8 research outputs found
Asymptotics, Geometry, and Soft Matter
This dissertation is concerned with two problems that lie at the interface of soft-matter physics, geometry, and asymptotic analysis, but otherwise have no bearing on one another. In the first problem, I consider the equilibrium thermal fluctuations of deformable mechanical frameworks. These frameworks have served as highly idealized representations of mechanical structures that underlie a plethora of soft, few-body systems at the submicron scale such as colloidal clusters and DNA origami. When the holonomic constraints in a framework cease to be linearly independent, singularities can appear in its configuration space, where it becomes energetically softer. Consequently, the framework\u27s free-energy landscape becomes dominated by the neighborhoods of points corresponding to these singularities. In the second problem, I study the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a uniaxially-curved shell as concrete examples. Waves propagating on such structures have multiple components owing to the curvature-mediated coupling of the tangential and normal components of the displacement field. Here, using the semiclassical approximation, I show that these waves form localized, bound states around points where the absolute curvature of the structure has a minimum. Both these problems exemplify the subtle interplay between the mechanical properties of soft materials and their geometry, which further sets the stage for many interesting consequences
On the Applicability of Low-Dimensional Models for Convective Flow Reversals at Extreme Prandtl Numbers
Constructing simpler models, either stochastic or deterministic, for
exploring the phenomenon of flow reversals in fluid systems is in vogue across
disciplines. Using direct numerical simulations and nonlinear time series
analysis, we illustrate that the basic nature of flow reversals in convecting
fluids can depend on the dimensionless parameters describing the system.
Specifically, we find evidence of low-dimensional determinism in flow reversals
occurring at zero Prandtl number, whereas we fail to find such signatures for
reversals at infinite Prandtl number. Thus, even in a single system, as one
varies the system parameters, one can encounter reversals that are
fundamentally different in nature. Consequently, we conclude that a single
general low-dimensional deterministic model cannot faithfully characterize flow
reversals for every set of parameter values.Comment: 9 pages, 4 figure
Synchronizing noisy nonidentical oscillators by transient uncoupling
Synchronization is the process of achieving identical dynamics among coupled
identical units. If the units are different from each other, their dynamics
cannot become identical; yet, after transients, there may emerge a functional
relationship between them -- a phenomenon termed "generalized synchronization."
Here, we show that the concept of transient uncoupling, recently introduced for
synchronizing identical units, also supports generalized synchronization among
nonidentical chaotic units. Generalized synchronization can be achieved by
transient uncoupling even when it is impossible by regular coupling. We
furthermore demonstrate that transient uncoupling stabilizes synchronization in
the presence of common noise. Transient uncoupling works best if the units stay
uncoupled whenever the driven orbit visits regions that are locally diverging
in its phase space. Thus, to select a favorable uncoupling region, we propose
an intuitive method that measures the local divergence at the phase points of
the driven unit's trajectory by linearizing the flow and subsequently
suppresses the divergence by uncoupling
Transient Uncoupling Induces Synchronization
Finding conditions that support synchronization is a fertile and active area
of research with applications across multiple disciplines. Here we present and
analyze a scheme for synchronizing chaotic dynamical systems by transiently
uncoupling them. Specifically, systems coupled only in a fraction of their
state space may synchronize even if fully coupled they do not. Although, for
many standard systems, coupling strengths need to be bounded to ensure
synchrony, transient uncoupling removes this bound and thus enables
synchronization in an infinite range of effective coupling strengths. The
presented coupling scheme thus opens up the possibility to induce synchrony in
(biological or technical) systems whose parameters are fixed and cannot be
modified continuously.Comment: 5 pages, 6 figure
Thermal Fluctuations of Singular Bar-Joint Mechanisms
A bar-joint mechanism is a deformable assembly of freely rotating joints
connected by stiff bars. Here we develop a formalism to study the equilibration
of common bar-joint mechanisms with a thermal bath. When the constraints in a
mechanism cease to be linearly independent, singularities can appear in its
shape space, which is the part of its configuration space after discarding
rigid motions. We show that the free-energy landscape of a mechanism at low
temperatures is dominated by the neighborhoods of points that correspond to
these singularities. We consider two example mechanisms with shape-space
singularities and find that they are more likely to be found in configurations
near the singularities than others. These findings are expected to help improve
the design of nanomechanisms for various applications.Comment: 6 + 21 pages; 3 + 7 figure