# 4.1 Brownian motion  (Page 4/5)

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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.

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

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.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles