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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply

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