scholarly journals The lifetime of a random set

2005 ◽  
Vol 42 (02) ◽  
pp. 566-580
Author(s):  
Peter C. Kiessler ◽  
Kanoktip Nimitkiatklai
Keyword(s):  
Random Walk ◽  
Bounded Region ◽  
Random Set ◽  
Time Required ◽  
Number Of Particles

We consider the lifetimes of systems that can be modeled as particles that move within a bounded region in ℝ n . Particles move within the set according to a random walk, and particles that leave the set are lost. We divide the set into equal cells and define the lifetime of the set as the time required for the number of particles in one of the cells to fall below a predetermined threshold. We show that the lifetime of the system, given a sufficiently large number of particles, is Weibull distributed.

scholarly journals The lifetime of a random set

2005 ◽  
Vol 42 (2) ◽  
pp. 566-580 ◽  
Author(s):  
Peter C. Kiessler ◽  
Kanoktip Nimitkiatklai
Keyword(s):  
Random Walk ◽  
Bounded Region ◽  
Random Set ◽  
Time Required ◽  
Number Of Particles

We consider the lifetimes of systems that can be modeled as particles that move within a bounded region in ℝn. Particles move within the set according to a random walk, and particles that leave the set are lost. We divide the set into equal cells and define the lifetime of the set as the time required for the number of particles in one of the cells to fall below a predetermined threshold. We show that the lifetime of the system, given a sufficiently large number of particles, is Weibull distributed.

scholarly journals Speed-detachment tradeoff and its effect on track bound transport of single motor protein

2018 ◽  
Author(s):  
Mohd Suhail Rizvi
Keyword(s):  
Experimental Data ◽  
Random Walk ◽  
Power Law ◽  
Average Distance ◽  
Motor Protein ◽  
Fixed Duration ◽  
Biological Cells ◽  
Single Motor ◽  
Stochastic Nature ◽  
Time Required

AbstractThe transportation of the cargoes in biological cells is primarily driven by the motor proteins on filamentous protein tracks. The stochastic nature of the motion of motor protein often leads to its spontaneous detachment from the track. Using the available experimental data, we demonstrate a tradeoff between the speed of the motor and its rate of spontaneous detachment from the track. Further, it is also shown that this speed-detachment relation follows a power law where its exponent dictates the nature of the motor protein processivity. We utilize this information to study the motion of motor protein on track using a random-walk model. We obtain the average distance travelled in fixed duration and average time required for covering a given distance by the motor protein. These analyses reveal non-monotonic dependence of the motor protein speed on its transport and, therefore, optimal motor speeds can be identified for the time and distance controlled conditions.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (04) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Amplitudes of Composition Fluctuations in the Simplest Chemical Reaction Equilibrium

2004 ◽  
Vol 218 (9) ◽  
pp. 1033-1040 ◽  
Author(s):  
M. Šolc ◽  
J. Hostomský
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Numerical Study ◽  
Mean Amplitude ◽  
Time Interval ◽  
The Mean ◽  
Composition Fluctuations ◽  
Number Of Particles

AbstractWe present a numerical study of equilibrium composition fluctuations in a system where the reaction X1 ⇔ X2 having the equilibrium constant equal to 1 takes place. The total number of reacting particles is N. On a discrete time scale, the amplitude of a fluctuation having the lifetime 2r reaction events is defined as the difference between the number of particles X1 in the microstate most distant from the microstate N/2 visited at least once during the fluctuation lifetime, and the equilibrium number of particles X1, N/2. On the discrete time scale, the mean value of this amplitude, m̅(r̅), is calculated in the random walk approximation. On a continuous time scale, the average amplitude of fluctuations chosen randomly and regardless of their lifetime from an ensemble of fluctuations occurring within the time interval (0,z), z → ∞, tends with increasing N to ~1.243 N0.25. Introducing a fraction of fluctuation lifetime during which the composition of the system spends below the mean amplitude m̅(r̅), we obtain a value of the mean amplitude of equilibrium fluctuations on the continuous time scale equal to ~1.19√N. The results suggest that using the random walk value m̅(r̅) and taking into account a) the exponential density of fluctuations lifetimes and b) the fact that the time sequence of reaction events represents the Poisson process, we obtain values of fluctuations amplitudes which differ only slightly from those derived for the Ehrenfest model.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (4) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

