arXiv subjects

David S. Dean

Publications and source records attributed to David S. Dean.

At least 55 records · Page 3Linked to original sources

arXiv2016

Universal ground state properties of free fermions in a $d$-dimensional trap

The ground state properties of $N$ spinless free fermions in a $d$-dimensional confining potential are studied. We find that any $n$-point correlation function has a simple determinantal structure that allows us to compute several properties exactly for large $N$. We show that the average density has a finite support with an edge, and near this edge the density exhibits a universal (valid for a wide class of potentials) scaling behavior for large $N$. The associated edge scaling function is computed exactly and generalizes to any $d$ the edge electron gas result of Kohn and Mattsson in $d=3$ [Phys. Rev. Lett. 81, 3487 (1998)]. In addition, we calculate the kernel (that characterizes any $n$-point correlation function) for large $N$ and show that, when appropriately scaled, it depends only on dimension $d$, but has otherwise universal scaling forms, at the edges. The edge kernel, for higher $d$, generalizes the Airy kernel in one dimension, well known from random matrix theory.

cond-mat.stat-mech↗

arXiv2015

Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time

We consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x) = mω^2 {x^2}/{2}$ at finite inverse temperature $β= 1/T$. The average density of fermions $ρ_N(x,T)$ at position $x$ is derived. For $N \gg 1$ and $β\sim {\cal O}(1/N)$, $ρ_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $β\ll 1/N$, and the Wigner semi-circle law at low temperature, for $β\gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $β{\hbar ω}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x \sim \pm \sqrt{2\hbar N/(mω)}$, are described by a limiting kernel $K^{\rm ff}_b(s,s')$ that depends continuously on $b$ and is a generalization of the Airy kernel, found in the Gaussian Unitary Ensemble of random matrices. Remarkably, exactly the same kernel $K^{\rm ff}_b(s,s')$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t= b^3$.

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arXiv2014

Approach to asymptotically diffusive behavior for Brownian particles in media with periodic diffusivities

We analyze the mean squared displacement of a Brownian particle in a medium with a spatially varying local diffusivity which is assumed to be periodic. When the system is asymptotically diffusive the mean squared displacement, characterizing the dispersion in the system, is, at late times, a linear function of time. A Kubo type formula is given for the mean squared displacement which allows the recovery of some known results for the effective diffusion constant $D_e$ in a direct way, but also allows an understanding of the asymptotic approach to the diffusive limit. In particular, as well as as computing the slope of a linear fit to the late time mean squared displacement, we find a formula for the constant where the fit intersects the y axis.

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arXiv2014

Diffusion in periodic, correlated random forcing landscapes

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically-extended (with period $L$) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent $H \in (0,1)$. While the periodicity ensures that the ultimate long-time behavior is diffusive, the generalised Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient $D_L$: Although one has the typical value $D^{\rm typ}_L \sim \exp(-βL^H)$, we show via an exact analytical approach that the positive moments ($k>0$) scale like $\langle D^k_L \rangle \sim \exp{[-c' (k βL^{H})^{1/(1+H)}]}$, and the negative ones as $\langle D^{-k}_L \rangle \sim \exp(a' (k βL^{H})^2)$, $c'$ and $a'$ being numerical constants and $β$ the inverse temperature. These results demonstrate that $D_L$ is strongly non-self-averaging. We further show that the probability distribution of $D_L$ has a log-normal left tail and a highly singular, one-sided log-stable right tail reminiscent of a Lifshitz singularity.

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arXiv2014

Fluctuation mediated interactions due to rigidity mismatch and their effect on miscibility of lipid mixtures in multicomponent membranes

We consider how membrane fluctuations can modify the miscibility of lipid mixtures, that is to say how the phase diagram of a boundary-constrained membrane is modified when the membrane is allowed to fluctuate freely in the case of zero surface tension. In order for fluctuations to have an effect, the different lipid types must have differing Gaussian rigidities. We show, somewhat paradoxically, that fluctuation-induced interactions can be treated approximately in a mean-field type theory. Our calculations predict that, depending on the difference in bending and Gaussian rigidity of the lipids, membrane fluctuations can either favor or disfavor mixing.

cond-mat.soft↗

arXiv2014

Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials : extracting information from transients

A Langevin process diffusing in a periodic potential landscape has a time dependent diffusion constant which means that its average mean squared displacement (MSD) only becomes linear at late times. The long time, or effective diffusion constant, can be estimated from the slope of a linear fit of the MSD at late times. Due to the cross over between a short time microscopic diffusion constant, which is independent of the potential, to the effective late time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a non-zero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.

