Brownian Motion - Science topic

@Neda Kaveh the images I uploaded were after they were left for 4 days and I didn't see much difference on day 5, the only difference was that I had more floating cells since they were overconfluent. I also checked the media and PBS and they were fine and I typically leave my media at 37 degrees for 30 mins or more before passaging

It is most probably mycoplasma contamination..

Yes

Bounded quadratic variation of a Brownian motion. where the supremum is taken over all possible partitions Π of the interval [0,T ] for all n. A function f is defined to have bounded variation if its total variation is finite. ... Brownian motion is almost surely nowhere differentiable.

Dear Omar!

Reliable measurements of non-spherical particles by the dynamic light scattering method are possible only when depolarized light is recorded. In this case, you can separate the contributions of translational and rotational diffusion to the autocorrelation function. A description of such measurements can be found in the article by Robert Pecora (https://www.researchgate.net/publication/226509433_Dynamic_Light_Scattering_Measurement_of_Nanometer_Particles_in_Liquids). For rod-shaped particles, one can try DLS measurements in oriented systems. That is, an external field (for example, an electric field) to orient the particles in a certain direction and determine the size as in the classical DLS. Perhaps, it will be possible to fix the difference, since for the rod the coefficients of translational diffusion are very different along and across the long axis, and rotational diffusion will be excluded (the axes of the particles are aligned in the same direction).

Good luck!

Best regards, Constantine Yerin

Nebras Mohamed

Particle size as a function of Brownian motion is determined by dynamic light scattering. If the particles are solid, they get approximately constant sizes. If the particles are from "soft matter", then difficulties arise. For measuring the size of surfactant micelles, see

The botanist Robert Brown, who is usually credited with the discovery of Brownian motion, discovered it while studying pollen grains of a plant suspended in water under a light microscope in 1827.

It's not possible to give an unambiguous answer. A neutral particle is made up of charged particles anyway. Water has an electrical dipole moment so it's the coupling between this dipole moment and the particle that matters. A neutral particle can only couple to the dipole moment of water through its dipole moment (if it doesn't vanish, as well-in which case it's the quadrupole moment that matters); an electrically charged particle can couple to the individual constituents of the dipole. So it can lose energy more easily and therefore one might say that it will be slower than a neutral particle that can't lose energy as easily, since it interacts less strongly with water molecules.

Hi Kaiser Ahmed Rocky

There are a number of factors and effects that can cause an irregular autocorrelation function to be measured.

In general a non exponential decay is caused by either non brownian motion being detected, or the average scattering intensity changing during the measurement. There are several potential causes for either of these.

The following webinar discusses how to interpret DLS data and may be of interest.

For your purpose you sould use the 2 phase nanofluid model

Prajal Chettri You get the diffusion coefficients directly and they can be plotted out - it's these that are transformed to PSD. Indeed in the Stokes-Einstein relationship there's a direct linear relation between hydrodynamic size and diffusion coefficient. so the plots will look identical (except that the x-axis will be different).

Thanks for your useful answers.

Apparently, you mean only external influences upon Brownian motion. The acceleration of the Brownian motion of particles in suspension is also influenced by factors such as shape (for example, if the particles have a long shape, then the translational Brownian motion is also superimposed by a rotational movement that increases the probability of collision of such particles; etc.); heterogeneity (in the sense of heterogeneity of the material); proportion of different particle sizes (depending on the characteristics of the suspension, different optimal proportions of particles occur; by the way, finding the optimal proportion of heterogeneous particle sizes under fixed other characteristics and conditions is a very interesting mathematical problem).

The above-listed factors, obviously, are not external factors of influence, – they are "internal" factors.

Further, from the external factors listed by you, the last two factors influencing the acceleration of Brownian motion.

The so-called “spacetime”, the fundamental basis of GRT is an abstract and esoteric geometrical/mathematical construct; supposedly with impossible physical, mechanical, metric etc. properties. This is a theory of mathematics but not a theory of physics!

For a more scientific and philosophical view of space and time please the following:

Fungi are not molecules, they are medium size airborne particles. You can't apply the term "molecular diffusion coefficient" for such particles. You may need to know the sedimentation rate.

I answer a long time after your question, Takumi Sase, but in case you are still working on it:

- The Hurst exponent of the standard Brownian motion (Bm) is 0.5. One can define fractional Brownian motions (fBm) of Hurst exponent H (in (0,1)). If H>0.5, the fBm is a fractional integral of order H-0.5 of a Bm. If H<0.5, the fBm is a fractional derivative of order 0.5-H of a Bm.

