arXiv subjects

Naftali R. Smith

Publications and source records attributed to Naftali R. Smith.

At least 37 records · Page 2Linked to original sources

arXiv2023

Striking universalities in stochastic resetting processes

Given a random process $x(τ)$ which undergoes stochastic resetting at a constant rate $r$ to a position drawn from a distribution ${\cal P}(x)$, we consider a sequence of dynamical observables $A_1, \dots, A_n$ associated to the intervals between resetting events. We calculate exactly the probabilities of various events related to this sequence: that the last element is larger than all previous ones, that the sequence is monotonically increasing, etc. Remarkably, we find that these probabilities are ``super-universal'', i.e., that they are independent of the particular process $x(τ)$, the observables $A_k$'s in question and also the resetting distribution ${\cal P}(x)$. For some of the events in question, the universality is valid provided certain mild assumptions on the process and observables hold (e.g., mirror symmetry).

cond-mat.stat-mech↗

arXiv2022

Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions

We consider an overdamped run-and-tumble particle in two dimensions, with self propulsion in an orientation that stochastically rotates by 90 degrees at a constant rate, clockwise or counter-clockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness $μ$, and possibly diffuses. We find the exact time-dependent distribution $P\left(x,y,t\right)$ of the particle's position, and in particular, the steady-state distribution $P_{\text{st}}\left(x,y\right)$ that is reached in the long-time limit. We also find $P\left(x,y,t\right)$ for a "free" particle, $μ=0$. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically-independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.

cond-mat.stat-mech↗

arXiv2022

Nonequilibirum steady state for harmonically-confined active particles

We study the full nonequilibirum steady state distribution $P_{\text{st}}\left(X\right)$ of the position $X$ of a damped particle confined in a harmonic trapping potential and experiencing active noise, whose correlation time $τ_c$ is assumed to be very short. Typical fluctuations of $X$ are governed by a Boltzmann distribution with an effective temperature that is found by approximating the noise as white Gaussian thermal noise. However, large deviations of $X$ are described by a non-Boltzmann steady-state distribution. We find that, in the limit $τ_c \to 0$, they display the scaling behavior $P_{\text{st}}\left(X\right)\sim e^{-s\left(X\right)/τ_{c}}$, where $s\left(X\right)$ is the large-deviation function. We obtain an expression for $s\left(X\right)$ for a general active noise, and calculate it exactly for the particular case of telegraphic (dichotomous) noise.

cond-mat.stat-mech↗

arXiv2022

Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition

We consider the relaxation (noise-free) statistics of the one-point height $H=h(x=0,t)$ where $h(x,t)$ is the evolving height of a one-dimensional Kardar-Parisi-Zhang (KPZ) interface, starting from a Brownian (random) initial condition. We find that, at short times, the distribution of $H$ takes the same scaling form $-\ln\mathcal{P}\left(H,t\right)=S\left(H\right)/\sqrt{t}$ as the distribution of H for the KPZ interface driven by noise, and we find the exact large-deviation function $S(H)$ analytically. At a critical value $H=H_c$, the second derivative of $S(H)$ jumps, signaling a dynamical phase transition (DPT). Furthermore, we calculate exactly the most likely history of the interface that leads to a given $H$, and show that the DPT is associated with spontaneous breaking of the mirror symmetry $x \leftrightarrow -x$ of the interface. In turn, we find that this symmetry breaking is a consequence of the non-convexity of a large-deviation function that is closely related to $S(H)$, and describes a similar problem but in half space. Moreover, the critical point $H_c$ is related to the inflection point of the large-deviation function of the half-space problem.

cond-mat.stat-mech↗

arXiv2022

Large deviations in chaotic systems: exact results and dynamical phase transition

Large deviations in chaotic dynamics have potentially significant and dramatic consequences. We study large deviations of series of finite lengths $N$ generated by chaotic maps. The distributions generally display an exponential decay with $N$, associated with large-deviation (rate) functions. We obtain the exact rate functions analytically for the doubling, tent, and logistic maps. For the latter two, the solution is given as a power series whose coefficients can be systematically calculated to any order. We also obtain the rate function for the cat map numerically, uncovering strong evidence for the existence of a remarkable singularity of it that we interpret as a second order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps.

nlin.CD↗

arXiv2022

Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees

The ``Brownian bees'' model describes an ensemble of $N=$~const independent branching Brownian particles. The conservation of $N$ is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of $N \gg 1$. At long times, the particle density approaches a spherically symmetric steady state solution with a compact support of radius $\bar{\ell}_0$. However, at finite $N$, the radius of this support, $L$, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni {\em et al}., Phys. Rev. E. {\bf104}, 054131 (2021)]. It is proportional to $N^{-1}\ln N$ at $N\to\infty$. We investigate here the tails of the probability density function (PDF), $P(L)$, of the swarm radius, when the absolute value of the radius fluctuation $ΔL=L-\bar{\ell}_0$ is sufficiently larger than the typical fluctuations' scale determined by the variance. For negative deviations the PDF can be obtained in the framework of the optimal fluctuation method (OFM). This part of the PDF displays the scaling behavior: $\ln P\propto - N ΔL^2\, \ln^{-1}(ΔL^{-2})$, demonstrating a logarithmic anomaly at small negative $ΔL$. For the opposite sign of the fluctuation, $ΔL > 0$, the PDF can be obtained with an approximation of a single particle, running away. We find that $\ln P \propto -N^{1/2}ΔL$. We consider in this paper only the case, when $|ΔL|$ is much less than the typical radius of the swarm at $N\gg 1$.

