arXiv subjects

David S. Dean

Publications and source records attributed to David S. Dean.

At least 37 records · Page 2Linked to original sources

arXiv2021

Generalized Taylor dispersion for translationally invariant microfluidic systems

We consider Taylor dispersion for tracer particles in micro-fluidic planar channels with strong confinement. In this context, the channel walls modify the local diffusivity tensor and also interactions between the tracer particles and the walls become important. We provide a simple and general formula for the effective diffusion constant along the channel as well as the first non-trivial finite time correction for arbitrary flows along the channel, arbitrary interaction potentials with the walls and arbitrary expressions for the diffusion tensor. The formula are in particular amenable to a straightforward numerical implementation, rendering them extremely useful for comparison with experiments. We present a number of applications, notably for systems which have parabolically varying diffusivity profiles, to systems with attractive interactions with the walls as well as electroosmotic flows between plates with differing surface charges within the Debye-Hückel approximation.

physics.flu-dyn↗

arXiv2021

Wigner function for noninteracting fermions in hard wall potentials

The Wigner function $W_N({\bf x}, {\bf p})$ is a useful quantity to characterize the quantum fluctuations of an $N$-body system in its phase space. Here we study $W_N({\bf x}, {\bf p})$ for $N$ noninteracting spinless fermions in a $d$-dimensional spherical hard box of radius $R$ at temperature $T=0$. In the large $N$ limit, the local density approximation (LDA) predicts that $W_N({\bf x}, {\bf p}) \approx 1/(2 π\hbar)^d$ inside a finite region of the $({\bf x}, {\bf p})$ plane, namely for $|{\bf x}| < R$ and $|{\bf p}| < k_F$ where $k_F$ is the Fermi momentum, while $W_N({\bf x}, {\bf p})$ vanishes outside this region, or "droplet", on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution $\hat ρ_N(p)$ and find a striking algebraic tail for very large momenta $\hat ρ_N(p) \propto 1/p^4$, well beyond $k_F$, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher $d$ and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension~$d$.

cond-mat.stat-mech↗

arXiv2021

Impurities in systems of noninteracting trapped fermions

We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

cond-mat.stat-mech↗

arXiv2020

Kernels for noninteracting fermions via a Green's function approach with applications to step potentials

The quantum correlations of $N$ noninteracting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green's function for the corresponding single particle Schrödinger equation. For smooth potentials the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large $N$ limit. It also recovers the density and the kernel of the so-called {\em Airy gas} at the edge. This method allows to analyse the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height $V_0$, we derive explicit expressions for the density and for the kernel. For large Fermi energy $μ>V_0$ it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the {\em critical point} of the square well potential when $μ=V_0$. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite $O(1)$ number of fermions on the shoulder, and moreover this number is independent of $μ$. In particular, the density exhibits an algebraic decay $\sim 1/x^2$, where $x$ is the distance from the jump. Furthermore, we show that the critical behaviour around $μ= V_0$ exhibits universality with respect with the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods-Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function.

cond-mat.stat-mech↗

arXiv2020

Position distribution in a generalised run and tumble process

We study a class of stochastic processes of the type $\frac{d^n x}{dt^n}= v_0\, σ(t)$ where $n>0$ is a positive integer and $σ(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a constant rate $γ$. For $n=1$, it reduces to the standard run and tumble process for active particles in one dimension. This process can be analytically continued to any $n>0$ including non-integer values. We compute exactly the mean squared displacement at time $t$ for all $n>0$ and show that at late times while it grows as $\sim t^{2n-1}$ for $n>1/2$, it approaches a constant for $n<1/2$. In the marginal case $n=1/2$, it grows very slowly with time as $\sim \ln t$. Thus the process undergoes a {\em localisation} transition at $n=1/2$. We also show that the position distribution $p_n(x,t)$ remains time-dependent even at late times for $n\ge 1/2$, but approaches a stationary time-independent form for $n<1/2$. The tails of the position distribution at late times exhibit a large deviation form, $p_n(x,t)\sim \exp\left[-γ\, t\, Φ_n\left(\frac{x}{x^*(t)}\right)\right]$, where $x^*(t)= v_0\, t^n/Γ(n+1)$. We compute the rate function $Φ_n(z)$ analytically for all $n>0$ and also numerically using importance sampling methods, finding excellent agreement between them. For three special values $n=1$, $n=2$ and $n=1/2$ we compute the exact cumulant generating function of the position distribution at all times $t$.