- The Bm is an integral of white noises (increments of a Bm are iid Gaussian variables, these increments are the white noise), so we couldsee the white noise as a derivative of Bm. The standard derivative is also "a fractional derivative of order 1". So, consistently with the previous paragraph, one could say that the Hurst exponent of the white noise is 0.5-1=-0.5.

- Alternatively, if one considers the white noise process X_t, the process is distributed in the same way at every time: X_t has the same distribution as X_s. Following Mandelbrot's definition of self-similarity (Y is H-self-similar, with H the Hurst exponent, if Y_t has the same distribution as (c^{-H})Y_{ct}), one concludes that X has a Hurst exponent equal to zero.

Galileo never thought that gravitation were going to permit a change of scale in the Universe and even he thought the inverse employing the effect between weight and bones resistance.

At present what we have is a map of what "could be the distribution of the dark matter in our Universe" using weak gravitational lensing. The scale changes don't play any role in this representations at all. The most important part corresponds to have a cross-correlated mass map based with maps of galaxy and cluster samples in the same dataset looking for the statistical significance. This is interesting but without any relationship with an scale physical consequence that were not known: on what scales are matter and geometry coupled by Einstein’s equations?. In standard cosmology, the FLRW metric is assumed to exactly describe the average growth of the Universe on arbitrarily large scales. However, in a generally inhomogeneous universe as described in timescape cosmology, this is no longer the case.

Dark matter follows to be nothing more than an hypothesis and what is a problem is if our gravitational models are enough for explaining the whole Universe making a comparition of such scale with ours.

Please go through the reference given below . This may help you to understand the problem you are facing

Biologicals. 2010 Mar; 38(2): 273–277.

Dear Researchers,

Have a nice weekend with your family....

Hi, it appears that no one has really answered your question, so I am going to take a stab at it and hopefully it helps. If you are using Esri products in the Geoprocessing tools there are the Spatial Analyst and Tracking Analyst extensions that are really helpful. You can build a model, through modeling building and test how it works. They do require some level of knowledge with ArcGIS and GIS. However, there are online tutorials for ArcGIS extensions, and YouTube videos, for free. I have seen in other forums people suggesting Hawths’ tool for spatial ecology, but I am not familiar with that software at all that I am aware of.

or if you read enough which most college students do check out the choices in YouTube videos

https://www.google.com/search?rlz=1C1AKJH_enUS704US704&ei=Ff6aXMHjIIGK5wK30KeIAQ&q=Tracking+analysis+site%3Ayoutube.com&oq=Tracking+analysis+site%3Ayoutube.com&gs_l=psy-ab.3...114159.118694..120182...0.0..0.66.931.15......0....1..gws-wiz.......0i71.BRqbCIRygBg

Also: a book is out on explaining Modeling suitability, movement and interaction

I hope this helps, Good luck with your project.

It's not advisable to filter lentivirus with a 0.22um filter - you'll lose some. That's why a 0.45um filter is generally recommended.

There are companies that will analyse cells and culture media for you to look for Mycoplasma and many other contaminants - that's probably the best way to know for sure.

Why did the wrong "entropy" appear ?

In summary , this was due to the following two reasons:

1) Physically, people didn't know Q=f(P, V, T).

2) Mathematically, people didn't know AΔB couldn‘t become AdB directely .

If people knew any one of them, the mistake of entropy would not happen in history.

Please read my paper and those answers of the questions related to my paper in my Projects.

The correlation plot is an exponential decay and thus is very low resolution (only 2 parameters needed to define it), so the '100 points' in your commentary doesn't imply more accuracy or precision - a better fit doesn't necessarily result. We also note that size standards in the < 100 nm range tend to be characterized by electron microscopy, so there is not a like-for-like comparison. We also note that the approximate size of a hydrogen atom is 73 pm (or about 0.07 nm), so the best we would hope to achieve would be quoting to +/- 0.1 nm. However, DLS measurements of materials like sucrose (sub-nm) by DLS are supported by other molecular data. See the attached.

We also have to be aware that we're relating a diffusion coefficient to size (via Stokes-Einstein) and thus covering stabilizers and surfactants are factored into the hydrodynamic size. And 'size' is an ambiguous term for irregular particles.....SAXS or electron microscopy shows the electron-dense region of the particle and thus will not consider or 'see' the adsorbed layers. I think 10 +/- 0.1 nm is probably the best we could hope to achieve (with all the notes and caveats above). However, we must be positive about the DLS ensemble technique - how else can we get information from billions of particles simultaneously with great sensitivity in the intensity distribution to the presence of tiny amounts of aggregate.

