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Related Concept Videos

Random Variables01:09

Random Variables

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...
Basic Continuous Time Signals01:22

Basic Continuous Time Signals

Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function, collectively referred to as singularity functions. Singularity functions are characterized by discontinuities or discontinuous derivatives.
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Wald-Wolfowitz Runs Test I

The Wald-Wolfowitz test, also known as the runs test, is a nonparametric statistical test used to assess the randomness of a sequence of two different types of elements (e.g., positive/negative values, successes/failures). It examines whether the order of the elements in a sequence is random or if there is a pattern or trend present. This nonparametric test applies to any ordered data despite the population and sample data distribution, even if a higher sample size is available.
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Sampling Continuous Time Signal01:11

Sampling Continuous Time Signal

In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
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Basic Discrete Time Signals01:16

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The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
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Using a Real-Time Locating System to Measure Walking Activity Associated with Wandering Behaviors Among Institutionalized Older Adults
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Continuous-time random walks at all times.

Anatoly B Kolomeisky1

  • 1Department of Chemistry, Rice University, Houston, Texas 77005-1892, USA. tolya@rice.edu

The Journal of Chemical Physics
|December 23, 2009
PubMed
Summary

This study introduces a new theoretical approach for continuous-time random walks (CTRW), providing a comprehensive description of dynamics at all times. The method yields explicit expressions for dynamic quantities in various CTRW models.

Area of Science:

  • Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Continuous-time random walks (CTRW) are crucial for modeling diverse phenomena.
  • Existing theoretical models often focus on long-time dynamics, neglecting early-time behavior.

Purpose of the Study:

  • To develop a novel theoretical framework for CTRW applicable across all timescales.
  • To derive explicit expressions for dynamic quantities in various CTRW models.
  • To analyze the approach to stationary states and generalized fluctuation theorems.

Main Methods:

  • Utilized a generalized master equations approach.
  • Derived Laplace transforms for all dynamic quantities.
  • Applied the method to homogeneous, periodic, and CTRW models with irreversible detachments.

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Main Results:

  • Obtained explicit expressions for dynamic quantities for various CTRW models.
  • Provided an effective description of CTRW dynamics at all times.
  • Analyzed the approach to stationary states and generalized fluctuation theorems.

Conclusions:

  • The generalized master equations approach offers a complete description of CTRW dynamics.
  • This framework enhances understanding of anomalous diffusion and related phenomena.
  • The results have implications for statistical physics and complex systems research.