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Time-dependent processes are often analysed using the power spectral density (PSD), calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble-average. Frequently, the available experimental data sets are too small for such ensemble averages, and hence it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from $S(f,T)$, the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable, parametrized by frequency $f$ and observation-time $T$, for a broad family of anomalous diffusions---fractional Brownian motion (fBm) with Hurst-index $H$---and derive exactly its probability density function. We show that $S(f,T)$ is proportional---up to a random numerical factor whose universal distribution we determine---to the ensemble-averaged PSD. For subdiffusion ($H<1/2$) we find that $S(f,T)sim A/f^{2H+1}$ with random-amplitude $A$. In sharp contrast, for superdiffusion $(H>1/2)$ $S(f,T)sim BT^{2H-1}/f^2$ with random amplitude $B$. Remarkably, for $H>1/2$ the PSD exhibits the same frequency-dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for $H>1/2$ the PSD is ageing and is dependent on $T$. Our predictions for both sub- and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels, and by extensive simulations.
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We study the correlations between the maxima $m$ and $M$ of a Brownian motion (BM) on the time intervals $[0,t_1]$ and $[0,t_2]$, with $t_2>t_1$. We determine exact forms of the distribution functions $P(m,M)$ and $P(G = M - m)$, and calculate the moments $mathbb{E}{left(M - mright)^k}$ and the cross-moments $mathbb{E}{m^l M^k}$ with arbitrary integers $l$ and $k$. We show that correlations between $m$ and $M$ decay as $sqrt{t_1/t_2}$ when $t_2/t_1 to infty$, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient $rho(m,M)$, the power spectrum of $M_t$, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.
We analyze the joint distributions and temporal correlations between the partial maximum $m$ and the global maximum $M$ achieved by a Brownian Bridge on the subinterval $[0,t_1]$ and on the entire interval $[0,t]$, respectively. We determine three probability distribution functions: The joint distribution $P(m,M)$ of both maxima; the distribution $P(m)$ of the partial maximum; and the distribution $Pi(G)$ of the gap between the maxima, $G = M-m$. We present exact results for the moments of these distributions and quantify the temporal correlations between $m$ and $M$ by calculating the Pearson correlation coefficient.
The ``Brownian bees model describes an ensemble of $N$ independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep a constant number of particles. In the limit of $Nto infty$, the spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady state solution with a compact support. Here we study fluctuations of the ``swarm of bees due to the random character of the branching Brownian motion in the limit of large but finite $N$. We consider a one-dimensional setting and focus on two fluctuating quantities: the swarm center of mass $X(t)$ and the swarm radius $ell(t)$. Linearizing a pertinent Langevin equation around the deterministic steady state solution, we calculate the two-time covariances of $X(t)$ and $ell(t)$. The variance of $X(t)$ directly follows from the covariance of $X(t)$, and it scales as $1/N$ as to be expected from the law of large numbers. The variance of $ell(t)$ behaves differently: it exhibits an anomalous scaling $ln N/N$. This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm where only a few particles are present, give a significant contribution to the variance. We argue that the variance of $ell(t)$ can be obtained from the covariance of $ell(t)$ by introducing a cutoff at the microscopic time $1/N$ where the continuum Langevin description breaks down. Our theoretical predictions are in good agreement with Monte-Carlo simulations of the microscopic model. Generalizations to higher dimensions are briefly discussed.
The Brownian bees model describes a system of $N$ independent branching Brownian particles. At each branching event the particle farthest from the origin is removed, so that the number of particles remains constant at all times. Berestycki et al. (2020) proved that, at $Nto infty$, the coarse-grained spatial density of this particle system lives in a spherically symmetric domain and is described by the solution of a free boundary problem for a deterministic reaction-diffusion equation. Further, they showed that, at long times, this solution approaches a unique spherically symmetric steady state with compact support: a sphere which radius $ell_0$ depends on the spatial dimension $d$. Here we study fluctuations in this system in the limit of large $N$ due to the stochastic character of the branching Brownian motion, and we focus on persistent fluctuations of the swarm size. We evaluate the probability density $mathcal{P}(ell,N,T)$ that the maximum distance of a particle from the origin remains smaller than a specified value $ell<ell_0$, or larger than a specified value $ell>ell_0$, on a time interval $0<t<T$, where $T$ is very large. We argue that $mathcal{P}(ell,N,T)$ exhibits the large-deviation form $-ln mathcal{P} simeq N T R_d(ell)$. For all $d$ we obtain asymptotics of the rate function $R_d(ell)$ in the regimes $ell ll ell_0$, $ellgg ell_0$ and $|ell-ell_0|ll ell_0$. For $d=1$ the whole rate function can be calculated analytically. We obtain these results by determining the optimal (most probable) density profile of the swarm, conditioned on the specified $ell$, and by arguing that this density profile is spherically symmetric with its center at the origin.
At fast timescales, the self-similarity of random Brownian motion is expected to break down and be replaced by ballistic motion. So far, an experimental verification of this prediction has been out of reach due to a lack of instrumentation fast and precise enough to capture this motion. With a newly developed detector, we have been able to observe the Brownian motion of a single particle in an optical trap with 75 MHz bandwidth and sub-{AA}ngstrom spatial precision. We report the first measurements of ballistic Brownian motion as well as the first determination of the velocity autocorrelation function of a Brownian particle. The data are in excellent agreement with theoretical predictions taking into account the inertia of the particle and the surrounding fluid as well as hydrodynamic memory effects.
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