cond-mat.stat-mech↗

arXiv2020

Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel

We study a system of $N$ noninteracting spinless fermions in a confining, double-well potential in one dimension. When the Fermi energy is close to the value of the potential at its local maximum we show that physical properties, such as the average density and the fermion position correlation functions, display a universal behavior that depends only on the local properties of the potential near its maximum. This behavior describes the merging of two Fermi gases, which are disjoint at sufficiently low Fermi energies. We describe this behavior in terms of a new correlation kernel that we compute analytically and we call it the inverted parabola kernel". As an application, we calculate the mean and variance of the number of particles in an interval of size $2L$ centered around the position of the local maximum, for sufficiently small $L$. Finally, we discuss the possibility of observing our results in experiments, as well as the extensions to nonzero temperature and to higher space dimensions.

cond-mat.stat-mech↗

arXiv2020

Thermal Casimir interactions for higher derivative field Lagrangians: generalized Brazovskii models

We examine the Casimir effect for free statistical field theories which have Hamiltonians with second order derivative terms. Examples of such Hamiltonians arise from models of non-local electrostatics, membranes with non-zero bending rigidities and field theories of the Brazovskii type that arise for polymer systems. The presence of a second derivative term means that new types of boundary conditions can be imposed, leading to a richer phenomenology of interaction phenomena. In addition zero modes can be generated that are not present in standard first derivative models, and it is these zero modes which give rise to long range Casimir forces. Two physically distinct cases are considered: (i) unconfined fields, usually considered for finite size embedded inclusions in an infinite fluctuating medium, here in a two plate geometry the fluctuating field exists both inside and outside the plates, (ii) confined fields, where the field is absent outside the slab confined between the two plates. We show how these two physically distinct cases are mathematically related and discuss a wide range of commonly applied boundary conditions. We concentrate our analysis to the critical region where the underlying bulk Hamiltonian has zero modes and show that very exotic Casimir forces can arise, characterised by very long range effects and oscillatory behavior that can lead to strong metastability in the system.

cond-mat.stat-mech↗

arXiv2020

The effect of driving on model C interfaces

We consider the effect of uniform driving on the interface between two phases which are described by model C dynamics. The non-driven system has a classical Gaussian interface described by capillary wave theory. The model under driving retains Gaussian statistics but the interface statistics are modified by driving, notably the height fluctuations are suppressed and the correlation length of the fluctuations is increased. The model we introduce can also be used as a model for the effect of activity on interface dynamics.

cond-mat.stat-mech↗

arXiv2019

Path integrals for higher derivative actions

We consider Euclidean path integrals with higher derivative actions, including those that depend quadratically on acceleration, velocity and position. Such path integrals arise naturally in the study of stiff polymers, membranes with bending rigidity as well as a number of models for electrolytes. The approach used is based on the relation between quadratic path integrals and Gaussian fields and we also show how it can be extended to the evaluation of even higher order path integrals.

cond-mat.stat-mech↗

arXiv2019

Nonequilibrium dynamics of noninteracting fermions in a trap

We consider the real time dynamics of $N$ noninteracting fermions in $d=1$. They evolve in a trapping potential $V(x)$, starting from the equilibrium state in a potential $V_0(x)$. We study the time evolution of the Wigner function $W(x,p,t)$ in the phase space $(x,p)$, and the associated kernel which encodes all correlation functions. At $t=0$ the Wigner function for large $N$ is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale $e_N$ being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several $V_0(x)$. In the large $N$ limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time dependent rescaling. For non-harmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a non-trivial time and space dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the "classical" regime $\hbar \sim 1/N$. Using the diagonal ensemble we compute the Wigner function in the quantum case (large $N$, fixed $\hbar$) and show that it agrees with the GGE. We also obtain the higher order (non-local) correlations in the diagonal ensemble.

