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Thus, in summary a Brownian motion curve can be defined to be a set of random variables in a probability spacethat is characterized by the following three properties.

For all time h>0, the displacements X(t+h) – X(t) have Gaussian distribution.

The displacements X(t+h) – X(t), 0<t1<t2<… tn, are independent of previous distributions.

The mean displacement is zero.

A Brownian motion curve – time vs.x-coordinate of walk.
From a resultingcurve, it is evident that Brownian motion fulfills the conditions of the Markov property and can therefore be regarded as Markovian.In the field of theoretical probability, a stochastic process is Markovian if the conditional distribution of future states of theprocess is conditionally independent of that of its past states. In other words, given X(t), the values of X before time t areirrelevant in predicting the future behavior of X.

Moreover, the trajectory of X is continuous, and it is also recurrent, returning periodically to its origin at0. Because of these properties, the mathematical model for Brownian motion can serve as a sophisticated random number generator.Therefore, Brownian motion as a mathematical model is not exclusive to the context of random movement of small particles suspended influid; it can be used to describe a number of phenomena such as fluctuations in the stock market and the evolution of physicaltraits as preserved in fossil records.

When the simulated Brownian trajectory of a particle is plotted onto an x-y plane, the resulting curve can besaid to be self-similar, a term that is often used to describe fractals. The idea of self-similarity means that for every segmentof a given curve, there is either a smaller segment or a larger segment of the same curve that is similar to it. Likewise, afractal is defined to be a geometric pattern that is repeated at indefinitely smaller scales to produce irregular shapes andsurfaces that are impossible to derive by means of classical geometry.

Figure 5. The simulated trajectory of a particle in Brownian motion beginning atthe origin (0,0) on an x-y plane after 1 second, 3 seconds, and 10 seconds.Because of the fractal nature of Brownian motion curves, the properties ofBrownian motion can be applied to a wide variety of fields through the process of fractal analysis. Many methods for generatingfractal shapes have been suggested in computer graphics, but some of the most successful have been expansions of the randomdisplacement method, which generates a pattern derived from properties of the fractional Brownian motion model. Algorithms anddistribution functions that are based upon the Brownian motion model have been used to develop applications in medical imaging andin robotics as well as to make predictions in market analysis, in manufacturing, and in decision making at large.

Rectified brownian motion

A random Brownian “walk” method fractal.
In recent years, biomedical research has shown that Brownian motion may play a critical role in the transport ofenzymes and chemicals both into and out of cells in the human body.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti

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Source:  OpenStax, Nanomaterials and nanotechnology. OpenStax CNX. May 07, 2014 Download for free at http://legacy.cnx.org/content/col10700/1.13
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