Related Experiment Video
Updated: Aug 6, 2025
08:13
Using the Race Model Inequality to Quantify Behavioral Multisensory Integration Effects
Published on: May 10, 2019
6.4K
Counting of level crossings for inertial random processes: Generalization of the Rice formula
Jaume Masoliver1, Matteo Palassini1
1Department of Condensed Matter Physics and Institute of Complex Systems (UBICS) University of Barcelona, 08028 Barcelona, Catalonia, Spain.
Physical Review. E
|March 18, 2023
Summary
This study generalizes Rice
Area of Science:
- Physics
- Mathematics
- Stochastic Processes
Background:
- Level crossing counting is crucial for analyzing stochastic processes.
- Rice's approach provides a foundational method for this analysis.
- Generalizing these methods is essential for broader applications.
Purpose of the Study:
- To generalize the classical Rice formula for level crossings.
- To extend the formula to encompass all Gaussian processes.
- To analyze inertial stochastic processes in physics.
Main Methods:
- Reviewing and generalizing Rice's approach.
- Applying the generalized formula to specific second-order processes.
- Utilizing numerical simulations for illustration.
Main Results:
- A generalized Rice formula applicable to all Gaussian processes.
- Exact crossing intensities derived for Brownian motion, random acceleration, and noisy harmonic oscillators.
- Analysis of long- and short-time dependencies of crossing intensities.
Conclusions:
- The generalized Rice formula effectively quantifies level crossings for inertial stochastic processes.
- The findings provide exact solutions and insights into process dynamics.
- Numerical simulations validate the theoretical results.
Related Concept Videos
01:09Random Variables
12.8K
A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
12.8K
00:59Propagation of Uncertainty from Random Error
752
An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
752
01:49Probability Laws
41.1K
Overview
41.1K
01:24Reynolds Transport Theorem
1.3K
The Reynolds transport theorem provides a framework to relate the time rate of change of an extensive property within a system to that in a control volume, which is crucial for analyzing fluid dynamics. Extensive properties, such as mass, velocity, acceleration, temperature, and momentum, can be expressed in terms of the mass of a fluid portion. These properties are called extensive because they depend on the system's size, while intensive properties are their corresponding values per unit...
1.3K
01:14Central Limit Theorem
15.5K
The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
The sample size, n, that...
The sample size, n, that...
15.5K
01:04Random Error
1.0K
Random or indeterminate errors originate from various uncontrollable variables, such as variations in environmental conditions, instrument imperfections, or the inherent variability of the phenomena being measured. Usually, these errors cannot be predicted, estimated, or characterized because their direction and magnitude often vary in magnitude and direction even during consecutive measurements. As a result, they are difficult to eliminate. However, the aggregate effect of these errors can be...
1.0K