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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

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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

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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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