systems with double-well potential subjected to L\'evy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an {\it analytically solvable model} to study the occurrence of phase transitions driven by L\'evy stable noise.Comment: submitted to EP

Studies on the embryology of Squilla oratoria de Haan

Resume 1) The egg of Squilla oratoria DE HAAN is centrolecithal and undergoes partial cleavages resulting in rudimentary primary yolk pyramids. 2) The germinal disk is first represented by a pair of optic lobes and a ventral plate, which are afterward connected by paired, lateral ectoderm thickenings to form a V-shape. The V is then transformed into an O by the appearance of a transverse band between the optic lobes of both sides. 3) A small blastopore is formed. Of the mesendoderm cells derived from the blastopore by cell immigrations, those attached to the lower surface of the lateral ectoderm thickenings are differentiated into a U-shaped, naupliar mesoderm band. This inesoderm band joins the preante:mulary mesoderm derived from the optic lobe, and grows into a complete ring conforming to the shape of the germinal disk. 4) The extra-blastoporic immigrants consist of a preantennulary mesoderm, mesodermal yolk cells and a part of the naupliar mesoderm. The greater part of the preantennulary mesoderm cells disintegrate sooner or later, without forming any distinct structure. The mesodermal yolk cells also degenerate after taking part in the dissolution of the deutoplasm. A discussion as regards the mutual relationship between these elements, with the conclusion that the formation of the preantennulary mesoderm represents the initial step of the extra-blastoporic cell sinking from the whole egg surface, is included. 5) The endodermal elements consist of a compact cell mass differentiated from the posterior part of the mesendoderm layer and the endodermal yolk cells immigrated from the blastopore. The yolk cells, after migrating through the most peripheral part of the yolk, scatter all over its surface. The endoderm plate is nothing but a mass of yolk cells. which remain without scattering. 6) Eight mesoteloblasts derived from the blastoporic lip are attached to the inner surface of the thoracico-abdominal process, making four groups. The ectoteloblasts are differentiated from the ordinary blastoderm cells in a later stage than the mesoteloblasts. In the final condition they consist of 21 cells forming a complete ring around the thoracico-abdominal process. 7) Both the ectoderm and the mesoderm are derived from the teloblasts in all of the post-naupliar segments.· The dorsal ectoderm, however, is non-teloblastic in only a few anterior segments. Differentiation of segments proceeds from the front toward the back. 8) The telson mesoderm is formed by the cells sunk from the telson ectoderm which is derived from the peri-blastoporic ectoderm . 9) The anus is the remnant of the blastopore. In accordance with the change of the caudal furca, the anus is displaced from the dorsal side of the telson to the ventral border between this and the last abdominal segment. 10) There is a distinct nauplius stage. Of the meta-naupliar segments, those from the m:txillula to the second maxilliped are laid on the germinal disk, the following segments together forming a thoracico-abdominal process. Two maxiliiped segments, however, are later separated from the cephalon with the development of the carapace fold, and join the trunk segments. Externally, six abdominal segments are formed. 11) The ganglionic cells are proliferated from the neuroblasts occupying the most superficial part of the centra1 nervous system. The giant ganglionic cells arise from the ordinary ganglionic cells and not directly from neuroblasts. The development of the cerebrum is described. The tritocerebra of both sides are conne~ted by a transverse nerve-fibre bundle behind the stomodaeum. The ganglia of the segments from the mandible to the second maxilliped first exhibit a typical ladder-like shape. Of these ganglia, the anterior three constitute a sub-resophageal ganglion by more or less complete fusion, while the posterior two are transferred from the cephalon to the thoracico-abdomimil process with the constriction of the segments. The inter-ganglionic cell groups take part in the constriction of the consecutive segments. The seventh abdominal ganglion is clearly indicated by the presence of such a cell group as well as of a pair of nerve fibre masses. 12) The development of the compound eye is traced. The ganglion opticum is derived from the ectoderm of the optic lobe lateral to the protocerebrum; it is not an outgrowth of the cerebrum. 13) The ganglion visceralum is differentiated from the anterior wall of the stomodaeum. 14) A median dorsal organ is formed. In close connection with the activity of this organ, the embryo undergoes one ecdysis. 15) The mid-gut epithelium is formed by the gradual expansion of the anterior and posterior endoderm plates over the yolk sac. These plates, however, extend only on the ventral side of the yolk sac before hatching. The posterior plate is produced by the concentration of the scattered yolk cells toward the periphery of the. plate differentiated from the mesendoderm, while the anterior plate is formed by yolk cells alone. 16) The greater part of the intestine develops from the outgrowth of the posterior endoderm epithelium, the proctodaeum occupying only the rectum. 17) The posterior liver lobes are produced from the posterior endoderm plate as a pair of blind tubes and extend as far backward as the telson. The anterior liver lobes and the lateral mid-gut cceca are rather incompletely developed, being ~eparated by shallow superficial grooves of the yolk sac. These two pairs of diverticula are only partially covered by the endoderm epithelium, and develop into more or less distinct coeca during larval life. They later seem to be completely absorbed again by the mid-gut. 18) The product of each division of the mesoteloblast is equivalent to one mesodermal-segment. The mesoderm of the seventh abdominal segment is derived from the posteriorly situated daughter cell produced by the last division of the teloblast. In' accordance with the grouping of teloblasts, the trunk mesoderm is separated into two ventral and two dorsal bands. Each band is further separated into segmentally arranged blocks, the somites. The ccelom develops in no stage and in no segment. 19) The dorsal mesoderm gives rise to the extensor and the oblique muscles of the trunk, the anterior and posterior limb muscles, as well as to the mesodermal inclusion of the limb. The -ventral mesoderm grows into the flexor. The connective tissue investing the intestine -and the liver lobes are principally constructed from the dorsal mesoderm. The germ cell does not appear until hatching. A brief account is also given of the fate of the naupliar mesoderm. 20) The heart wall and the pericardial floor are morphologically one unit. They arise from the dorsal mesoderm as a pair of membranes stretching between it and the intestine. The dilated and elongated parts of the heart are formed by the subsequent union of these paired rudiments. 21) The anterior dorsal vessel has a two-fold origin; it is formed by the fusion of an anterior rudiment extending backward from the rostrum and a posterior one developing as a tubular outgrowth of the heart. The former is derived from the rearrangement of mesenchymatous cells which migrated from the anterior end of the naupliar mesoderm. 22) By the time of hatching, two pairs of lateral vessels are formed as hollow linear thickenings of the pericardial floor in front and behind the dilated part of the heart. 23) The antennal gland remains rudimentary without acquiring any intercellular lumen. The maxillar gland is not laid until hatching. The labral and anal glands are derived from the peristomodaeal and the telson mesoderm respectively. 23) Comparisons are made bewteen Squilla and other orders of Malacostraca as regards the salient points of the embryonic development. These have led to the conclusion that the Stomatopoda are most closely related in their embryonic development to Nebaliacea, and further that Stomatopoda represent a rather primitive group separated from the main stem of Malacostraca very early, only next in order to Nebaliacea

