5,435 research outputs found
Lyapunov exponents from geodesic spread in configuration space
The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic
spread is here explicitly worked out at arbitrary dimension for the
configuration space manifold M_E = {q in R^N | V(q) < E} of a standard
Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As
the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC
equation can be used to study the degree of instability of the Hamiltonian
flow. It is found that the solutions of the JLC equation are closely resembling
the solutions of the standard tangent dynamics equation which is used to
compute Lyapunov exponents. Therefore the instability exponents obtained
through the JLC equation are in perfect quantitative agreement with usual
Lyapunov exponents. This work completes a previous investigation that was
limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure
Phase transitions as topology changes in configuration space: an exact result
The phase transition in the mean-field XY model is shown analytically to be
related to a topological change in its configuration space. Such a topology
change is completely described by means of Morse theory allowing a computation
of the Euler characteristic--of suitable submanifolds of configuration
space--which shows a sharp discontinuity at the phase transition point, also at
finite N. The present analytic result provides, with previous work, a new key
to a possible connection of topological changes in configuration space as the
origin of phase transitions in a variety of systems.Comment: REVTeX file, 5 pages, 1 PostScript figur
Topological aspects of geometrical signatures of phase transitions
Certain geometric properties of submanifolds of configuration space are
numerically investigated for classical lattice phi^4 models in one and two
dimensions. Peculiar behaviors of the computed geometric quantities are found
only in the two-dimensional case, when a phase transition is present. The
observed phenomenology strongly supports, though in an indirect way, a recently
proposed topological conjecture about a topology change of the configuration
space submanifolds as counterpart of a phase transition.Comment: REVTEX file, 4 pages, 5 figure
Hamiltonian dynamics and geometry of phase transitions in classical XY models
The Hamiltonian dynamics associated to classical, planar, Heisenberg XY
models is investigated for two- and three-dimensional lattices. Besides the
conventional signatures of phase transitions, here obtained through time
averages of thermodynamical observables in place of ensemble averages,
qualitatively new information is derived from the temperature dependence of
Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests
to consider other observables of geometric meaning tightly related with the
largest Lyapunov exponent. The numerical computation of these observables -
unusual in the study of phase transitions - sheds a new light on the
microscopic dynamical counterpart of thermodynamics also pointing to the
existence of some major change in the geometry of the mechanical manifolds at
the thermodynamical transition. Through the microcanonical definition of the
entropy, a relationship between thermodynamics and the extrinsic geometry of
the constant energy surfaces of phase space can be naturally
established. In this framework, an approximate formula is worked out,
determining a highly non-trivial relationship between temperature and topology
of the . Whence it can be understood that the appearance of a phase
transition must be tightly related to a suitable major topology change of the
. This contributes to the understanding of the origin of phase
transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22
PostScript figure
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
This paper deals with the problem of analytically computing the largest
Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is
succesfully reached within a theoretical framework that makes use of a
geometrization of newtonian dynamics in the language of Riemannian geometry. A
new point of view about the origin of chaos in these systems is obtained
independently of homoclinic intersections. Chaos is here related to curvature
fluctuations of the manifolds whose geodesics are natural motions and is
described by means of Jacobi equation for geodesic spread. Under general
conditions ane effective stability equation is derived; an analytic formula for
the growth-rate of its solutions is worked out and applied to the
Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent
agreement is found the theoretical prediction and the values of the Lyapunov
exponent obtained by numerical simulations for both models.Comment: RevTex, 40 pages, 8 PostScript figures, to be published in Phys. Rev.
E (scheduled for November 1996
Geometry of dynamics, Lyapunov exponents and phase transitions
The Hamiltonian dynamics of classical planar Heisenberg model is numerically
investigated in two and three dimensions. By considering the dynamics as a
geodesic flow on a suitable Riemannian manifold, it is possible to analytically
estimate the largest Lyapunov exponent in terms of some curvature fluctuations.
