2,515 research outputs found
The von Neumann entropy asymptotics in multidimensional fermionic systems
We study the von Neumann entropy asymptotics of pure translation-invariant
quasi-free states of d-dimensional fermionic systems. It is shown that the
entropic area law is violated by all these states: apart from the trivial
cases, the entropy of a cubic subsystem with edge length L cannot grow slower
than L^{d-1}ln L. As for the upper bound of the entropy asymptotics, the
zero-entropy-density property of these pure states is the only limit: it is
proven that arbitrary fast sub-L^d entropy growth is achievable.Comment: 10 page
Physiologically structured populations with diffusion and dynamic boundary conditions
We consider a linear size-structured population model with diffusion in the
size-space. Individuals are recruited into the population at arbitrary sizes.
The model is equipped with generalized Wentzell-Robin (or dynamic) boundary
conditions. This allows modelling of "adhesion" at extremely small or large
sizes. We establish existence and positivity of solutions by showing that
solutions are governed by a positive quasicontractive semigroup of linear
operators on the biologically relevant state space. This is carried out via
establishing dissipativity of a suitably perturbed semigroup generator. We also
show that solutions of the model exhibit balanced exponential growth, that is
our model admits a finite dimensional global attractor. In case of strictly
positive fertility we are able to establish that solutions in fact exhibit
asynchronous exponential growth
Steady states in hierarchical structured populations with distributed states at birth
We investigate steady states of a quasilinear first order hyperbolic partial
integro-differential equation. The model describes the evolution of a
hierarchical structured population with distributed states at birth.
Hierarchical size-structured models describe the dynamics of populations when
individuals experience size-specific environment. This is the case for example
in a population where individuals exhibit cannibalistic behavior and the chance
to become prey (or to attack) depends on the individual's size. The other
distinctive feature of the model is that individuals are recruited into the
population at arbitrary size. This amounts to an infinite rank integral
operator describing the recruitment process. First we establish conditions for
the existence of a positive steady state of the model. Our method uses a fixed
point result of nonlinear maps in conical shells of Banach spaces. Then we
study stability properties of steady states for the special case of a separable
growth rate using results from the theory of positive operators on Banach
lattices.Comment: to appear in Discrete and Continuous Dynamical Systems - Series
On the sharpness of the zero-entropy-density conjecture
The zero-entropy-density conjecture states that the entropy density, defined
as the limit of S(N)/N at infinity, vanishes for all translation-invariant pure
states on the spin chain. Or equivalently, S(N), the von Neumann entropy of
such a state restricted to N consecutive spins, is sublinear. In this paper it
is proved that this conjecture cannot be sharpened, i.e., translation-invariant
states give rise to arbitrary fast sublinear entropy growth. The proof is
constructive, and is based on a class of states derived from quasifree states
on a CAR algebra. The question whether the entropy growth of pure quasifree
states can be arbitrary fast sublinear was first raised by Fannes et al. [J.
Math. Phys. 44, 6005 (2003)]. In addition to the main theorem it is also shown
that the entropy asymptotics of all pure shift-invariant nontrivial quasifree
states is at least logarithmic.Comment: 11 pages, references added, corrected typo
Structured populations with distributed recruitment: from PDE to delay formulation
In this work first we consider a physiologically structured population model
with a distributed recruitment process. That is, our model allows newly
recruited individuals to enter the population at all possible individual
states, in principle. The model can be naturally formulated as a first order
partial integro-differential equation, and it has been studied extensively. In
particular, it is well-posed on the biologically relevant state space of
Lebesgue integrable functions. We also formulate a delayed integral equation
(renewal equation) for the distributed birth rate of the population. We aim to
illustrate the connection between the partial integro-differential and the
delayed integral equation formulation of the model utilising a recent spectral
theoretic result. In particular, we consider the equivalence of the steady
state problems in the two different formulations, which then leads us to
characterise irreducibility of the semigroup governing the linear partial
integro-differential equation. Furthermore, using the method of
characteristics, we investigate the connection between the time dependent
problems. In particular, we prove that any (non-negative) solution of the
delayed integral equation determines a (non-negative) solution of the partial
differential equation and vice versa. The results obtained for the particular
distributed states at birth model then lead us to present some very general
results, which establish the equivalence between a general class of partial
differential and delay equation, modelling physiologically structured
populations.Comment: 28 pages, to appear in Mathematical Methods in the Applied Science
- …