Related Experiment Videos
Power-law behavior in a cascade process with stopping events: a solvable model.
Ken Yamamoto1, Yoshihiro Yamazaki
1Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo, Japan.
Summary
This study introduces a stochastic fracture model yielding power-law distributions. The model shows that fracture probability influences the distribution exponent, with size-dependent probability leading to exponents below -1.
Area of Science:
- Physics
- Materials Science
- Geophysics
Background:
- Fracture mechanics often involves complex fragmentation processes.
- Stochastic models are crucial for understanding probabilistic events in material failure.
- Power-law distributions are observed in various natural and engineered systems.
Purpose of the Study:
- To develop and analytically solve a stochastic cascade fracture model.
- To investigate the influence of fracture probability on resulting power-law distributions.
- To explore the applicability of the proposed model to real-world phenomena.
Main Methods:
- Formulation of a stochastic model for cascade fracture.
- Inclusion of a probability of ceasing fracture at each stage.
- Analytical solution to derive power-law distributions.
- Analysis of cases with constant and size-dependent fracture probabilities.
Main Results:
- A power-law-like distribution is derived from the stochastic model.
- When fracture probability is constant, the exponent ranges from -1 to 0, dependent on probability and fracture point distribution.
- When fracture probability is size-dependent, the exponent is consistently less than -1, independent of fracture point distribution.
Conclusions:
- The proposed stochastic model successfully generates power-law distributions.
- Fracture probability, especially when size-dependent, significantly impacts the exponent of the power-law distribution.
- The model offers a framework for analyzing fragmentation phenomena with probabilistic elements.
Related Concept Videos
01:21Design Example: Creating a Hydraulic Model of a Dam Spillway
Scaled hydraulic models of dam spillways provide a practical way to replicate and study the intricate flow dynamics of these structures. Often built to a 1:15 ratio, these models allow for observing critical water behavior, such as velocity distribution, flow patterns, and energy dissipation.
01:25Modeling with Differential Equations
Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
01:29Exponential Equations with Logarithms: Problem Solving
In ecological studies, exponential models are often used to predict how populations grow over time under favorable conditions. These models assume that the growth rate is proportional to the current population, leading to continuous and compounding increases.The model expresses the population as a function of time, combining the initial population with a growth factor raised to an exponent involving the growth rate and time. To estimate how long it takes for a population to reach a specific...
01:26Exponential Equations for Modeling Growth
Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is the relative...
01:30Typical Model Studies
Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.
01:14Pharmacodynamic Models: Link Model and Systems Pharmacodynamic Model
The link model is a fundamental pharmacokinetic-pharmacodynamic (PK–PD) approach to account for delayed drug responses when the observed effect does not immediately correlate with the drug's plasma concentration peak. This delay is mathematically addressed by introducing an effect compartment concentration, Ce, which is kinetically linked to the plasma concentration, Cp, via a first-order rate constant, ke0. The linkage allows for a more accurate prediction of drug effects over time. A higher...