The certificate for the RM8011 standard from NIST is enlightening in this respect.

Hi,

I don't have any reference to give you, but, as I understand your question, you have two independent diffusion processes:

* There is a particle (2) diffusing with constant D_2 with respect to a medium.

* There is a medium (1) diffusing with constant D_1 with respect to the desired frame of reference (0).

* Let x_21 be the coordinate of the particle w.r.t. the medium.

* Let x_10 be the coordinate of the medium w.r.t. the frame of reference.

* The probability density function (PDF) for the particle w.r.t. the medium is

c_21(x_21, t)=1/(4(pi)D_2t)^1/2 e^( - (x_21)^2/4D_2t)

* The PDF for the medium w.r.t. the frame of refernce is

c_10(x_10, t)=1/(4(pi)D_1t)^1/2 e^( - (x_10)^2/4D_1t)

* The position of the particle w.r.t. the frame of reference is

x = x_21 + x_10 => x_10 = x - x_21

* If the diffusion processes are independent, we have

c(x, t) = integral c_21(x_21, t) * c_10(x - x_21, t) d x_21,

with the integral taken of the domain of x_21.

I don't know if there is an analytical solution to this convolution, but you can easily find out with Maple/Mathematica/Matlab.

Good luck,

/Stefan

It turns out that you can invert a Perlin noise perfectly rather easily, if the input is a Perlin itself. Or fit a set of gradient vectors in a least squares sense if the input is not generated from a Perlin procedure. Then the inverted gradients can be used to build distributions.

Why do not use the power spectrum method (PSD) to get both stiffness (k) and sensitivity(S)?

The PSD is calculated by Fourier transforming the QPD signal in Volts and is fitted with a lorentzian function to determine two constants: the plateau and the corner frequency, which define S and k.

Although EspressoMD also looks inviting, I'm currently going with OpenMM. This one has a fairly good userguide to start creating customised forcefields, defined in XML. Have made a simple set of CG-particles.

NB for future readers: As of writing, openMM matches molecule topologies to forcefield descriptions by matching bond networks with correct element names.

Docs:

Forums:

Hi,

You can stick to maximum likelihood estimation, and depending on the requirements of your project (speed,limited number of data entries etc. etc.) you can then adjust or search for a faster solution.

I would recommend to check the sde package in R and search for articles limilar to this one: https://arxiv.org/pdf/1408.2441.pdf

Best regards,

Stan

Dear Robert, in Newton-like cooling law models, some authors proposed an exponent = 4/3 for the difference of Temperatures (see attached paper, page 21). Gianluca

try using adaptive bayesian forecasting (ref Springer---Bayesian Forecasting and Dynamic Models

Authors: West, Mike, Harrison, Jeff )

Hi, the precision of the integration can be achieved by any classical integrator (ode45, dopri8, etc...) with variable step. On output  you will obtain the relative and absolute precisions you put on entry. When increasing the order of the integrator  more calculus will be done at each step but the number of steps required to achieve a given precision will be lower. This is the only difference. Thus, if your problem is not stiff you can use any classical ode solver whatever the complexity of your dynamics.

The paper that Yashawantha shares does not give an answer to this question, but does discuss the velocity of sedimentation, which is a process which should be considered in verifying that a dispersed system is stable. 

In terms of describing the Brownian motion of particles in suspension, MSD is generally rather than velocity as it makes sense to consider how far we may expect a particle to have moved by random events over time. Root mean square velocity may be calculated in describing the dynamics of gas particles, but there the dynamics are different. 

Using only accelerometers will not bring you anywhere. First you should correct for gravity, then the double integration will explode any inaccuracy. The solution is to combine accelerometers with other sensors, like gyroscopes and magnetic sensors, maybe a GPS receiver.

Such a device 'Xsens' is commercially available.

 Hi, perhaps, H=0.54 implies that the signal follows only the Wiener process, namly standard Brownian motion. So, I think you should do surrogate test.

I hope my suggestion will help you. :)

@Bablu K. Ghosh

Thank you for the references. The missing part of all these studies is indeed an absence of the formal analysis of the interaction of electromagnetic waves with the biological matter. An appropriate assessment should be performed using the Maxwell equations. You can find quite good discussion of the relevant methodology in

I hope that this direction of research could be accommodated in your study.

