658 research outputs found
Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices
We consider the AC transport in a quantum RC circuit made of a coherent
chaotic cavity with a top gate. Within a random matrix approach, we study the
joint distribution for the mesoscopic capacitance and
the charge relaxation resistance , where is the geometric capacitance
and the quantum capacitance. We study the limit of a large number of
conducting channels with a Coulomb gas method. We obtain and show that the relative
fluctuations are of order both for and , with strong
correlations . The detailed analysis
of large deviations involves a second order phase transition in the Coulomb
gas. The two dimensional phase diagram is obtained.Comment: LaTex, 6 pages, 3 pdf figures ; v2 : minor corrections & Refs. adde
Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model
We establish the connection between a multichannel disordered model --the 1D
Dirac equation with matricial random mass-- and a random matrix
model corresponding to a deformation of the Laguerre ensemble. This allows us
to derive exact determinantal representations for the density of states and
identify its low energy () behaviour
. The vanishing of the exponent
for specific values of the averaged mass over disorder ratio
corresponds to phase transitions of topological nature characterised by the
change of a quantum number (Witten index) which is deduced straightforwardly in
the matrix model.Comment: 7+4 pages, 9+1 pdf figures ; v2: paper reorganised, discussion of
non-isotropic case adde
One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
We study the one-dimensional Schr\"odinger equation with a disordered
potential of the form where
is a Gaussian white noise with mean and variance , and
is a random superposition of delta functions distributed uniformly on the real
line with mean density and mean strength . Our study is motivated by
the close connection between this problem and classical diffusion in a random
environment (the Sinai problem) in the presence of random absorbers~:
models the force field acting on the diffusing particle and models
the absorption properties of the medium in which the diffusion takes place. The
focus is on the calculation of the complex Lyapunov exponent , where is the integrated density of
states per unit length and the reciprocal of the localisation length.
By using the continuous version of the Dyson-Schmidt method, we find an exact
formula, in terms of a Hankel function, in the particular case where the
strength of the delta functions is exponentially-distributed with mean .
Building on this result, we then solve the general case -- in the low-energy
limit -- in terms of an infinite sum of Hankel functions. Our main result,
valid without restrictions on the parameters of the model, is that the
integrated density of states exhibits the power law behaviour
N(E) \underset{E\to0+}{\sim} E^\nu \hspace{0.5cm} \mbox{where }
\nu=\sqrt{\mu^2+2\rho/g}\:.
This confirms and extends several results obtained previously by approximate
methods.Comment: LaTeX, 44 pages, 17 pdf figure
Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap
We study statistical properties of non-interacting identical bosons or
fermions in the canonical ensemble. We derive several general representations
for the -point correlation function of occupation numbers
. We demonstrate that it can be expressed as a ratio
of two determinants involving the (canonical) mean occupations
, ..., , which can themselves be conveniently
expressed in terms of the -body partition functions (with ). We
draw some connection with the theory of symmetric functions, and obtain an
expression of the correlation function in terms of Schur functions. Our
findings are illustrated by revisiting the problem of Bose-Einstein
condensation in a 1D harmonic trap, for which we get analytical results. We get
the moments of the occupation numbers and the correlation between ground state
and excited state occupancies. In the temperature regime dominated by quantum
correlations, the distribution of the ground state occupancy is shown to be a
truncated Gumbel law. The Gumbel law, describing extreme value statistics, is
obtained when the temperature is much smaller than the Bose-Einstein
temperature.Comment: RevTex, 13 pages, 6 pdf figures ; v2: minor corrections (eqs. 40,41
added
Truncated linear statistics associated with the top eigenvalues of random matrices
Given a certain invariant random matrix ensemble characterised by the joint
probability distribution of eigenvalues , many
important questions have been related to the study of linear statistics of
eigenvalues , where is a known
function. We study here truncated linear statistics where the sum is restricted
to the largest eigenvalues: .