cond-mat.stat-mech↗

arXiv2018

The role of non-conservative scattering forces and damping on Brownian particles in optical traps

We consider a model of a particle trapped in a harmonic optical trap but with the addition of a non-conservative radiation induced force. This model is known to correctly describe experimentally observed trapped particle statistics for a wide range of physical parameters such as temperature and pressure. We theoretically analyse the effect of non-conservative force on the underlying steady state distribution as well as the power spectrum for the particle position. We compute perturbatively the probability distribution of the resulting non-equilibrium steady states for all dynamical regimes, underdamped through to overdamped and give expressions for the associated currents in phase space (position and velocity). We also give the spectral density of the trapped particle's position in all dynamical regimes and for any value of the non-conservative force. Signatures of the presence non-conservative forces are shown to be particularly strong for in the underdamped regime at low frequencies.

cond-mat.stat-mech↗

arXiv2018

Noninteracting fermions in a trap and random matrix theory

We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions $N \gg 1$, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension $d=1$ and at zero temperature $T=0$, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the "Airy kernel") and hard edge (described by the "Bessel kernel") universality classes. We further discuss extensions of these results to higher dimensions $d\geq 2$ and to finite temperature. Finally, we discuss correlations in the phase space, i.e., in the space of positions and momenta, characterized by the so called Wigner function.

cond-mat.stat-mech↗

arXiv2018

Wigner function of noninteracting trapped fermions

We study analytically the Wigner function $W_N({\bf x},{\bf p})$ of $N$ noninteracting fermions trapped in a smooth confining potential $V({\bf x})$ in $d$ dimensions. At zero temperature, $W_N({\bf x},{\bf p})$ is constant over a finite support in the phase space $({\bf x},{\bf p})$ and vanishes outside. Near the edge of this support, we find a universal scaling behavior of $W_N({\bf x},{\bf p})$ for large $N$. The associated scaling function is independent of the precise shape of the potential as well as the spatial dimension $d$. We further generalize our results to finite temperature $T>0$. We show that there exists a low temperature regime $T \sim e_N/b$ where $e_N$ is an energy scale that depends on $N$ and the confining potential $V({\bf x})$, where the Wigner function at the edge again takes a universal scaling form with a $b$-dependent scaling function. This temperature dependent scaling function is also independent of the potential as well as the dimension $d$. Our results generalize to any $d\geq 1$ and $T \geq 0$ the $d=1$ and $T=0$ results obtained by Bettelheim and Wiegman [Phys. Rev. B ${\bf 84}$, 085102 (2011)].

cond-mat.stat-mech↗

arXiv2017

A Gaussian theory for fluctuations in simple liquids

Assuming an effective quadratic Hamiltonian, we derive an approximate, linear stochastic equation of motion for the density-fluctuations in liquids, composed of overdamped Brownian particles. From this approach, time dependent two point correlation functions (such as the intermediate scattering function) are derived. We show that this correlation function is exact at short times, for any interaction and, in particular, for arbitrary external potentials so that it applies to confined systems. Furthermore, we discuss the relation of this approach to previous ones, such as dynamical density functional theory as well as the formally exact treatment. This approach, inspired by the well known Landau-Ginzburg Hamiltonians, and the corresponding "Model B" equation of motion, may be seen as its microscopic version, containing information about the details on the particle level.

