arXiv subjects

Aurélien Grabsch

Publications and source records attributed to Aurélien Grabsch.

34 records · Page 2Linked to original sources

arXiv2022

Exact time dependence of the cumulants of a tracer position in a dense lattice gas

We develop a general method to calculate the exact time dependence of the cumulants of the position of a tracer particle in a dense lattice gas of hardcore particles. More precisely, we calculate the cumulant generating function associated with the position of a tagged particle at arbitrary time, and at leading order in the density of vacancies on the lattice. In particular, our approach gives access to the short-time dynamics of the cumulants of the tracer position -- a regime in which few results are known. The generality of our approach is demonstrated by showing that it goes beyond the case of a symmetric 1D random walk, and covers the important situations of (i) a biased tracer; (ii) comb-like structures; and (iii) $d$-dimensional situations.

cond-mat.stat-mech↗

arXiv2022

Exact closure and solution for spatial correlations in single-file diffusion

Single-file transport, where particles diffuse in narrow channels while not overtaking each other, is a fundamental model for the tracer subdiffusion observed in confined systems, such as zeolites or carbon nanotubes. This anomalous behavior originates from strong bath-tracer correlations in 1D, which, despite extensive effort, have however remained elusive, because they involve an infinite hierarchy of equations. Here, for the Symmetric Exclusion Process, a paradigmatic model of single-file diffusion, we break the hierarchy and unveil a closed exact equation satisfied by these correlations, which we solve. Beyond quantifying the correlations, the central role of this key equation as a novel tool for interacting particle systems is further demonstrated by showing that it applies to out-of equilibrium situations, other observables and other representative single-file systems.

cond-mat.stat-mech↗

arXiv2021

General truncated linear statistics for the top eigenvalues of random matrices

Invariant ensemble, which are characterised by the joint distribution of eigenvalues $P(λ_1,\ldots,λ_N)$, play a central role in random matrix theory. We consider the truncated linear statistics $L_K = \sum_{n=1}^K f(λ_n)$ with $1 \leq K \leq N$, $λ_1 > λ_2 > \cdots > λ_N$ and $f$ a given function. This quantity has been studied recently in the case where the function $f$ is monotonous. Here, we consider the general case, where this function can be non-monotonous. Motivated by the physics of cold atoms, we study the example $f(λ)=λ^2$ in the Gaussian ensembles of random matrix theory. Using the Coulomb gas method, we obtain the distribution of the truncated linear statistics, in the limit $N \to \infty$ and $K \to \infty$, with $κ= K/N$ fixed. We show that the distribution presents two essential singularities, which arise from two infinite order phase transitions for the underlying Coulomb gas. We further argue that this mechanism is universal, as it depends neither on the choice of the ensemble, nor on the function $f$.

cond-mat.stat-mech↗

arXiv2021

Generalised density profiles in single-file systems

Single-file diffusion refers to the motion in narrow channels of particles which cannot bypass each other. These strong correlations between particles lead to tracer subdiffusion, which has been observed in contexts as varied as transport in porous media, zeolites or confined colloidal suspensions, and theoretically studied in numerous works. Most approaches to this celebrated many-body problem were restricted to the description of the tracer only, whose essential properties, such as large deviation functions or two-time correlation functions, were determined only recently. Here, we go beyond this standard description by introducing and determining analytically generalised density profiles (GDPs) in the frame of the tracer. In addition to controlling the statistical properties of the tracer, these quantities fully characterise the correlations between the tracer position and the bath particles density. Considering the hydrodynamic limit of the problem, we unveil universal scaling properties of the GDPs with space and time, and a non-monotonic dependence with the distance to the tracer despite the absence of any asymmetry. Our analytical approach provides exact results for the GDPs of paradigmatic models of single-file diffusion, such as Brownian particles with hardcore repulsion, the Symmetric Exclusion Process and the Random Average Process. The range of applicability of our approach is further illustrated by considering extensions to general interactions between particles and out-of-equilibrium situations.