The agreement between numerical and analytical values for Lyapunov exponents is
very good in a wide range of temperatures. Moreover, in the three dimensional
case, in correspondence with the second order phase transition, the curvature
fluctuations exibit a singular behaviour which is reproduced in an abstract
geometric model suggesting that the phase transition might correspond to a
change in the topology of the manifold whose geodesics are the motions of the
system.Comment: REVTeX, 10 pages, 5 PostScript figures, published versio
Accurate and efficient constrained molecular dynamics of polymers using Newton's method and special purpose code
In molecular dynamics simulations we can often increase the time step by imposing constraints on bond lengths and bond angles. This allows us to extend the length of the time interval and therefore the range of physical phenomena that we can afford to simulate. We examine the existing algorithms and software for solving nonlinear constraint equations in parallel and we explain why it is necessary to advance the state-of-the-art. We present ILVES-PC, a new algorithm for imposing bond constraints on proteins accurately and efficiently. It solves the same system of differential algebraic equations as the celebrated SHAKE algorithm, but ILVES-PC solves the nonlinear constraint equations using Newton’s method rather than the nonlinear Gauss-Seidel method. Moreover, ILVES-PC solves the necessary linear systems using a specialized linear solver that exploits the structure of the protein. ILVES-PC can rapidly solve constraint equations as accurately as the hardware will allow. The run-time of ILVES-PC is proportional to the number of constraints. We have integrated ILVES-PC into GROMACS and simulated proteins of different sizes. Compared with SHAKE, we have achieved speedups of up to 4.9× in single-threaded executions and up to 76× in shared-memory multi-threaded executions. Moreover, ILVES-PC is more accurate than P-LINCS algorithm. Our work is a proof-of-concept of the utility of software designed specifically for the simulation of polymers
Increased plasma levels of mitochondrial DNA and pro-inflammatory cytokines in patients with progressive multiple sclerosis
The role of damage-associated molecular patterns in multiple sclerosis (MS) is under investigation. Here, we studied the contribution of circulating high mobility group box protein 1 (HMGB1) and mitochondrial DNA (mtDNA) to neuroinflammation in progressive MS. We measured plasmatic mtDNA, HMGB1 and pro-inflammatory cytokines in 38 secondary progressive (SP) patients, 35 primary progressive (PP) patients and 42 controls. Free mtDNA was higher in SP than PP. Pro-inflammatory cytokines were increased in progressive patients. In PP, tumor necrosis factor-α correlated with MS Severity Score. Thus, in progressive patients, plasmatic mtDNA and pro-inflammatory cytokines likely contribute to the systemic inflammatory status
The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems
The Fermi-Pasta-Ulam -model of harmonic oscillators with cubic
anharmonic interactions is studied from a statistical mechanical point of view.
Systems of N= 32 to 128 oscillators appear to be large enough to suggest
statistical mechanical behavior. A key element has been a comparison of the
maximum Lyapounov coefficient of the FPU -model and
that of the Toda lattice. For generic initial conditions, is
indistinguishable for the two models up to times that increase with decreasing
energy (at fixed N). Then suddenly a bifurcation appears, which can be
discussed in relation to the breakup of regular, soliton-like structures. After
this bifurcation, the of the FPU model appears to approach a
constant, while the of the Toda lattice appears to approach
zero, consistent with its integrability. This suggests that for generic initial
conditions the FPU -model is chaotic and will therefore approach
equilibrium and equipartition of energy. There is, however, a threshold energy
density , below which trapping occurs; here the
dynamics appears to be regular, soliton-like and the approach to equilibrium -
if any - takes longer than observable on any available computer. Above this
threshold the system appears to behave in accordance with statistical
mechanics, exhibiting an approach to equilibrium in physically reasonable
times. The initial conditions chosen by Fermi, Pasta and Ulam were not generic
and below threshold and would have required possibly an infinite time to reach
equilibrium.Comment: 24 pages, REVTeX, 8 PostScript figures. Published versio
Methods to study splicing from high-throughput RNA Sequencing data
The development of novel high-throughput sequencing (HTS) methods for RNA
(RNA-Seq) has provided a very powerful mean to study splicing under multiple
conditions at unprecedented depth. However, the complexity of the information
to be analyzed has turned this into a challenging task. In the last few years,
a plethora of tools have been developed, allowing researchers to process
RNA-Seq data to study the expression of isoforms and splicing events, and their
relative changes under different conditions. We provide an overview of the
methods available to study splicing from short RNA-Seq data. We group the
methods according to the different questions they address: 1) Assignment of the
sequencing reads to their likely gene of origin. This is addressed by methods
that map reads to the genome and/or to the available gene annotations. 2)
Recovering the sequence of splicing events and isoforms. This is addressed by
transcript reconstruction and de novo assembly methods. 3) Quantification of
events and isoforms. Either after reconstructing transcripts or using an
annotation, many methods estimate the expression level or the relative usage of
isoforms and/or events. 4) Providing an isoform or event view of differential
splicing or expression. These include methods that compare relative
event/isoform abundance or isoform expression across two or more conditions. 5)
Visualizing splicing regulation. Various tools facilitate the visualization of
the RNA-Seq data in the context of alternative splicing. In this review, we do
not describe the specific mathematical models behind each method. Our aim is
rather to provide an overview that could serve as an entry point for users who
need to decide on a suitable tool for a specific analysis. We also attempt to
propose a classification of the tools according to the operations they do, to
facilitate the comparison and choice of methods.Comment: 31 pages, 1 figure, 9 tables. Small corrections adde
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