170 research outputs found
Extinction of metastable stochastic populations
We investigate extinction of a long-lived self-regulating stochastic
population, caused by intrinsic (demographic) noise. Extinction typically
occurs via one of two scenarios depending on whether the absorbing state n=0 is
a repelling (scenario A) or attracting (scenario B) point of the deterministic
rate equation. In scenario A the metastable stochastic population resides in
the vicinity of an attracting fixed point next to the repelling point n=0. In
scenario B there is an intermediate repelling point n=n_1 between the
attracting point n=0 and another attracting point n=n_2 in the vicinity of
which the metastable population resides. The crux of the theory is WKB method
which assumes that the typical population size in the metastable state is
large. Starting from the master equation, we calculate the quasi-stationary
probability distribution of the population sizes and the (exponentially long)
mean time to extinction for each of the two scenarios. When necessary, the WKB
approximation is complemented (i) by a recursive solution of the
quasi-stationary master equation at small n and (ii) by the van Kampen
system-size expansion, valid near the fixed points of the deterministic rate
equation. The theory yields both entropic barriers to extinction and
pre-exponential factors, and holds for a general set of multi-step processes
when detailed balance is broken. The results simplify considerably for
single-step processes and near the characteristic bifurcations of scenarios A
and B.Comment: 19 pages, 7 figure
Extinction in Lotka-Volterra model
Competitive birth-death processes often exhibit an oscillatory behavior. We
investigate a particular case where the oscillation cycles are marginally
stable on the mean-field level. An iconic example of such a system is the
Lotka-Volterra model of predator-prey competition. Fluctuation effects due to
discreteness of the populations destroy the mean-field stability and eventually
drive the system toward extinction of one or both species. We show that the
corresponding extinction time scales as a certain power-law of the population
sizes. This behavior should be contrasted with the extinction of models stable
in the mean-field approximation. In the latter case the extinction time scales
exponentially with size.Comment: 11 pages, 17 figure
On population extinction risk in the aftermath of a catastrophic event
We investigate how a catastrophic event (modeled as a temporary fall of the
reproduction rate) increases the extinction probability of an isolated
self-regulated stochastic population. Using a variant of the Verhulst logistic
model as an example, we combine the probability generating function technique
with an eikonal approximation to evaluate the exponentially large increase in
the extinction probability caused by the catastrophe. This quantity is given by
the eikonal action computed over "the optimal path" (instanton) of an effective
classical Hamiltonian system with a time-dependent Hamiltonian. For a general
catastrophe the eikonal equations can be solved numerically. For simple models
of catastrophic events analytic solutions can be obtained. One such solution
becomes quite simple close to the bifurcation point of the Verhulst model. The
eikonal results for the increase in the extinction probability caused by a
catastrophe agree well with numerical solutions of the master equation.Comment: 11 pages, 11 figure
Algebraic Aspects of Abelian Sandpile Models
The abelian sandpile models feature a finite abelian group G generated by the
operators corresponding to particle addition at various sites. We study the
canonical decomposition of G as a product of cyclic groups G = Z_{d_1} X
Z_{d_2} X Z_{d_3}...X Z_{d_g}, where g is the least number of generators of G,
and d_i is a multiple of d_{i+1}. The structure of G is determined in terms of
toppling matrix. We construct scalar functions, linear in height variables of
the pile, that are invariant toppling at any site. These invariants provide
convenient coordinates to label the recurrent configurations of the sandpile.
For an L X L square lattice, we show that g = L. In this case, we observe that
the system has nontrivial symmetries coming from the action of the cyclotomic
Galois group of the (2L+2)th roots of unity which operates on the set of
eigenvalues of the toppling matrix. These eigenvalues are algebraic integers,
whose product is the order |G|. With the help of this Galois group, we obtain
an explicit factorizaration of |G|. We also use it to define other simpler,
though under-complete, sets of toppling invariants.Comment: 39 pages, TIFR/TH/94-3
Large fluctuations in stochastic population dynamics: momentum space calculations
Momentum-space representation renders an interesting perspective to theory of
large fluctuations in populations undergoing Markovian stochastic gain-loss
processes. This representation is obtained when the master equation for the
probability distribution of the population size is transformed into an
evolution equation for the probability generating function. Spectral
decomposition then brings about an eigenvalue problem for a non-Hermitian
linear differential operator. The ground-state eigenmode encodes the stationary
distribution of the population size. For long-lived metastable populations
which exhibit extinction or escape to another metastable state, the
quasi-stationary distribution and the mean time to extinction or escape are
encoded by the eigenmode and eigenvalue of the lowest excited state. If the
average population size in the stationary or quasi-stationary state is large,
the corresponding eigenvalue problem can be solved via WKB approximation
amended by other asymptotic methods. We illustrate these ideas in several model
examples.Comment: 20 pages, 9 figures, to appear in JSTA
Monge's transport problem in the Heisenberg group
We prove the existence of solutions to Monge transport problem between two
compactly supported Borel probability measures in the Heisenberg group equipped
with its Carnot-Caratheodory distance assuming that the initial measure is
absolutely continuous with respect to the Haar measure of the group
Laser Spectroscopic Studies of the E 1ÂŁ+ State of the MgO Molecule
The E1ÎŁ+ âRydberg' state of 24Mg16O has been characterized by two-color resonance-enhanced two-photon ionization (R2PI) spectroscopy in the 36 000â40 000 cmâ1 region. Several rotationally resolved bands, assigned consistently to 24Mg16O(E1ÎŁ+âX1ÎŁ+) vibronic transitions, have been analyzed. The effective BvâČ(vâČ=0â8) constants determined exhibit an unusual variation with vâČ. Possible causes of this variation are discussed. Estimated spectroscopic constants for the E1ÎŁ+ state are reported
Schwarzschild Atmospheric Processes: A Classical Path to the Quantum
We develop some classical descriptions for processes in the Schwarzschild
string atmosphere. These processes suggest relationships between macroscopic
and microscopic scales. The classical descriptions developed in this essay
highlight the fundamental quantum nature of the Schwarzschild atmospheric
processes.Comment: to appear in Gen. Rel. Gra
Quantum-classical transition in Scale Relativity
The theory of scale relativity provides a new insight into the origin of
fundamental laws in physics. Its application to microphysics allows us to
recover quantum mechanics as mechanics on a non-differentiable (fractal)
spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as
geodesic equations in this framework. A development of the intrinsic properties
of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads
us to a derivation of the Dirac equation within the scale-relativity paradigm.
The complex form of the wavefunction in the Schrodinger and Klein-Gordon
equations follows from the non-differentiability of the geometry, since it
involves a breaking of the invariance under the reflection symmetry on the
(proper) time differential element (ds - ds). This mechanism is generalized
for obtaining the bi-quaternionic nature of the Dirac spinor by adding a
further symmetry breaking due to non-differentiability, namely the differential
coordinate reflection symmetry (dx^mu - dx^mu) and by requiring invariance
under parity and time inversion. The Pauli equation is recovered as a
non-relativistic-motion approximation of the Dirac equation.Comment: 28 pages, no figur
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