cond-mat.stat-mech↗

arXiv2020

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 $L$ and a reflecting boundary. The reflection of a wave of frequency $ω$ is described by the scattering matrix $\mathcal{S}(ω)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $\mathcal{Q}=-\mathrm{i}\, \mathcal{S}^\dagger\partial_ω\mathcal{S}$ 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, $\mathcal{S}=\mathrm{e}^{2\mathrm{i}kL}\mathcal{U}_L\mathcal{U}_R$ (with $\mathcal{U}_L=\mathcal{U}_R^\mathrm{T}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $\widetilde{\mathcal{Q}} =\mathcal{U}_R\,\mathcal{Q}\,\mathcal{U}_R^\dagger = (2L/v)\,\mathbf{1}_N -\mathrm{i}\,\mathcal{U}_L^\dagger\partial_ω\big(\mathcal{U}_L\mathcal{U}_R\big)\,\mathcal{U}_R^\dagger$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $\widetilde{\mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $L\to\infty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $\mathcal{Q}$'s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $\langle\mathrm{tr}(\mathcal{Q})\rangle$, $\langle\mathrm{tr}(\mathcal{Q}^2)\rangle$ and $\langle\big[\mathrm{tr}(\mathcal{Q})\big]^2\rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=\lim_{N\to\infty}(1/N)\,\mathrm{tr}\big\{\big( z\,\mathbf{1}_N - N\,\mathcal{Q}\big)^{-1}\big\}$ in the large $N$ limit.

math-ph↗

arXiv2019

Wigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts

We consider wave propagation in a complex structure coupled to a finite number $N$ of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, related to the energy (or frequency) derivative of the scattering matrix $\mathcal{S}$. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix $\mathcal{Q}_s=-\mathrm{i}\hbar\,\mathcal{S}^{-1/2}\big(\partial_\varepsilon\mathcal{S}\big)\,\mathcal{S}^{-1/2}$, and obtain the joint distribution of $\mathcal{S}$ and $\mathcal{Q}_s$ for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) $0<T\leq1$. We derive two representations of the distribution of $\mathcal{Q}_s$ in terms of matrix integrals specified by the Dyson symmetry index $β=1,2,4$ (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay $τ_\mathrm{W}=(1/N)\,\mathrm{tr}\big\{\mathcal{Q}_s\big\}$, which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution $\mathscr{P}_{N,β}(τ)$ of the Wigner time delay in the weak coupling limit $NT\ll1$ and identify three different asymptotic regimes.

math-ph↗

arXiv2019

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 $\mathcal{Q}$ for a chaotic cavity with uniform absorption, coupled via $N$ perfect channels. In the unitary class $β=2$ 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 $N$ 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 $τ_{\mathrm{W}} = \mathrm{tr}[\mathcal{Q}]/N$ in the presence of absorption.

math-ph↗

arXiv2019

Extremes of $2d$ Coulomb gas: universal intermediate deviation regime

In this paper, we study the extreme statistics in the complex Ginibre ensemble of $N \times N$ random matrices with complex Gaussian entries, but with no other symmetries. All the $N$ eigenvalues are complex random variables and their joint distribution can be interpreted as a $2d$ Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential $v(r) \propto r^2$. We study the statistics of the eigenvalue with the largest modulus $r_{\max}$ in the complex plane. The typical and large fluctuations of $r_{\max}$ 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 $v(r)$ as long as it is spherically symmetric and increases faster than $\ln r^2$ for large $r$ with an unbounded support. If the confining potential $v(r)$ 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 $v(r) \sim \ln r^2$ for $r \gg 1$ with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.

cond-mat.stat-mech↗

arXiv2018

Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap

We study statistical properties of $N$ non-interacting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratio of two $p\times p$ determinants involving the (canonical) mean occupations $\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\leq N$). 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.

cond-mat.stat-mech↗

arXiv2018

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 $P(λ_1,\ldots,λ_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L=\sum_{i=1}^Nf(λ_i)$, where $f(λ)$ is a known function. We study here truncated linear statistics where the sum is restricted to the $N_1<N$ largest eigenvalues: $\tilde{L}=\sum_{i=1}^{N_1}f(λ_i)$. 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 $f(λ)=\sqrtλ$. Using the Coulomb gas technique, we study the $N\to\infty$ limit with $N_1/N$ fixed. We show that the constraint that $\tilde{L}=\sum_{i=1}^{N_1}f(λ_i)$ 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 $f(λ)$ 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.

