On an exit problem in a one-dimensional random walk in the diffusion approximation

1991 ◽  
Vol 69 (10) ◽  
pp. 1284-1285
Author(s):  
Amal K. Das
Keyword(s):  
Diffusion Coefficient ◽  
Random Walk ◽  
Diffusion Approximation ◽  
Brownian Particle ◽  
Linear Chain ◽  
One Dimensional ◽  
Random Walker ◽  
Exit Problem ◽  
Traversal Time

We address a nonstandard random-walk problem in which the random walker, a Brownian particle, enters a linear chain at x = 0 and exits at x = L under the constraint that allows the particle to return to an arbitrarily close neighbourhood of the entrance point, but does not allow the entrance point to be touched back. In the diffusion approximation, the traversal time T*, calculated in an unconventional way, is found to be T/3, where T is the usual diffusion traversal time, L2/2D, D being the diffusion coefficient.

Searching for a one-dimensional random walker

1974 ◽  
Vol 11 (01) ◽  
pp. 86-93 ◽  
Author(s):  
Bernard J. McCabe
Keyword(s):  
Random Walk ◽  
Wiener Process ◽  
One Dimensional ◽  
Random Walker ◽  
Search Plan ◽  
Crossing Time ◽  
Definition Of ◽  
First Crossing Time ◽  
Finite Moments ◽  
Bernoulli Random Walk

Let {xk } k ≧ − r be a simple Bernoulli random walk with x –r = 0. An integer valued threshold ϕ = {ϕ k } k≧1 is called a search plan if |ϕ k+1−ϕ k |≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ 1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.

Searching for a one-dimensional random walker

1974 ◽  
Vol 11 (1) ◽  
pp. 86-93 ◽  
Author(s):  
Bernard J. McCabe
Keyword(s):  
Random Walk ◽  
Wiener Process ◽  
One Dimensional ◽  
Random Walker ◽  
Search Plan ◽  
Crossing Time ◽  
Definition Of ◽  
First Crossing Time ◽  
Finite Moments ◽  
Bernoulli Random Walk

Let {xk}k ≧ − r be a simple Bernoulli random walk with x–r = 0. An integer valued threshold ϕ = {ϕk}k≧1 is called a search plan if |ϕk+1−ϕk|≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.

scholarly journals Anomalous Diffusion with an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 506
Author(s):  
Sho Nakade ◽  
Kazuki Kanki ◽  
Satoshi Tanaka ◽  
Tomio Petrosky
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Constant ◽  
Mean Square Displacement ◽  
Second Law Of Thermodynamics ◽  
Molecular Chain ◽  
Chain Model ◽  
Phase Mixing ◽  
One Dimensional ◽  
The One ◽  
Negative Diffusion

An interesting anomaly in the diffusion process with an apparently negative diffusion coefficient defined through the mean-square displacement in a one-dimensional quantum molecular chain model is shown. Nevertheless, the system satisfies the H-theorem so that the second law of thermodynamics is satisfied. The reason why the “diffusion constant” becomes negative is due to the effect of the phase mixing process, which is a characteristic result of the one-dimensionality of the system. We illustrate the situation where this negative “diffusion constant” appears.

A one-dimensional chloride-bridged PtII/PtIVmixed-valence complex with a 4-[(4-hydroxyphenyl)diazenyl]benzenesulfonate counter-ion

2015 ◽  
Vol 71 (12) ◽  
pp. 1033-1036 ◽  
Author(s):  
Nobuyuki Matsushita ◽  
Ayako Taira
Keyword(s):  
Valence State ◽  
Title Compound ◽  
Linear Chain ◽  
Mixed Valence ◽  
Chain Structure ◽  
Structural Parameter ◽  
One Dimensional ◽  
Square Planar ◽  
Cl Atoms ◽  
Linear Chain Structure

The title compound,catena-poly[[[bis(ethylenediamine-κ2N,N′)platinum(II)]- μ-chlorido-[bis(ethylenediamine)platinum(IV)]-μ-chlorido] tetrakis{4-[(4-hydroxyphenyl)diazenyl]benzenesulfonate} dihydrate], {[PtIIPtIVCl2(C2H8N2)4](HOC6H4N=NC6H4SO3)4·2H2O}n, has a linear chain structure composed of square-planar [Pt(en)2]2+(en is ethylenediamine) and elongated octahedraltrans-[PtCl2(en)2]2+cations stacked alternately, bridged by Cl atoms, along thebaxis. The Pt atoms are located on an inversion centre, while the Cl atoms are disordered over two sites and form a zigzag ...Cl—PtIV—Cl...PtII... chain, with a PtIV—Cl bond length of 2.3140 (14) Å, an interatomic PtII...Cl distance of 3.5969 (15) Å and a PtIV—Cl...PtIIangle of 170.66 (6)°. The structural parameter indicating the mixed-valence state of the Pt atom, expressed by δ = (PtIV—Cl)/(PtII...Cl), is 0.643.

One-dimensional magnetism of a linear chain compound containing yttrium(III) and a nitronyl nitroxide radical

1989 ◽  
Vol 28 (16) ◽  
pp. 3230-3234 ◽  
Author(s):  
Cristiano Benelli ◽  
Andrea Caneschi ◽  
Dante Gatteschi ◽  
Luca Pardi ◽  
Paul Rey
Keyword(s):  
Linear Chain ◽  
Nitroxide Radical ◽  
Nitronyl Nitroxide ◽  
Nitronyl Nitroxide Radical ◽  
One Dimensional ◽  
Chain Compound

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (4) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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