cond-mat.stat-mech↗

arXiv2022

Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, $u(x,t=0)=W δ(x)$. We characterize the process by the heat, transferred to the right of a specified point $x=X$ by time $T$, $$ J=\int_X^\infty u(x,t=T)\,dx\,, $$ and study the full probability distribution $\mathcal{P}(J,X,T)$. The particular case of $X=0$ has been recently solved [Bettelheim \textit{et al}. Phys. Rev. Lett. \textbf{128}, 130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$ as a function of $X$ and $T$ has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate $\mathcal{P}(J,X,T)$ by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov-Shabat inverse scattering method. We also discuss asymptotics of $\mathcal{P}(J,X,T)$ which we extract from the exact solution, and also obtain by applying two different perturbation methods directly to the MFT equations.

cond-mat.stat-mech↗

arXiv2022

Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

We study the fluctuations of the area $A(t)= \int_0^t x(τ)\, dτ$ under a self-similar Gaussian process (SGP) $x(τ)$ with Hurst exponent $H>0$ (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate $r$. Typical fluctuations of $A(t)$ scale as $\sim \sqrt{t}$ for large $t$ and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of $A(t)$. In the long-time limit $t\to\infty$, we find that the full distribution of the area takes the form $P_{r}\left(A|t\right)\sim\exp\left[-t^αΦ\left(A/t^β\right)\right]$ with anomalous exponents $α=1/(2H+2)$ and $β= (2H+3)/(4H+4)$ in the regime of moderately large fluctuations, and a different anomalous scaling form $P_{r}\left(A|t\right)\sim\exp\left[-tΨ\left(A/t^{\left(2H+3\right)/2}\right)\right]$ in the regime of very large fluctuations. The associated rate functions $Φ(y)$ and $Ψ(w)$ depend on $H$ and are found exactly. Remarkably, $Φ(y)$ has a singularity that we interpret as a first-order dynamical condensation transition, while $Ψ(w)$ exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of $Φ(y)$ around the origin $y=0$ correctly describes the typical, Gaussian fluctuations of $A(t)$. Despite these anomalous scalings, we find that all of the cumulants of the distribution $P_{r}\left(A|t\right)$ grow linearly in time, $\langle A^n\rangle_c\approx c_n \, t$, in the long-time limit. For the case of reset Brownian motion (corresponding to $H=1/2$), we develop a recursive scheme to calculate the coefficients $c_n$ exactly and use it to calculate the first 6 nonvanishing cumulants.

cond-mat.stat-mech↗

arXiv2022

Counting statistics for non-interacting fermions in a rotating trap

We study the ground state of $N \gg 1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $Ω>0$. The support of the density of the Fermi gas is a disk of radius $R_e$. We calculate the variance of the number of fermions ${\cal N}_R$ inside a disk of radius $R$ centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and interesting behaviours in two different scaling regimes: (i) $Ω/ ω<1 $ and (ii) $1 - Ω/ ω= O(1/N)$, where $ω$ is the angular frequency of the oscillator. In the first regime (i) we find that ${\rm Var}\,{\cal N}_{R}\simeq\left(A\log N+B\right)\sqrt{N}$ and we calculate $A$ and $B$ as functions of $R/R_e$, $Ω$ and $ω$. We also predict the higher cumulants of ${\cal N}_{R}$ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime (ii), the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling $k$ successive Landau levels, as found in previous studies. Here, we show that ${\rm Var}\,{\cal N}_{R}$ is a discontinuous piecewise linear function of $\sim (R/R_e) \sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any $k$. We argue that a similar piecewise linear behavior extends to all the cumulants of ${\cal N}_{R}$ and to the entanglement entropy. We show that these results match smoothly at large $k$ with the above results for $Ω/ω=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of ${\rm Var}\,{\cal N}_{R}$ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the $z$ direction.

cond-mat.stat-mech↗

arXiv2022

Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process

We study the full distribution of $A=\int_{0}^{T}x^{n}\left(t\right)dt$, $n=1,2,\dots$, where $x\left(t\right)$ is an Ornstein-Uhlenbeck process. We find that for $n>2$ the long-time ($T \to \infty$) scaling form of the distribution is of the anomalous form $P\left(A;T\right)\sim e^{-T^μf_{n}\left(ΔA/T^ν\right)}$ where $ΔA$ is the difference between $A$ and its mean value, and the anomalous exponents are $μ=2/\left(2n-2\right)$, and $ν=n/\left(2n-2\right)$. The rate function $f_n\left(y\right)$, that we calculate exactly, exhibits a first-order dynamical phase transition which separates between a homogeneous phase that describes the Gaussian distribution of typical fluctuations, and a "condensed" phase that describes the tails of the distribution. We also calculate the most likely realizations of $\mathcal{A}(t)=\int_{0}^{t}x^{n}\left(s\right)ds$ and the distribution of $x(t)$ at an intermediate time $t$ conditioned on a given value of $A$. Extensions and implications to other continuous-time systems are discussed.