scholarly journals On the tail of the branching random walk local time

2020 ◽  
Author(s):  
Omer Angel ◽  
Tom Hutchcroft ◽  
Antal Járai
Keyword(s):  
Random Walk ◽  
Local Time ◽  
Single Particle ◽  
Branching Random Walk ◽  
Exponential Moment ◽  
Offspring Distribution ◽  
Critical Branching Random Walk ◽  
Number Of Particles

Abstract Consider a critical branching random walk on $$\mathbb Z^d$$ Z d , $$d\ge 1$$ d ≥ 1 , started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

Total number of particles in a bounded region estimated directly with the nucleator: Granulosa cell number in ovarian follicles

1993 ◽  
Vol 168 (2) ◽  
pp. 724-731 ◽  
Author(s):  
Peter V. Bagger ◽  
Lotte Bang ◽  
Morten D. Christiansen ◽  
Hans Jørgen G. Gundersen ◽  
Lene Mortensen
Keyword(s):  
Granulosa Cell ◽  
Ovarian Follicles ◽  
Cell Number ◽  
Bounded Region ◽  
Number Of Particles

scholarly journals On the application of the depth-averaged random walk method to solute transport simulations

2019 ◽  
Vol 22 (1) ◽  
pp. 33-45 ◽  
Author(s):  
Fan Yang ◽  
Dongfang Liang ◽  
Xuefei Wu ◽  
Yang Xiao
Keyword(s):  
Random Walk ◽  
Solute Transport ◽  
Sampling Technique ◽  
Meshfree Method ◽  
Chaotic Mixing ◽  
Random Walk Method ◽  
Parametric Studies ◽  
Artificial Diffusion ◽  
Shallow Basin ◽  
Number Of Particles

Abstract Most numerical studies on solute mixing rely on mesh-based methods, and complicated schemes have been developed to enhance numerical stability and reduce artificial diffusion. This paper systematically studies the depth-averaged random walk scheme, which is a meshfree method with the merits of being highly robust and free of numerical diffusion. First, the model is used to solve instantaneous release problems in uniform flows. Extensive parametric studies are carried out to investigate the influences of the number of particles and the size of time steps. The predictions are shown to be independent of time steps but are sensitive to the particle numbers. Second, the model is applied to the solute transport problem along an estuary subject to extensive wetting and drying during tidal oscillations. Finally, the model is used to investigate the wind-induced chaotic mixing in a shallow basin. The effect of diffusion on the chaotic mixing is investigated. This study proposes a generic sampling method to interpret the output of the random walk method and highlights the importance of accurately taking diffusion into account in analysing the transport phenomena. The sampling technique also offers a guideline for estimating the total number of particles needed in the application.

scholarly journals Heavy-tailed branching random walks on multidimensional lattices. A moment approach

2020 ◽  
pp. 1-22
Author(s):  
Anastasiya Rytova ◽  
Elena Yarovaya
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Lattice Point ◽  
Positive Eigenvalue ◽  
Infinite Variance ◽  
Finite Variance ◽  
Branching Random Walks ◽  
Heavy Tailed ◽  
Number Of Particles

We study a continuous-time branching random walk (BRW) on the lattice ℤ d , d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point β c  > 0 is found for every d ≥ 1. In particular, if β > β c the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > β c called supercritical. Classification of the BRW treated as subcritical (β < β c ) or critical (β = β c ) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤ d and of the particle population on ℤ d according to the ratio d/α.

Sign in / Sign up

Export Citation Format

Share Document