Low-dimensional chaos induced by frustration in a non-monotonic system

We report a novel mechanism for the occurrence of chaos at the macroscopic level induced by the frustration of interaction, namely frustration-induced chaos, in a non-monotonic sequential associative memory model. We succeed in deriving exact macroscopic dynamical equations from the microscopic dynamics in the case of the thermodynamic limit and prove that two order parameters dominate this large-degree-of-freedom system. Two-parameter bifurcation diagrams are obtained from the order-parameter equations. Then we analytically show that the chaos is low-dimensional at the macroscopic level when the system has some degree of frustration, but that the chaos definitely does not occur without the frustration.Comment: 2 figure

Oscillator neural network model with distributed native frequencies

We study associative memory of an oscillator neural network with distributed native frequencies. The model is based on the use of the Hebb learning rule with random patterns ($\xi_i^{\mu}=\pm 1$), and the distribution function of native frequencies is assumed to be symmetric with respect to its average. Although the system with an extensive number of stored patterns is not allowed to get entirely synchronized, long time behaviors of the macroscopic order parameters describing partial synchronization phenomena can be obtained by discarding the contribution from the desynchronized part of the system. The oscillator network is shown to work as associative memory accompanied by synchronized oscillations. A phase diagram representing properties of memory retrieval is presented in terms of the parameters characterizing the native frequency distribution. Our analytical calculations based on the self-consistent signal-to-noise analysis are shown to be in excellent agreement with numerical simulations, confirming the validity of our theoretical treatment.Comment: 9 pages, revtex, 6 postscript figures, to be published in J. Phys.

Pattern-recalling processes in quantum Hopfield networks far from saturation

As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is itself an uncertain, noisy unit with a finite unnegligible error in the input-output relation. To model the situation artificially, a kind of 'heat bath' that surrounds neurons is introduced. The heat bath, which is a source of noise, is specified by the 'temperature'. Several studies concerning the pattern-recalling processes of the Hopfield model governed by the Glauber-dynamics at finite temperature were already reported. However, we might extend the 'thermal noise' to the quantum-mechanical variant. In this paper, in terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo method (the quantum MCMC), we analytically derive macroscopically deterministic equations of order parameters such as 'overlap' in a quantum-mechanical variant of the Hopfield neural networks (let us call "quantum Hopfield model" or "quantum Hopfield networks"). For the case in which non-extensive number $p$ of patterns are embedded via asymmetric Hebbian connections, namely, $p/N \to 0$ for the number of neuron $N \to \infty$ ('far from saturation'), we evaluate the recalling processes for one of the built-in patterns under the influence of quantum-mechanical noise.Comment: 10 pages, 3 figures, using jpconf.cls, Proc. of Statphys-Kolkata VI