Abstract
A quasisymmetry is a special symmetry that enhances the ability of a magnetic field to trap charged particles. Quasisymmetric magnetic fields may allow the realization of next generation ...fusion reactors (stellarators) with superior performance when compared with tokamak designs. Nevertheless, the existence of such magnetic configurations lacks mathematical proof due to the complexity of the governing equations. Here, we prove the existence of weakly quasisymmetric magnetic fields by constructing explicit examples. This result is achieved by a tailored parametrization of both magnetic field and hosting toroidal domain, which are optimized to fulfill quasisymmetry. The obtained solutions hold in a toroidal volume, are smooth, possess nested flux surfaces, are not invariant under continuous Euclidean isometries, have a non-vanishing current, exhibit a vanishing rotational transform, and fit within the framework of anisotropic magnetohydrodynamics. Due to the vanishing rotational transform, these solutions are however not suitable for particle confinement.
Abstract
The Boltzmann distribution of an ideal gas is determined by the Hamiltonian function generating single particle dynamics. Systems with higher complexity often exhibit topological ...constraints, which are independent of the Hamiltonian and may affect the shape of the distribution function as well. Here, we study a further source of heterogeneity, the curvature of spacetime arising from the general theory of relativity. The present construction relies on three assumptions: first, the statistical ensemble is made of particles obeying geodesic equations, which define the phase space of the system. Next, the metric coefficients are time-symmetric, implying that, if thermodynamic equilibrium is achieved, all physical observables are independent of coordinate time. Finally, ergodicity is enforced with respect to proper time, so that ambiguity in the choice of a time variable for the statistical ensemble is removed. Under these hypothesis, we derive the distribution function of thermodynamic equilibrium, and verify that it reduces to the Boltzmann distribution in the non-relativistic limit. We further show that spacetime curvature affects physical observables, even far from the source of the metric. Two examples are analyzed: an ideal gas in Schwarzschild spacetime and a charged gas in Kerr–Newman spacetime. In the Schwarzschild case, conservation of macroscopic constraints, such as angular momentum, combined with relativistic distortion of the distribution function can produce configurations with decreasing density and growing azimuthal rotation velocity far from the event horizon of the central mass. In the Kerr–Newman case, it is found that kinetic energy associated with azimuthal rotations is an increasing function of the radial coordinate, and it eventually approaches a constant value corresponding to non-relativistic equipartition, even though spatial particle density decreases.
Chloroplasts of plants and algae are currently believed to originate from a cyanobacterial endosymbiont, mainly based on the shared proteins involved in the oxygenic photosynthesis and gene ...expression system. The phylogenetic relationship between the chloroplast and cyanobacterial genomes was important evidence for the notion that chloroplasts originated from cyanobacterial endosymbiosis. However, studies in the post-genomic era revealed that various substances (glycolipids, peptidoglycan, etc.) shared by cyanobacteria and chloroplasts are synthesized by different pathways or phylogenetically unrelated enzymes. Membranes and genomes are essential components of a cell (or an organelle), but the origins of these turned out to be different. Besides, phylogenetic trees of chloroplast-encoded genes suggest an alternative possibility that chloroplast genes could be acquired from at least three different lineages of cyanobacteria. We have to seriously examine that the chloroplast genome might be chimeric due to various independent gene flows from cyanobacteria. Chloroplast formation could be more complex than a single event of cyanobacterial endosymbiosis. I present the “host-directed chloroplast formation” hypothesis, in which the eukaryotic host cell that had acquired glycolipid synthesis genes as an adaptation to phosphate limitation facilitated chloroplast formation by providing glycolipid-based membranes (pre-adaptation). The origins of the membranes and the genome could be different, and the origin of the genome could be complex.