cond-mat.stat-mech↗

arXiv2021

Full counting statistics for interacting trapped fermions

We study $N$ spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index $β$. In the fermion model $β$ controls the strength of the interaction, $β=2$ corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the Fermi gas. We predict that for general $β$ the variance of ${\cal N}_{\cal D}$ grows as $A_β \log N + B_β$ for $N \gg 1$ and we obtain a formula for $A_β$ and $B_β$. This is based on an explicit calculation for $β\in\left\{ 1,2,4\right\} $ and on a conjecture that we formulate for general $β$. This conjecture further allows us to obtain a universal formula for the higher cumulants of ${\cal N}_{\cal D}$. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter $K = 2/β$, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter $β$. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

cond-mat.stat-mech↗

arXiv2021

Counting statistics for non-interacting fermions in a $d$-dimensional potential

We develop a first-principle approach to compute the counting statistics in the ground-state of $N$ noninteracting spinless fermions in a general potential in arbitrary dimensions $d$ (central for $d>1$). In a confining potential, the Fermi gas is supported over a bounded domain. In $d=1$, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the support. We show that the variance of ${\cal N}_{\cal D}$ grows as $N^{(d-1)/d} (A_d \log N + B_d)$ for large $N$, and obtain the explicit dependence of $A_d, B_d$ on the potential and on the size of ${\cal D}$ (for a spherical domain in $d>1$). This generalizes the free-fermion results for microscopic domains, given in $d=1$ by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem $\cal{D}$, in any dimension, supported by exact results for $d=1$.

cond-mat.stat-mech↗

arXiv2021

Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution

A conditioned stochastic process can display a very different behavior from the unconditioned process. In particular, a conditioned process can exhibit non-Gaussian fluctuations even if the unconditioned process is Gaussian. In this work, we revisit the Ferrari-Spohn model of a Brownian bridge conditioned to avoid a moving wall, which pushes the system into a large-deviation regime. We extend this model to an arbitrary number $N$ of non-crossing Brownian bridges. We obtain the joint distribution of the distances of the Brownian particles from the wall at an intermediate time in the form of the determinant of an $N\times N$ matrix whose entries are given in terms of the Airy function. We show that this distribution coincides with that of the positions of $N$ spinless noninteracting fermions trapped by a linear potential with a hard wall. We then explore the $N \gg 1$ behavior of the system. For simplicity we focus on the case where the wall's position is given by a semicircle as a function of time, but we expect our results to be valid for any concave wall function.

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

Approximate analytical solutions to the condensation-coagulation equation of aerosols

We present analytical solutions to the steady state injection-condensation-coagulation equation of aerosols in the atmosphere. These solutions are appropriate under different limits but more general than previously derived analytical solutions. For example, we provide an analytic solution to the coagulation limit plus a condensation correction. Our solutions are then compared with numerical results. We show that the solutions can be used to estimate the sensitivity of the cloud condensation nuclei number density to the nucleation rate of small condensation nuclei and to changes in the formation rate of sulfuric acid.

physics.ao-ph↗

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

The Airy distribution: experiment, large deviations and additional statistics

The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations - the Donsker-Varadhan formalism and the optimal fluctuation method - manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area, and measure its mean in the experiment. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, and we identify the reason for this coincidence.

cond-mat.stat-mech↗

arXiv2019

Geometrical optics of constrained Brownian motion: three short stories

The optimal fluctuation method -- essentially geometrical optics -- gives a deep insight into large deviations of Brownian motion. Here we illustrate this point by telling three short stories about Brownian motions, "pushed" into a large-deviation regime by constraints. In story 1 we compute the short-time large deviation function (LDF) of the winding angle of a Brownian particle wandering around a reflecting disk in the plane. Story 2 addresses a stretched Brownian motion above absorbing obstacles in the plane. We compute the short-time LDF of the position of the surviving Brownian particle at an intermediate point. Story 3 deals with survival of a Brownian particle in 1+1 dimension against absorption by a wall which advances according to a power law $x_{\text{w}}\left(t\right)\sim t^γ$, where $γ>1/2$. We also calculate the LDF of the particle position at an earlier time, conditional on the survival by a later time. In all three stories we uncover singularities of the LDFs which have a simple geometric origin and can be interpreted as dynamical phase transitions. We also use the small-deviation limit of the geometrical optics to reconstruct the distribution of \emph{typical} fluctuations. We argue that, in stories 2 and 3, this is the Ferrari-Spohn distribution.

cond-mat.stat-mech↗