653 research outputs found
Purification of quantum trajectories
We prove that the quantum trajectory of repeated perfect measurement on a
finite quantum system either asymptotically purifies, or hits upon a family of
`dark' subspaces, where the time evolution is unitary.Comment: 10 page
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents
A number of results for reactions involving subdiffusive species all with the
same anomalous exponent gamma have recently appeared in the literature and can
often be understood in terms of a subordination principle whereby time t in
ordinary diffusion is replaced by t^gamma. However, very few results are known
for reactions involving different species characterized by different anomalous
diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive
particle surrounded by a sea of (sub)diffusive traps in one dimension. We find
rigorous results for the asymptotic survival probability of the particle in
most cases, with the exception of the case of a particle that diffuses normally
while the anomalous diffusion exponent of the traps is smaller than 2/3.Comment: To appear in Phys. Rev.
Simulations for trapping reactions with subdiffusive traps and subdiffusive particles
While there are many well-known and extensively tested results involving
diffusion-limited binary reactions, reactions involving subdiffusive reactant
species are far less understood. Subdiffusive motion is characterized by a mean
square displacement with . Recently we
calculated the asymptotic survival probability of a (sub)diffusive
particle () surrounded by (sub)diffusive traps () in one
dimension. These are among the few known results for reactions involving
species characterized by different anomalous exponents. Our results were
obtained by bounding, above and below, the exact survival probability by two
other probabilities that are asymptotically identical (except when
and ). Using this approach, we were not able to
estimate the time of validity of the asymptotic result, nor the way in which
the survival probability approaches this regime. Toward this goal, here we
present a detailed comparison of the asymptotic results with numerical
simulations. In some parameter ranges the asymptotic theory describes the
simulation results very well even for relatively short times. However, in other
regimes more time is required for the simulation results to approach asymptotic
behavior, and we arrive at situations where we are not able to reach asymptotia
within our computational means. This is regrettably the case for
and , where we are therefore not able to prove
or disprove even conjectures about the asymptotic survival probability of the
particle.Comment: 15 pages, 10 figures, submitted to Journal of Physics: Condensed
Matter; special issue on Chemical Kinetics Beyond the Textbook: Fluctuations,
Many-Particle Effects and Anomalous Dynamics, eds. K.Lindenberg, G.Oshanin
and M.Tachiy
Quantum state estimation and large deviations
In this paper we propose a method to estimate the density matrix \rho of a
d-level quantum system by measurements on the N-fold system. The scheme is
based on covariant observables and representation theory of unitary groups and
it extends previous results concerning the estimation of the spectrum of \rho.
We show that it is consistent (i.e. the original input state \rho is recovered
with certainty if N \to \infty), analyze its large deviation behavior, and
calculate explicitly the corresponding rate function which describes the
exponential decrease of error probabilities in the limit N \to \infty. Finally
we discuss the question whether the proposed scheme provides the fastest
possible decay of error probabilities.Comment: LaTex2e, 40 pages, 2 figures. Substantial changes in Section 4: one
new subsection (4.1) and another (4.2 was 4.1 in the previous version)
completely rewritten. Minor changes in Sect. 2 and 3. Typos corrected.
References added. Accepted for publication in Rev. Math. Phy
The target problem with evanescent subdiffusive traps
We calculate the survival probability of a stationary target in one dimension
surrounded by diffusive or subdiffusive traps of time-dependent density. The
survival probability of a target in the presence of traps of constant density
is known to go to zero as a stretched exponential whose specific power is
determined by the exponent that characterizes the motion of the traps. A
density of traps that grows in time always leads to an asymptotically vanishing
survival probability. Trap evanescence leads to a survival probability of the
target that may be go to zero or to a finite value indicating a probability of
eternal survival, depending on the way in which the traps disappear with time
Trapping in complex networks
We investigate the trapping problem in Erdos-Renyi (ER) and Scale-Free (SF)
networks. We calculate the evolution of the particle density of
random walkers in the presence of one or multiple traps with concentration .
We show using theory and simulations that in ER networks, while for short times
, for longer times exhibits a more
complex behavior, with explicit dependence on both the number of traps and the
size of the network. In SF networks we reveal the significant impact of the
trap's location: is drastically different when a trap is placed on a
random node compared to the case of the trap being on the node with the maximum
connectivity. For the latter case we find
\rho(t)\propto\exp\left[-At/N^\frac{\gamma-2}{\gamma-1}\av{k}\right] for all
, where is the exponent of the degree distribution
.Comment: Appendix adde
Entropy production and fluctuation relations for a KPZ interface
We study entropy production and fluctuation relations in the restricted
solid-on-solid growth model, which is a microscopic realization of the KPZ
equation. Solving the one dimensional model exactly on a particular line of the
phase diagram we demonstrate that entropy production quantifies the distance
from equilibrium. Moreover, as an example of a physically relevant current
different from the entropy, we study the symmetry of the large deviation
function associated with the interface height. In a special case of a system of
length L=4 we find that the probability distribution of the variation of height
has a symmetric large deviation function, displaying a symmetry different from
the Gallavotti-Cohen symmetry.Comment: 21 pages, 5 figure
Phase transitions and configuration space topology
Equilibrium phase transitions may be defined as nonanalytic points of
thermodynamic functions, e.g., of the canonical free energy. Given a certain
physical system, it is of interest to understand which properties of the system
account for the presence of a phase transition, and an understanding of these
properties may lead to a deeper understanding of the physical phenomenon. One
possible approach of this issue, reviewed and discussed in the present paper,
is the study of topology changes in configuration space which, remarkably, are
found to be related to equilibrium phase transitions in classical statistical
mechanical systems. For the study of configuration space topology, one
considers the subsets M_v, consisting of all points from configuration space
with a potential energy per particle equal to or less than a given v. For
finite systems, topology changes of M_v are intimately related to nonanalytic
points of the microcanonical entropy (which, as a surprise to many, do exist).
In the thermodynamic limit, a more complex relation between nonanalytic points
of thermodynamic functions (i.e., phase transitions) and topology changes is
observed. For some class of short-range systems, a topology change of the M_v
at v=v_t was proved to be necessary for a phase transition to take place at a
potential energy v_t. In contrast, phase transitions in systems with long-range
interactions or in systems with non-confining potentials need not be
accompanied by such a topology change. Instead, for such systems the
nonanalytic point in a thermodynamic function is found to have some
maximization procedure at its origin. These results may foster insight into the
mechanisms which lead to the occurrence of a phase transition, and thus may
help to explore the origin of this physical phenomenon.Comment: 22 pages, 6 figure
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