With my best regards,

Janusz Pudykiewicz

From your question I gather that you are not an expert in (pseudo)random numbers yet. If I am right, a good introduction is in the book "Numerical Recipees" (for any computer language) or in the book of Daan Frenkel and Berend Smit. There are more sources, but these two are easily accessible.The best I read to date is chapter 3 of Donald Knuth's volume 2: "Seminumerical algorithms" from the series "The Art of Computer Programming".

An answer to your question might be available at random.org where "real" random numbers are available. Whether that is sufficient for your purpose needs to be checked so you will have to do statistical tests to prove the correctness of your results. Some of such tests are described in the mentioned books.

Take a look at this thesis: http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=2440&context=etd

and Chapter 2 by Zhang in Nanomaterial: Impacts on Cell Biology and Medicine

(Most of what you need can be viewed via Google Books here without buying the book - you'll find the references to Smoluchowski and aggregation kinetics)

Dear Neetish,

I have no paper for your system, I do not know in detail, however, problem of flocculation goes back to the general basic balance of attractive and repulsive forces, which can be quite complex. If attractive forces dominate structure builds up leading to increased viscosity.

Hi Colin,

Contamination is the most challenging and commonly faced problem for all cell culturists. For identification of contamination, i.e. to be sure if at all there is something contaminating your cell culture you may try these:

1. Try not to discard the spent media but keep it in a plate (may be in a 60mm one), if the media turns yellow or cloudy even then, you know it is contamination! To determine what the contaminating organism is, microscopy is the best way.

2. Fix the cells and stain with DAPI. If you see DAPI staining in small dots outside nucleus, you know that it is staining the DNA of your contaminant.

3. Trypan Blue staining showing small dots with dark outline and lighter inner space indicates presence of contaminants.

4. For simple light microscopy, under 40X magnification, if you see black dots stuck on the plate which doesn't seem to go even if you change plate daily, and also has some kind of brownian motion, its bacterial cells that you are looking at.

To prevent contamination-

1. If you have already identified it to be mycoplasma, then there are many preventive measures which might help you. Like you can use 0.1 micron filter to filter sterilize your media and other stuffs of cell culture.

2. I heard Plasmocin from Invivogen is very effective to combat mycoplasma infection. You can follow the link for further information.

3. Also, even though your cells look healthy, their morphology or growth rate doesn't change, appearance of black dots indicate that you need to check for contamination. 

4. Colour change of phenol red is not sufficient enough to determine the presence of contaminant, as rate of colour change depends on the confluency and number of cells seeded. 

Hope it helps!

Hi

I believe all boils down to fundamental assumptions on length and time scales when formulating the wave equation. Simply put, you state a case where you ignore the micro and focus on the macro properties.

An acoustic wave propagates in a medium that can be described by gross quantities rather than at the molecular level. For things to meet this assumption the average gas particles need to bump into each other quite frequently. If memory serves me correctly, I believe for air at 20 c and ambient static pressure, the average particle contact time is in the ballpark 1E-8s. This would not be the case at near vacuum conditions.

I suggest you look at some books starting from the very fundamentals to better answer your questions, e.g. this excellent book

or this one by A.D. Pierce

If your interest lies in molecular vibration, I can tell you that the N2 molecule vibrates at one of its molecular natural frequencies and may affect acoustic properties. This is so when the N2 molecular vibration is taken up by an adsorber, such as active coal. The N2 molecule then sticks to the surface of the active coal and the loss of molecular vibration energy can be seen as acoustic absorption. The effect is visible up to 80-100 Hz for ordinary air and the phenomenon has been exploited in loudspeakers. http://kef.com/html/en/innovation/ace/index.html

Note the difference between the adsorber (active coal) providing acoustic absorption (damping, i.e. a conversion from vibration to heat).

Sincerely

Claes

Rafael,

I am answering your questions 1 and 3 based on my past imaging experience:

1) What parameters are important for recording a time lapse?

If you use bright field, then you record the positions vs. time of your particles (Xi, Yi, ti) for 2D diffusion or (Xi, Yi, Zi, ti) for 3D diffusion. If you use fluorescence then you record peak intensity spots vs time associated with the particles. Since your particles are micron sized you can go with either (if the particles are fluorescent); otherwise you go with bright field using a 100x objective of large numerical aperture NA=1.0-1.4 for good resolution).

3) How does one retrieve diffusion coefficient values from the tracks?

You collect sufficient mean square displacements (MSD) vs time lapse delta(t) for some 100-500 particles until the histogram appears having a smooth contour, and then calculate the average diffusion coefficient D for your particles based on MSD=4D*delta(t) for 2D diffusion or MSD=6D*delta(t) for 3D diffusion.