Motivated by the analysis of the statistical physics of fluctuating
one-dimensional interfaces, we consider the case of the Laguerre ensemble of
random matrices with . Using the Coulomb gas
technique, we study the limit with fixed. We show that the
constraint that is fixed drives an
infinite order phase transition in the underlying Coulomb gas. This transition
corresponds to a change in the density of the gas, from a density defined on
two disjoint intervals to a single interval. In this latter case the density
presents a logarithmic divergence inside the bulk. Assuming that
is monotonous, we show that these features arise for any random matrix ensemble
and truncated linear statitics, which makes the scenario described here robust
and universal.Comment: LaTeX, 30 pages, 20 pdf figures. Updated version: a typo has been
corrected in Eq. (3.30) and more details are provided in the Appendi
Fluctuations of observables for free fermions in a harmonic trap at finite temperature
We study a system of 1D noninteracting spinless fermions in a confining trap
at finite temperature. We first derive a useful and general relation for the
fluctuations of the occupation numbers valid for arbitrary confining trap, as
well as for both canonical and grand canonical ensembles. Using this relation,
we obtain compact expressions, in the case of the harmonic trap, for the
variance of certain observables of the form of sums of a function of the
fermions' positions, . Such observables are also
called linear statistics of the positions. As anticipated, we demonstrate
explicitly that these fluctuations do depend on the ensemble in the
thermodynamic limit, as opposed to averaged quantities, which are ensemble
independent. We have applied our general formalism to compute the fluctuations
of the number of fermions on the positive axis at finite
temperature. Our analytical results are compared to numerical simulations. We
discuss the universality of the results with respect to the nature of the
confinement.Comment: 36 pages, 6 pdf figure
Distribution of the Wigner-Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases
Within the random matrix theory approach to quantum scattering, we derive the
distribution of the Wigner-Smith time delay matrix for a chaotic
cavity with uniform absorption, coupled via perfect channels. In the
unitary class we obtain a compact expression for the distribution of
the full matrix in terms of a matrix integral. In the other symmetry classes we
derive the joint distribution of the eigenvalues. We show how the large
properties of this distribution can be analysed in terms of two interacting
Coulomb gases living on two different supports. As an application of our
results, we study the statistical properties of the Wigner time delay
in the presence of absorption.Comment: 27 pages, 4 pdf figure
Extremes of Coulomb gas: universal intermediate deviation regime
In this paper, we study the extreme statistics in the complex Ginibre
ensemble of random matrices with complex Gaussian entries, but
with no other symmetries. All the eigenvalues are complex random variables
and their joint distribution can be interpreted as a Coulomb gas with a
logarithmic repulsion between any pair of particles and in presence of a
confining harmonic potential . We study the statistics of the
eigenvalue with the largest modulus in the complex plane. The
typical and large fluctuations of around its mean had been studied
before, and they match smoothly to the right of the mean. However, it remained
a puzzle to understand why the large and typical fluctuations to the left of
the mean did not match. In this paper, we show that there is indeed an
intermediate fluctuation regime that interpolates smoothly between the large
and the typical fluctuations to the left of the mean. Moreover, we compute
explicitly this "intermediate deviation function" (IDF) and show that it is
universal, i.e. independent of the confining potential as long as it is
spherically symmetric and increases faster than for large with an
unbounded support. If the confining potential has a finite support, i.e.
becomes infinite beyond a finite radius, we show via explicit computation that
the corresponding IDF is different. Interestingly, in the borderline case where
the confining potential grows very slowly as for
with an unbounded support, the intermediate regime disappears and there is a
smooth matching between the central part and the left large deviation regime.Comment: 36 pages, 7 figure
Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
We consider a multichannel wire with a disordered region of length and a
reflecting boundary. The reflection of a wave of frequency is
described by the scattering matrix , encoding the
probability amplitudes to be scattered from one channel to another. The
Wigner-Smith time delay matrix is another important matrix
encoding temporal aspects of the scattering process. In order to study its
statistical properties, we split the scattering matrix in terms of two unitary
matrices,
(with in the presence of TRS), and
introduce a novel symmetrisation procedure for the Wigner-Smith matrix:
,
where is the wave vector and the group velocity. We demonstrate that
can be expressed under the form of an exponential
functional of a matrix Brownian motion. For semi-infinite wires, ,
using a matricial extension of the Dufresne identity, we recover
straightforwardly the joint distribution for 's eigenvalues of
Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length , the
exponential functional representation is used to calculate the first moments
,
and
. Finally we derive a
partial differential equation for the resolvent
in the large limit.Comment: 30 pages, LaTe
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