3,845 research outputs found
The Legendre Transform in Non-additive Thermodynamics and Complexity
We present an argument which purports to show that the use of the standard
Legendre transform in non-additive Statistical Mechanics is not appropriate.
For concreteness, we use as paradigm, the case of systems which are
conjecturally described by the (non-additive) Tsallis entropy. We point out the
form of the modified Legendre transform that should be used, instead, in the
non-additive thermodynamics induced by the Tsallis entropy. We comment on more
general implications of this proposal for the thermodynamics of "complex
systems".Comment: 23 pages. LaTeX2e. No figure
Homogenization of cohesive fracture in masonry structures
We derive a homogenized mechanical model of a masonry-type structure
constituted by a periodic assemblage of blocks with interposed mortar joints.
The energy functionals in the model under investigation consist in (i) a linear
elastic contribution within the blocks, (ii) a Barenblatt's cohesive
contribution at contact surfaces between blocks and (iii) a suitable unilateral
condition on the strain across contact surfaces, and are governed by a small
parameter representing the typical ratio between the length of the blocks and
the dimension of the structure. Using the terminology of Gamma-convergence and
within the functional setting supplied by the functions of bounded deformation,
we analyze the asymptotic behavior of such energy functionals when the
parameter tends to zero, and derive a simple homogenization formula for the
limit energy. Furthermore, we highlight the main mathematical and mechanical
properties of the homogenized energy, including its non-standard growth
conditions under tension or compression. The key point in the limit process is
the definition of macroscopic tensile and compressive stresses, which are
determined by the unilateral conditions on contact surfaces and the geometry of
the blocks
Well-posedness of Wasserstein Gradient Flow Solutions of Higher Order Evolution Equations
A relaxed notion of displacement convexity is defined and used to establish
short time existence and uniqueness of Wasserstein gradient flows for higher
order energy functionals. As an application, local and global well-posedness of
different higher order non-linear evolution equations are derived. Examples
include the thin-film equation and the quantum drift diffusion equation in one
spatial variable
Dealing with moment measures via entropy and optimal transport
A recent paper by Cordero-Erausquin and Klartag provides a characterization
of the measures on which can be expressed as the moment measures
of suitable convex functions , i.e. are of the form (\nabla u)\_\\#e^{- u}
for and finds the corresponding by a
variational method in the class of convex functions. Here we propose a purely
optimal-transport-based method to retrieve the same result. The variational
problem becomes the minimization of an entropy and a transport cost among
densities and the optimizer turns out to be . This
requires to develop some estimates and some semicontinuity results for the
corresponding functionals which are natural in optimal transport. The notion of
displacement convexity plays a crucial role in the characterization and
uniqueness of the minimizers
A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity
We consider a one dimensional transport model with nonlocal velocity given by
the Hilbert transform and develop a global well-posedness theory of probability
measure solutions. Both the viscous and non-viscous cases are analyzed. Both in
original and in self-similar variables, we express the corresponding equations
as gradient flows with respect to a free energy functional including a singular
logarithmic interaction potential. Existence, uniqueness, self-similar
asymptotic behavior and inviscid limit of solutions are obtained in the space
of probability measures with finite second
moments, without any smallness condition. Our results are based on the abstract
gradient flow theory developed in \cite{Ambrosio}. An important byproduct of
our results is that there is a unique, up to invariance and translations,
global in time self-similar solution with initial data in
, which was already obtained in
\textrm{\cite{Deslippe,Biler-Karch}} by different methods. Moreover, this
self-similar solution attracts all the dynamics in self-similar variables. The
crucial monotonicity property of the transport between measures in one
dimension allows to show that the singular logarithmic potential energy is
displacement convex. We also extend the results to gradient flow equations with
negative power-law locally integrable interaction potentials
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
Measure valued solutions of sub-linear diffusion equations with a drift term
In this paper we study nonnegative, measure valued solutions of the initial
value problem for one-dimensional drift-diffusion equations when the nonlinear
diffusion is governed by an increasing function with . By using tools of optimal transport, we will show
that this kind of problems is well posed in the class of nonnegative Borel
measures with finite mass and finite quadratic momentum and it is the
gradient flow of a suitable entropy functional with respect to the so called
-Wasserstein distance. Due to the degeneracy of diffusion for large
densities, concentration of masses can occur, whose support is transported by
the drift. We shall show that the large-time behavior of solutions depends on a
critical mass , which can be explicitely characterized in terms of
and of the drift term. If the initial mass is less then ,
the entropy has a unique minimizer which is absolutely continuous with respect
to the Lebesgue measure. Conversely, when the total mass of the solutions
is greater than the critical one, the steady state has a singular part in which
the exceeding mass is accumulated.Comment: 30 page
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