99 research outputs found
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
Phase transitions and edge scaling of number variance in Gaussian random matrices
We consider Gaussian random matrices, whose average density of
eigenvalues has the Wigner semi-circle form over . For
such matrices, using a Coulomb gas technique, we compute the large behavior
of the probability that
eigenvalues lie within the box . This probability scales as
, where is the Dyson index of the ensemble
and is a -independent rate function that we compute
exactly. We identify three regimes as is varied: (i) (bulk), (ii) on a scale of
(edge) and (iii) (tail). We find a
dramatic non-monotonic behavior of the number variance as a function
of : after a logarithmic growth in the bulk (when ), decreases abruptly as approaches the edge of
the semi-circle before it decays as a stretched exponential for .
This "drop-off" of at the edge is described by a scaling function
which smoothly interpolates between the bulk (i) and the
tail (iii). For we compute explicitly in terms of the
Airy kernel. These analytical results, verified by numerical simulations,
directly provide for the full statistics of particle-number
fluctuations at zero temperature of 1d spinless fermions in a harmonic trap.Comment: 5 pag., 3 fig, published versio
Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties
We study the dynamics of a one-dimensional run and tumble particle subjected
to confining potentials of the type , with . The
noise that drives the particle dynamics is telegraphic and alternates between
values. We show that the stationary probability density has a
rich behavior in the -plane. For , the distribution has a
finite support in and there is a critical line that
separates an active-like phase for where diverges
at , from a passive-like phase for where
vanishes at . For , the stationary density collapses to a
delta function at the origin, . In the marginal case ,
we show that, for , the stationary density is a
symmetric exponential, while for , it again is a delta
function . For the special cases and , we obtain
exactly the full time-dependent distribution , that allows us to study
how the system relaxes to its stationary state. In addition, in these two
cases, we also study analytically the full distribution of the first-passage
time to the origin. Numerical simulations are in complete agreement with our
analytical predictions.Comment: 17 pages, 12 figure
Disordered free fermions and the Cardy Ostlund fixed line at low temperature
Using functional RG, we reexamine the glass phase of the 2D random-field Sine
Gordon model. It is described by a line of fixed points (FP) with a
super-roughening amplitude as
temperature is varied. A speculation is that this line is identical to the
one found in disordered free-fermion models via exact results from ``nearly
conformal'' field theory. This however predicts , contradicting
numerics. We point out that this result may be related to failure of
dimensional reduction, and that a functional RG method incorporating higher
harmonics and non-analytic operators predicts a non-zero which
compares reasonably with numerics.Comment: 8 pages, 3 figures, only material adde
The longest excursion of stochastic processes in nonequilibrium systems
We consider the excursions, i.e. the intervals between consecutive zeros, of
stochastic processes that arise in a variety of nonequilibrium systems and
study the temporal growth of the longest one l_{\max}(t) up to time t. For
smooth processes, we find a universal linear growth \simeq
Q_{\infty} t with a model dependent amplitude Q_\infty. In contrast, for
non-smooth processes with a persistence exponent \theta, we show that <
l_{\max}(t) > has a linear growth if \theta
\sim t^{1-\psi} if \theta > \theta_c. The amplitude Q_{\infty} and the exponent
\psi are novel quantities associated to nonequilibrium dynamics. These
behaviors are obtained by exact analytical calculations for renewal and
multiplicative processes and numerical simulations for other systems such as
the coarsening dynamics in Ising model as well as the diffusion equation with
random initial conditions.Comment: 4 pages,2 figure
Condensation of the roots of real random polynomials on the real axis
We introduce a family of real random polynomials of degree n whose
coefficients a_k are symmetric independent Gaussian variables with variance
= e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly
the mean number of real roots for large n. As \alpha is varied, one finds
three different phases. First, for 0 \leq \alpha \sim
(\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase
where grows algebraically with a continuously varying exponent,
\sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for
\alpha > 2, one finds a third phase where \sim n. This family of real
random polynomials thus exhibits a condensation of their roots on the real line
in the sense that, for large n, a finite fraction of their roots /n are
real. This condensation occurs via a localization of the real roots around the
values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure
Simple solvable models with strong long-range correlations
We consider a set of independent and identically distributed (i.i.d)
random variables whose common distribution has a
parameter (or a set of parameters) which itself is random with its own
distribution. For a fixed value of this parameter , the variables are
independent and we call them conditionally independent and identically
distributed (c.i.i.d). However, once integrated over the distribution of the
parameter , the variables get strongly correlated, yet retaining a
solvable structure for various observables, such as for the sum and the
extremes of 's. This provides a simple recipe to generate a class of
solvable strongly correlated systems. We illustrate how this recipe works via
three physical examples where particles on a line perform independent (i)
Brownian motions, (ii) ballistic motions with random initial velocities, and
(iii) L\'evy flights, but they get strongly correlated via simultaneous
resetting to the origin. Our results are verified in numerical simulations.
This recipe can be used to generate an endless variety of solvable strongly
correlated systems.Comment: 24 pages, 9 figure
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