Inelastic collapse of a ball bouncing on a randomly vibrating platform

Resetting by rescaling: Exact results for a diffusing particle in one dimension

In this paper, we study a simple model of a diffusive particle on a line, undergoing a stochastic resetting with rate r, via rescaling its current position by a factor a, which can be either positive or negative. For |a|<1, the position distribution becomes stationary at long times and we compute this limiting distribution exactly for all |a|<1. This symmetric distribution has a Gaussian shape near its peak at x=0, but decays exponentially for large |x|. We also studied the mean first-passage time (MFPT) T(0) to a target located at a distance L from the initial position (the origin) of the particle. As a function of the initial position x, the MFPT T(x) satisfies a nonlocal second order differential equation and we have solved it explicitly for 0≤a<1. For -1 Collapse

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Affiliation(s)

Marco Biroli LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
Yannick Feld Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 91191 Gif-sur-Yvette, France
Alexander K Hartmann Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
Satya N Majumdar LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
Grégory Schehr Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France

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First-passage area distribution and optimal fluctuations of fractional Brownian motion

We study the probability distribution P(A) of the area A=∫_{0}^{T}x(t)dt swept under fractional Brownian motion (fBm) x(t) until its first passage time T to the origin. The process starts at t=0 from a specified point x=L. We show that P(A) obeys exact scaling relation P(A)=D^{1/2H}/L^{1+1/H}Φ_{H}(D^{1/2H}A/L^{1+1/H}), where 0 Collapse

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Alexander K Hartmann Institut für Physik, Universitåt Oldenburg - 26111 Oldenburg, Germany
Baruch Meerson Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

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DNA breathing dynamics under periodic forcing: Study of several distribution functions of relevant Brownian functionals

In this paper, we study DNA breathing dynamics in the presence of an external periodic force by proposing and inspecting several probability distribution functions (PDFs) of relevant Brownian functionals which specify the bubble lifetime, reactivity, and average size. We model the bubble dynamics process by an overdamped Langevin equation of broken base pairs on the Poland-Scheraga free energy landscape. Introducing an effective time-independent description for timescales larger than T[over ̃]=2π/ω (where ω is the frequency of external periodic force) and using an elegant backward Fokker-Planck method we derive closed form expressions of several PDFs associated with such stochastic processes. For instance, with an initial bubble size of x_{0}, we derive the following analytical expressions: (i) the PDF P(t_{f}|x_{0}) of the first passage time t_{f} which specifies the lifetime of the DNA breathing process, (ii) the PDF P(A|x_{0}) of the area A until the first passage time, and it provides much valuable information about the average bubble size and reactivity of the process, and (iii) the PDF P(M) associated with the maximum bubble size M of the breathing process before complete denaturation. Our analysis is limited to two limits: (a) large bubble size and (b) small bubble size. We further confirm our analytical predictions by computing the same PDFs with direct numerical simulations of the corresponding Langevin equations. We obtain very good agreement of our theoretical predictions with the numerically simulated results. Finally, several nontrivial scaling behaviors in the asymptotic limits for the above-mentioned PDFs are predicted, which can be verified further from experimental observation. Our main conclusion is that the large bubble dynamics is unaffected by the rapidly oscillating force, but the small bubble dynamics is significantly affected by the same periodic force.

DNA breathing dynamics: analytic results for distribution functions of relevant Brownian functionals

We investigate DNA breathing dynamics by suggesting and examining several Brownian functionals associated with bubble lifetime and reactivity. Bubble dynamics is described as an overdamped random walk in the number of broken base pairs. The walk takes place on the Poland-Scheraga free-energy landscape. We suggest several probability distribution functions that characterize the breathing process, and adopt the recently studied backward Fokker-Planck method and the path decomposition method as elegant and flexible tools for deriving these distributions. In particular, for a bubble of an initial size x₀, we derive analytical expressions for (i) the distribution P(t{f}|x₀) of the first-passage time t{f}, characterizing the bubble lifetime, (ii) the distribution P(A|x₀) of the area A until the first-passage time, providing information about the effective reactivity of the bubble to processes within the DNA, (iii) the distribution P(M) of the maximum bubble size M attained before the first-passage time, and (iv) the joint probability distribution P(M,t{m}) of the maximum bubble size M and the time t{m} of its occurrence before the first-passage time. These distributions are analyzed in the limit of small and large bubble sizes. We supplement our analytical predictions with direct numericalsimulations of the related Langevin equation, and obtain a very good agreement in the appropriate limits. The nontrivial scaling behavior of the various quantities analyzed here can, in principle, be explored experimentally.

Dynamical properties of an harmonic oscillator impacting a vibrating wall

The dynamics of a spring-mass system under repeated impact with a vibrating wall is investigated using the static wall approximation. The evolution of the harmonic oscillator is described by two coupled difference equations. These equations are solved numerically, and in some cases exact analytical expressions have also been found. For a periodically vibrating wall, Fermi acceleration is only found at resonance. There, the average rebounding velocity increases linearly with the number of collisions. Near resonance, the average rebounding velocity grows initially with the number of collisions and eventually reaches a plateau. In the vicinity of resonance, the motion of the oscillator exhibits scaling properties over a range of frequency ratios. The presence of dissipation at resonance destroys the Fermi-acceleration process and induces scaling behavior similar to that at near resonance. For a moving wall with a random amplitude at collisions, Fermi acceleration is observed independently of the ratio between the wall and oscillator frequencies. In this case the average rebounding velocity grows with the square root of the number of collisions with the wall. Also, in this latter case, dissipation suppresses the Fermi-acceleration mechanism and induces a scaling behavior with the same universality class as that of the dissipative bouncing ball model with random external perturbations.

Completely inelastic ball

This Rapid Communication presents an analytical study of the bouncing of a completely inelastic ball on a vertically vibrated plate. The interplay of saddle-node and period-doubling bifurcations leads to an intricate structure of the bifurcation diagram with uncommon properties, such as an infinity of bifurcation cascades in a finite range of the control parameter Gamma. A pseudochaotic behavior, consisting in arbitrarily long and complex periodic sequences, is observed through this generic system.