Large deviations in statistics of the convex hull of passive and active particles: A theoretical study

Abstract

We investigate analytically the distribution tails of the area A and perimeter L of a convex hull for different types of planar random walks. For N noninteracting Brownian motions of duration T we find that the large-L and A tails behave as $\mathcal{P}\left(L\right)\sim e^{-b_N{L}^2/DT}$ and $\mathcal{P}\left(A\right)\sim e^{-c_{N}A/DT}$, while the small-$L$ and $A$ tails behave as $\mathcal{P}\left(L\right)\sim e^{-d_{N}DT/L^2}$ and $\mathcal{P}\left(A\right)\sim e^{-e_{N}DT/A}$, where $D$ is the diffusion coefficient. We calculated all of the coefficients ($b_N,c_N,d_N,e_N$) exactly. Strikingly, we find that $b_N$ and $c_N$ are independent of N, for $N\ge 3$ and $N\ge 4$, respectively. We find that the large-L (A) tails are dominated by a single, most probable realization that attains the desired L (A). The left tails are dominated by the survival probability of the particles inside a circle of appropriate size. For active particles and at long times, we find that large-L and A tails are given by $\mathcal{P}\left(L\right)\sim e^{-T{\Psi }_N^{\text{per}}\left(L/T\right)}$ and $\mathcal{P}\left(A\right)\sim e^{-T{\Psi }_N^{\text{area}}\left(\sqrt{A}/T\right)}$ respectively. We calculate the large deviation functions ${\Psi }_N$ exactly and find that they exhibit multiple singularities. We interpret these as dynamical phase transitions of first order. We extended several of these results to dimensions $d>2$. Our analytic predictions display excellent agreement with existing results that were obtained from extensive numerical simulations.

15 pages, 9 figures

References in corpus (18)

• Novel type of phase transition in a system of self-driven particles
• The large deviation approach to statistical mechanics
• Motility-Induced Phase Separation
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Dynamic first-order phase transition in kinetically constrained models of glasses
• Activity driven fluctuations in living cells
• Dynamical symmetry breaking and phase transitions in driven diffusive systems
• Spatial extent of an outbreak in animal epidemics
• Current fluctuations in stochastic systems with long-range memory
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Convex Hulls of Random Walks: Large-Deviation Properties
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Nonequilibirum steady state for harmonically-confined active particles
• Active Brownian Particles in a Circular Disk with an Absorbing Boundary
• Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study
• Nonequilibrium steady state of trapped active particles
• Geometrical optics of large deviations of Brownian motion in inhomogeneous media
• Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process

Cited by in corpus (4)

• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Large deviations in statistics of the local time and occupation time for a run and tumble particle
• Nonequilibrium steady state of Brownian motion in an intermittent potential
• The Tracy-Widom distribution at large Dyson index