scholarly journals Description of Chemical Transport in Laboratory Rock Cores Using the Continuous Random Walk Formalism

2020 ◽  
Vol 56 (9) ◽  
Author(s):  
Takeshi Kurotori ◽  
Christopher Zahasky ◽  
Sally M. Benson ◽  
Ronny Pini
Keyword(s):  
Random Walk ◽  
Chemical Transport ◽  
Rock Cores ◽  
Continuous Random Walk

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

A (not so) random walk through hydrological space and time

2021 ◽  
Author(s):  
Brian Berkowitz
Keyword(s):  
Random Walk ◽  
Percolation Theory ◽  
Statistical Physics ◽  
Chemical Transport ◽  
Self Organization ◽  
Philosophical Perspective ◽  
Guiding Principle ◽  
Patterns And Processes ◽  
Temporal Measures ◽  
The Relationship

<p>A key philosophical perspective in science is that nature obeys general laws. Identification of these laws involves integration of system conceptualization, observation, experimentation and quantification. This perspective was a guiding principle of John Dalton’s research as he searched for patterns and common behaviors; he performed a broad range of experiments in chemistry and physics, and he entered over 200,000 observations in his meteorological diary during a period of 57 years. In this spirit, we examine general concepts based largely on statistical physics – universality, criticality, self-organization, and the relationship between spatial and temporal measures – and demonstrate how they meaningfully describe patterns and processes of fluid flow and chemical transport in hydrological systems. We discuss examples that incorporate random walks, percolation theory, fractals, and thermodynamics in analyses of hydrological systems – aquifers, soil environments and catchments – to quantify what appear to be universal dynamic behaviors and characterizations.</p>

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (01) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain,obtained by restarting the original chain at a fixed state after each absorption. The limiting age,A(j), is the weak limit of the timegivenXn=j(n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems forA(J) (J →∞) are given for these examples.

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (1) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

The maximum and time to absorption of a left-continuous random walk

1976 ◽  
Vol 13 (3) ◽  
pp. 444-454 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Joint Distribution ◽  
Distribution Of The Maximum ◽  
Conditional Limit Theorem ◽  
Continuous Random Walk ◽  
Conditional Limit

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (01) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (2) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

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