270 research outputs found
Minimal Brownian Ratchet: An Exactly Solvable Model
We develop an exactly-solvable three-state discrete-time minimal Brownian
ratchet (MBR), where the transition probabilities between states are
asymmetric. By solving the master equations we obtain the steady-state
probabilities. Generally the steady-state solution does not display detailed
balance, giving rise to an induced directional motion in the MBR. For a reduced
two-dimensional parameter space we find the null-curve on which the net current
vanishes and detailed balance holds. A system on this curve is said to be
balanced. On the null-curve, an additional source of external random noise is
introduced to show that a directional motion can be induced under the zero
overall driving force. We also indicate the off-balance behavior with biased
random noise.Comment: 4 pages, 4 figures, RevTex source, General solution added. To be
appeared in Phys. Rev. Let
Mean curvature flow with triple junctions in higher space dimensions
We consider mean curvature flow of n-dimensional surface clusters. At
(n-1)-dimensional triple junctions an angle condition is required which in the
symmetric case reduces to the well-known 120 degree angle condition. Using a
novel parametrization of evolving surface clusters and a new existence and
regularity approach for parabolic equations on surface clusters we show local
well-posedness by a contraction argument in parabolic Hoelder spaces.Comment: 31 pages, 2 figure
Incompatible sets of gradients and metastability
We give a mathematical analysis of a concept of metastability induced by
incompatibility. The physical setting is a single parent phase, just about to
undergo transformation to a product phase of lower energy density. Under
certain conditions of incompatibility of the energy wells of this energy
density, we show that the parent phase is metastable in a strong sense, namely
it is a local minimizer of the free energy in an neighbourhood of its
deformation. The reason behind this result is that, due to the incompatibility
of the energy wells, a small nucleus of the product phase is necessarily
accompanied by a stressed transition layer whose energetic cost exceeds the
energy lowering capacity of the nucleus. We define and characterize
incompatible sets of matrices, in terms of which the transition layer estimate
at the heart of the proof of metastability is expressed. Finally we discuss
connections with experiment and place this concept of metastability in the
wider context of recent theoretical and experimental research on metastability
and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea
Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source
This paper deals with the long-time behavior of solutions of nonlinear
reaction-diffusion equations describing formation of morphogen gradients, the
concentration fields of molecules acting as spatial regulators of cell
differentiation in developing tissues. For the considered class of models, we
establish existence of a new type of ultra-singular self-similar solutions.
These solutions arise as limits of the solutions of the initial value problem
with zero initial data and infinitely strong source at the boundary. We prove
existence and uniqueness of such solutions in the suitable weighted energy
spaces. Moreover, we prove that the obtained self-similar solutions are the
long-time limits of the solutions of the initial value problem with zero
initial data and a time-independent boundary source
On the structure of phase transition maps for three or more coexisting phases
This paper is partly based on a lecture delivered by the author at the ERC
workshop "Geometric Partial Differential Equations" held in Pisa in September
2012. What is presented here is an expanded version of that lecture.Comment: 23 pages, 6 figure
Characterization of Generalized Young Measures Generated by Symmetric Gradients
This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer\ue2\u80\u93Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The \ue2\u80\u9clocal\ue2\u80\u9d proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti\ue2\u80\u99s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences
Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids
Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold
(M,g). In this paper we study the restrictions on the topology and geometry of
the fibres (the level sets) of the solutions f to (P1). We give a technique
based on certain remarkable property of the fibres (the analytic representation
property) for going from the initial PDE to a global analytical
characterization of the fibres (the equilibrium partition condition). We study
this analytical characterization and obtain several topological and geometrical
properties that the fibres of the solutions must possess, depending on the
topology of M and the metric tensor g. We apply these results to the classical
problem in physics of classifying the equilibrium shapes of both Newtonian and
relativistic static self-gravitating fluids. We also suggest a relationship
with the isometries of a Riemannian manifold.Comment: 36 pages. In this new version the analytic representation hypothesis
is proved. Please address all correspondence to D. Peralta-Sala
Distributed model predictive control of linear systems with coupled constraints based on collective neurodynamic optimization
© Springer Nature Switzerland AG 2018. Distributed model predictive control explores an array of local predictive controllers that synthesize the control of subsystems independently yet they communicate to efficiently cooperate in achieving the closed-loop control performance. Distributed model predictive control problems naturally result in sequential distributed optimization problems that require real-time solution. This paper presents a collective neurodynamic approach to design and implement the distributed model predictive control of linear systems in the presence of globally coupled constraints. For each subsystem, a neurodynamic model minimizes its cost function using local information only. According to the communication topology of the network, neurodynamic models share information to their neighbours to reach consensus on the optimal control actions to be carried out. The collective neurodynamic models are proven to guarantee the global optimality of the model predictive control system
Flows of granular material in two-dimensional channels
Secondary cone-type crushing machines are an important part of the aggregate production process. These devices process roughly crushed material into aggregate of greater consistency and homogeneity. We apply a continuum model for granular materials (`A Constitutive Law For Dense Granular Flows', Nature 441, p727-730, 2006) to flows of granular material in representative two-dimensional channels, applying a cyclic applied crushing stress in lieu of a moving boundary. Using finite element methods we solve a sequence of quasi-steady fluid problems within the framework of a pressure dependent particle size problem in time. Upon approximating output quantity and particle size we adjust the frequency and strength of the crushing stroke to assess their impact on the output
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