cond-mat.stat-mech↗

arXiv2018

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, $\mathcal{L}=\sum_n h(x_n)$. 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 $\mathcal{N}_+$ 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.

cond-mat.stat-mech↗

arXiv2017

Truncated linear statistics associated with the eigenvalues of random matrices II. Partial sums over proper time delays for chaotic quantum dots

Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues $\{λ_1,\cdots,λ_N\}$. We study the distribution of truncated linear statistics of the form $\tilde{L}=\sum_{i=1}^p f(λ_i)$ with $p<N$. This problem has been considered by us in a previous paper when the $p$ eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the "thinned ensembles" introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function for $\tilde{L}$. Large deviations of linear statistics $L=\sum_{i=1}^N f(λ_i)$ are usually dominated by the energy of the Coulomb gas, which scales as $\sim N^2$, implying that the relative fluctuations are of order $1/N$. For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect. Because the entropy scales as $\sim N$, the relative fluctuations are of order $1/\sqrt{N}$. Our analysis relies on the mapping on a problem of $p$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures. We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here ($f(λ)=1/λ$), the corresponding phase diagram is separated in three different phases.

cond-mat.stat-mech↗

arXiv2016

Distribution of spectral linear statistics on random matrices beyond the large deviation function -- Wigner time delay in multichannel disordered wires

An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(λ_1,\cdots,λ_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N)\sum_if(λ_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $N\to\infty$ limit, $L$ scales as $L\sim N^η$, where the scaling exponent $η$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-η}\,L)\simeq A_{β,N}(s)\,\exp\big\{-(βN^2/2)\,Φ(s)\big\}$, where $β\in\{1,\,2,\,4\}$ is the Dyson index. The Coulomb gas technique naturally provides the large deviation function $Φ(s)$, which can be efficiently obtained thanks to a "thermodynamic identity" introduced earlier. We conjecture the pre-exponential function $A_{β,N}(s)$. We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and $L$ has infinite moments)~: this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function $A_{β,N}(s)$, which ensures the decay of the distribution for large argument.

cond-mat.stat-mech↗

arXiv2016

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 $N\times N$ 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 ($\varepsilon\to0$) behaviour $ρ(\varepsilon)\sim|\varepsilon|^{α-1}$. The vanishing of the exponent $α$ for $N$ specific values of the averaged mass over disorder ratio corresponds to $N$ phase transitions of topological nature characterised by the change of a quantum number (Witten index) which is deduced straightforwardly in the matrix model.

cond-mat.dis-nn↗

arXiv2015

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 $C_μ=(1/C+1/C_q)^{-1}$ and the charge relaxation resistance $R_q$, where $C$ is the geometric capacitance and $C_q$ the quantum capacitance. We study the limit of a large number of conducting channels $N$ with a Coulomb gas method. We obtain $\langle R_q\rangle\simeq h/(Ne^2)=R_\mathrm{dc}$ and show that the relative fluctuations are of order $1/N$ both for $C_q$ and $R_q$, with strong correlations $\langle δC_qδR_q\rangle/\sqrt{\langle δC_q^2\rangle\,\langle δR_q^2\rangle}\simeq+0.707$. The detailed analysis of large deviations involves a second order phase transition in the Coulomb gas. The two dimensional phase diagram is obtained.

cond-mat.mes-hall↗

arXiv2013

One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers

We study the one-dimensional Schrödinger equation with a disordered potential of the form $V (x) = ϕ(x)^2+ϕ'(x) + κ(x) $ where $ϕ(x)$ is a Gaussian white noise with mean $μg$ and variance $g$, and $κ(x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $ρ$ and mean strength $v$. 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~: $ϕ(x)$ models the force field acting on the diffusing particle and $κ(x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $ Ω(E) = γ(E) - \mathrm{i} πN(E) $, where $N$ is the integrated density of states per unit length and $γ$ 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 $v=2g$. 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 $$ N(E) \underset{E\to0+}{\sim} E^ν\hspace{0.5cm} \mbox{where } ν=\sqrt{μ^2+2ρ/g}\:. $$ This confirms and extends several results obtained previously by approximate methods.

cond-mat.dis-nn↗