The paradigm “cyanobacterial origin of chloroplasts” is currently viewed as an established fact. However, we may have to re-consider the origin of chloroplast membranes, because membranes are not ...replicated by their own. It is the genes for lipid biosynthetic enzymes that are inherited. In the current understandings, these enzymes became encoded by the nuclear genome as a result of endosymbiotic gene transfer from the endosymbiont. However, we previously showed that many enzymes involved in the synthesis of chloroplast peptidoglycan and glycolipids did not originate from cyanobacteria. Here I present results of comprehensive phylogenetic analysis of chloroplast enzymes involved in fatty acid and lipid biosynthesis, as well as additional chloroplast components related to photosynthesis and gene expression. Four types of phylogenetic relationship between chloroplast enzymes (encoded by the chloroplast and nuclear genomes) and cyanobacterial counterparts were found: type 1, chloroplast enzymes diverged from inside of cyanobacterial clade; type 2, chloroplast and cyanobacterial enzymes are sister groups; type 3, chloroplast enzymes originated from homologs of bacteria other than cyanobacteria; type 4, chloroplast enzymes diverged from eukaryotic homologs. Estimation of evolutionary distances suggested that the acquisition times of chloroplast enzymes were diverse, indicating that multiple gene transfers accounted for the chloroplast enzymes analyzed. Based on the results, I try to relax the tight logic of the endosymbiotic origin of chloroplasts involving a single endosymbiotic event by proposing alternative hypotheses. The hypothesis of host-directed chloroplast formation proposes that glycolipid synthesis ability had been acquired by the eukaryotic host before the acquisition of chloroplast ribosomes. Chloroplast membrane system could have been provided by the host, whereas cyanobacteria contributed to the genes for the genetic and photosynthesis systems, at various times, either before or after the formation of chloroplast membranes. The origin(s) of chloroplasts seems to be more complicated than the single event of primary endosymbiosis.
•The major postulate of Margulis tried to explain the evolution of various types of mitosis.•She considered endosymbiotic origin of plastids known evidence justifying exclusion of plastids from the ...classification of eukaryotes.•She pretended to make her hypothesis testable or falsifiable as a sign of good science, but her hypothesis was finally entirely falsified.•The meaning of endosymbiosis changed with time so that Margulis understood symbiosis in a different way as we do.•The success of Margulis was a result of integration of different disciplines, political situations, and rise of philosophy of science.
Fifty years ago, Lynn Margulis proposed a comprehensive hypothesis on the origin of eukaryotic cells with an emphasis on the origin of mitosis. This hypothesis postulated that the eukaryotic cell is a composite of different parts as a result of the symbiosis of various different bacteria. In this hypothesis, she integrated previously proposed ideas that mitochondria and chloroplasts were descendants of endosymbionts that originated from aerobic bacteria and blue-green algae (now cyanobacteria), respectively. However, the major part of her hypothesis, which she believed to be original, was the origin of mitosis. The core of her postulate involved a chromosome partition mechanism dependent on DNA-microtubule binding, which originated from a hypothetical centriole-DNA complex, with an ability to replicate. Surprisingly, her complete lack of real experimental works in the cytoskeleton, cell motility, or paleontology did not prevent this 29-year-old junior scientist from assembling archival knowledge and constructing a narrative on the evolution of all organisms. Whether the centriole-DNA complex originated from a spirochete or not was a minor anecdote in this initial postulate. Unfortunately, this hypothesis on the origin of mitosis, which she believed to be a holistic unity, testable by experiments, was entirely refuted. Despite falsification of her original narrative as a whole, her success as a founder of endosymbiotic theory on the origin of mitochondria and chloroplasts is undoubted. We will discuss the reasons for her success in terms of the historical situation in the latter half of the 20th century.
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Abstract
Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The ...properties of the Poisson bracket are encapsulated in the symplectic $2$-form, a closed second-order differential form. Due to closure, the symplectic $2$-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie–Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic $2$-form is replaced by a symplectic $3$-form. We show that, using the closure of the symplectic $3$-form, a result analogous to the classical Lie–Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic $3$-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third-order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic $3$-form.
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