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arXiv2014

Relaxation of the thermal Casimir force between net neutral plates containing Brownian charges

We investigate the dynamics of thermal Casimir interactions between plates described within a living conductor model, with embedded mobile anions and cations, whose density field obeys a stochastic partial differential equation which can be derived starting from the Langevin equations of the individual particles. This model describes the thermal Casimir interaction in the same way that the fluctuating dipole model describes van der Waals interactions. The model is analytically solved in a Debye-Hückel-like approximation. We identify several limiting dynamical regimes where the time dependence of the thermal Casimir interactions can be obtained explicitly. Most notably we find a regime with diffusive scaling, even though the charges are confined to the plates and do not diffuse into the intervening space, which makes the diffusive scaling difficult to anticipate and quite unexpected on physical grounds.

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arXiv2013

Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionising areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. However, the price paid for having a direct visualisation of a single particle trajectory with high temporal and spatial resolution is a consequent lack of statistics. This naturally calls for reliable analytical tools which will allow one to extract the properties specific to a statistical ensemble from just a single trajectory. In this article we briefly survey different analytical methods currently used to determine the ensemble average diffusion coefficient from single particle data and then focus specifically on weighted least-squares estimators, seeking the weight functions for which such estimators are ergodic. Finally, we address the question of the effects of disorder on such estimators.

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arXiv2013

On ergodic least-squares estimators of the generalized diffusion coefficient for fractional Brownian motion

We analyse a class of estimators of the generalized diffusion coefficient for fractional Brownian motion $B_t$ of known Hurst index $H$, based on weighted functionals of the single time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true, ensemble-averaged, generalized diffusion coefficient to any necessary precision from a single trajectory data, but at expense of a progressively higher experimental resolution. Convergence is fastest around $H\simeq0.30$, a value in the subdiffusive regime.

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arXiv2013

Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

In this paper we study the distribution function $P(u_α)$ of the estimators $u_α \sim T^{-1} \int^T_0 \, ω(t) \, {\bf B}^2_{t} \, dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\bf B}_{t}$. We pursue here the optimisation further by considering a family of weight functions of the form $ω(t) = (t_0 + t)^{-α}$, where $t_0$ is a time lag and $α$ is an arbitrary real number, and seeking such values of $α$ for which the estimators most efficiently filter out the fluctuations. We calculate $P(u_α)$ exactly for arbitrary $α$ and arbitrary spatial dimension $d$, and show that only for $α= 2$ the distribution $P(u_α)$ converges, as $ε= t_0/T \to 0$, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with $α= 2$ possess an ergodic property, so that the ensemble averaged diffusion coefficient can be obtained with any necessary precision from a single trajectory data, but at the expense of a progressively higher experimental resolution. For any $α\neq 2$ the distribution attains, as $ε\to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.

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arXiv2012

Optimal fits of diffusion constants from single time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.

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arXiv2012

Aspects of One-Dimensional Coulomb Gases

In this short review, we discuss recent advances in exact solutions of models based on a one- dimensional (1D) Coulomb gas by means of field-theoretic functional integral methods. The exact solutions can be used to assess the accuracy of various approximations such as the weak coupling Poisson-Boltzmann theory as well as the strong coupling theory of Coulomb gases. We consider three different 1D models: the Coulomb fluid configuration in the case of the soap film model consisting of positively and negatively charged particles between adsorbing boundaries, counterions between two charged surfaces, and an ionic liquid lattice capacitor with positively and negatively charged particles on a lattice between one positive and one negative bounding surface.