cond-mat.soft↗

arXiv2016

Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature

We consider $N$ non-interacting fermions in an isotropic $d$-dimensional harmonic trap. We compute analytically the cumulative distribution of the maximal radial distance of the fermions from the trap center at zero temperature. While in $d=1$ the limiting distribution (in the large $N$ limit), properly centered and scaled, converges to the squared Tracy-Widom distribution of the Gaussian Unitary Ensemble in Random Matrix Theory, we show that for all $d>1$, the limiting distribution converges to the Gumbel law. These limiting forms turn out to be universal, i.e., independent of the details of the trapping potential for a large class of isotropic trapping potentials. We also study the position of the right-most fermion in a given direction in $d$ dimensions and, in the case of a harmonic trap, the maximum momentum, and show that they obey similar Gumbel statistics. Finally, we generalize these results to low but finite temperature.

cond-mat.stat-mech↗

arXiv2016

Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations

We study a system of $N$ non-interacting spin-less fermions trapped in a confining potential, in arbitrary dimensions $d$ and arbitrary temperature $T$. The presence of the trap introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so called "bulk regime"), physical properties of the fermions have traditionally been understood using the Local Density Approximation. However, this approximation drastically fails near the edge where the density vanishes. In this paper we show that, even near the edge, novel universal properties emerge, independently of the details of the confining potential. We show that for large $N$, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any $n$-point correlation function can be expressed as an $n \times n$ determinant whose entry is called the kernel. Near the edge, we derive the large $N$ scaling form of the kernels. In $d=1$ and $T=0$, this reduces to the so called Airy kernel, that appears in the Gaussian Unitary Ensemble (GUE) of random matrix theory. In $d=1$ and $T>0$ we show a remarkable connection between our kernel and the one appearing in the $1+1$-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently our result provides a finite $T$ generalization of the Tracy-Widom distribution, that describes the fluctuations of the rightmost fermion at $T=0$. In $d>1$ and $T \geq 0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the $1d$ Airy kernel at $T=0$ obtained in random matrix theory. Some of our finite temperature results should be testable in present-day cold atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.

cond-mat.stat-mech↗

arXiv2016

Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model

The Sinai model of a tracer diffusing in a quenched Brownian potential is a much studied problem exhibiting a logarithmically slow anomalous diffusion due to the growth of energy barriers with the system size. However, if the potential is random but periodic, the regime of anomalous diffusion crosses over to one of normal diffusion once a tracer has diffused over a few periods of the system. Here we consider a system in which the potential is given by a Brownian Bridge on a finite interval $(0,L)$ and then periodically repeated over the whole real line, and study the power spectrum $S(f)$ of the diffusive process $x(t)$ in such a potential. We show that for most of realizations of $x(t)$ in a given realization of the potential, the low-frequency behavior is $S(f) \sim {\cal A}/f^2$, i.e., the same as for standard Brownian motion, and the amplitude ${\cal A}$ is a disorder-dependent random variable with a finite support. Focusing on the statistical properties of this random variable, we determine the moments of ${\cal A}$ of arbitrary, negative or positive order $k$, and demonstrate that they exhibit a multi-fractal dependence on $k$, and a rather unusual dependence on the temperature and on the periodicity $L$, which are supported by atypical realizations of the periodic disorder. We finally show that the distribution of ${\cal A}$ has a log-normal left tail, and exhibits an essential singularity close to the right edge of the support, which is related to the Lifshitz singularity. Our findings are based both on analytic results and on extensive numerical simulations of the process $x(t)$.

cond-mat.stat-mech↗

arXiv2016

Nonequilibrium Tuning of the Thermal Casimir Effect

In net-neutral systems correlations between charge fluctuations generate strong attractive thermal Casimir forces and engineering these forces to optimize nanodevice performance is an important challenge. We show how the normal and lateral thermal Casimir forces between two plates containing Brownian charges can be modulated by decorrelating the system through the application of an electric field, which generates a nonequilibrium steady state with a constant current in one or both plates, reducing the ensuing fluctuation-generated normal force while at the same time generating a lateral drag force. This hypothesis is confirmed by detailed numerical simulations as well as an analytical approach based on stochastic density functional theory.

cond-mat.stat-mech↗