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arXiv2012

Coulomb interactions between disordered charge distributions

We review some of the recent results in the context of the Coulomb interaction between dielectric surfaces which are randomly charged but remain net-neutral on the average. Such surfaces are found to interact in vacuum with attractive long-range normal forces due to the presence of the quenched charge disorder which can compete with the standard Casimir-Van der Waals forces. They can also interact with random lateral forces and torques, which exhibit zero mean values but finite and long-range root-mean-square values. These effects can play an important role in Casimir experiments as well as in the interaction between solid surfaces as well as biomolecular surfaces that are often covered by disordered charge distributions.

cond-mat.soft↗

arXiv2012

Electrostatic interactions mediated by polarizable counterions: weak and strong coupling limits

We investigate the statistical mechanics of an inhomogeneous Coulomb fluid composed of charged particles with static polarizability. We derive the weak- and the strong-coupling approximations and evaluate the partition function in a planar dielectric slab geometry with charged boundaries. We investigate the density profiles and the disjoining pressure for both approximations. Comparison to the case of non-polarizable counterions shows that polarizability brings important differences in the counterion density distribution as well as the counterion mediated electrostatic interactions between charged dielectric interfaces.

cond-mat.soft↗

arXiv2012

Sample-to-sample torque fluctuations in a system of coaxial randomly charged surfaces

Polarizable randomly charged dielectric objects have been recently shown to exhibit long-range lateral and normal interaction forces even when they are effectively net neutral. These forces stem from an interplay between the quenched statistics of random charges and the induced dielectric image charges. This type of interaction has recently been evoked to interpret measurements of Casimir forces in vacuo, where a precise analysis of such disorder-induced effects appears to be necessary. Here we consider the torque acting on a randomly charged dielectric surface (or a sphere) mounted on a central axle next to another randomly charged surface and show that although the resultant mean torque is zero, its sample-to-sample fluctuation exhibits a long-range behavior with the separation distance between the juxtaposed surfaces and that, in particular, its root-mean-square value scales with the total area of the surfaces. Therefore, the disorder-induced torque between two randomly charged surfaces is expected to be much more pronounced than the disorder-induced lateral force and may provide an effective way to determine possible disorder effects in experiments, in a manner that is independent of the usual normal force measurement.

cond-mat.soft↗

arXiv2012

Wrapping Transition and Wrapping-Mediated Interactions for Discrete Binding along an Elastic Filament: An Exact Solution

The wrapping equilibria of one and two adsorbing cylinders are studied along a semi-flexible filament (polymer) due to the interplay between elastic rigidity and short-range adhesive energy between the cylinder and the filament. We show that statistical mechanics of the system can be solved exactly using a path integral formalism which gives access to the full effect of thermal fluctuations, going thus beyond the usual Gaussian approximations which take into account only the contributions from the minimal energy configuration and small fluctuations about this minimal energy solution. We obtain the free energy of the wrapping-unwrapping transition of the filament around the cylinders as well as the effective interaction between two wrapped cylinders due to thermal fluctuations of the elastic filament. A change of entropy due to wrapping of the filament around the adsorbing cylinders as they move closer together is identified as an additional source of interactions between them. Such entropic wrapping effects should be distinguished from the usual entropic configuration effects in semi-flexible polymers. Our results may be applicable to the problem of adsorption of proteins as well as synthetic nano-particles on semi-flexible polymers such as DNA.

cond-mat.soft↗

arXiv2012

The one dimensional Coulomb lattice fluid capacitor

The one dimensional Coulomb lattice fluid in a capacitor configuration is studied. The model is formally exactly soluble via a transfer operator method within a field theoretic representation of the model. The only interactions present in the model are the one dimensional Coulomb interaction between cations and anions and the steric interaction imposed by restricting the maximal occupancy at any lattice site to one particle. Despite the simplicity of the model, a wide range of intriguing physical phenomena arise, some of which are strongly reminiscent of those seen in experiments and numerical simulations of three dimensional ionic liquid based capacitors. Notably we find regimes where over-screening and density oscillations are seen near the capacitor plates. The capacitance is also shown to exhibit strong oscillations as a function of applied voltage. It is also shown that the corresponding mean field theory misses most of these effects. The analytical results are confirmed by extensive numerical simulations.

cond-mat.soft↗

arXiv2012

Optimal estimates of the diffusion coefficient of a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way of extracting diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate. Furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.

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