Here is a trivial paper of mine on the topic many years back, FYI:

Hope that helps.

Thanks once again Mr. Jose...

The thermocouples, which I am using, have the bead size of 1 - 2 mm, while the Brownian velocity of nanoparticles is about the order of manometers. Whether it is ok to use such type of thermocouples at nanoscale?

Some books

The Science of Fractal Images

Authors: Michael F. Barnsley, Robert L. Devaney, Benoit B. Mandelbrot, Heinz-Otto Peitgen, Dietmar Saupe, Richard F. Voss

Editors: Heinz-Otto Peitgen, Dietmar Saupe

ISBN: 978-1-4612-8349-2 (Print) 978-1-4612-3784-6 (Online)

Get Access......http://link.springer.com/book/10.1007/978-1-4612-3784-6

Dear Dr.: Lawler:

This is an interesting question, with several very interesting sub-questions. But there are a great number of concepts involved at each part, as mentioned. It is possible that an approach more constructive be based on a specific parts exploration. Then, at nanoparticles superparamagnetics, in addition to the superparamagnetism concepts, nanoparticles can be interacting ( high density of nanoparticle) or non-interacting, also can be single domain or non-single domains. At any combination, concept of critical radii of nanoparticle, total magnetic moment of nanoparticle and Exchange integral are realistic parameters.

In this way, each part of the major question is below reproduced and in the sequence each part of the answer is approached, as follow:

“I am wondering about how an aqueous suspension of superparamagnetic nanoparticles would behave if an externally applied uniform magnetic field were applied. With no magnetic field, I assume the particles will move in a random motion as they collide with each other and the water molecules--normal Brownian motion.”

Typically, fluids contain a very low nanoparticles concentration, this feature make impossible mechanical interaction between nanoparticles. Depending on the temperature and fluid medium density and viscosity, the phenomenon called Brownian motion can occurs with major or minor intensity, in fact another factor can be affect the Brownian motion as nanoparticle surface wettability and capability of the fluid of interact with surface from superficial charge-density of nanoparticle, consider see:

“But what if the particles were inside a solenoid type of apparatus? For example, the suspension was inside a glass capillary tube with a current carrying wire wrapped lengthwise around it. My question is, the instant you apply current and establish that magnetic field, how will the motion of the SP nanoparticles change? ”

It is necessary a clear vision of Magnetic behavior of matter and contribution of size effects or finite size effects on the magnetic-behaviour of the matter, consider see:

According to the Neel model there is a critical size of nanoparticles for that magnetic single domain state can be reached via Exchange interaction. In this sense, the superparamagnetic state can be further characterized as having a contribution of exchange interaction tending to null, but not null, by comparison with contribution of thermal energy.

A possible interpretation these characteristics phenomena is that a magnetic moment liquid ascribed to lattice atoms can be operational but its magnitude change (stable magnetic order is loosen) in the time domain due thermal effects or thermal fluctuation.

Seems that there are several models to the comprehension of the behavior of collection of nanoparticle based on a small set of assumption, someone complex intrinsically. The simplest model involves a collection of monodisperse of nanoparticles sizes and single domain nanoparticles.

In a broad sense, superparamagnetics nanoparticles will move in the fluid as a function of magnetization induced by magnetic field of solenoid. The movement of collection of particles depending on the direction of applied field, in fact at opposite direction is expected.

“Without the magnetic field, each particle has 6 degrees of freedom (translational motion about x, y, z axes and rotational motion also about x, y, z). If the x axis be taken as passing through the center of the horizontal capillary tube, then which of the 6 types of motion will be affected once the solenoid is activated?”

Well, I expect that understood the question. In fact, neither all nanoparticle exhibits spherical symmetry, typically oblate or more symmetric nanoparticle exist. Supposing that above discussed be satisfied, finite size effect, one domain (single domain) or two or more domains can be developed in a spherical nanoparticle. The magnetic moment of all atom belong to this domain will give a total magnetic moment with more or less contribution of exchange integral. After this, there is a vector magnetization that turn restricts the idea of several degrees of freedom. In this sense, the direction parallel and anti-parallel to the magnetization vector are now most relevant at nanoparticle since vector magnetic field stemming of solenoid is aligned to vector magnetization stemming of nanoparticles. Again, here, naturally, there is not magnetic interaction between nanoparticles.

“My thought is that translational motion will not change. But the particle will be restricted to only rotating about the x axis (aligning itself with the magnetic field) and no longer rotate in the y and z. Is that right?”

Is complicated to comment the non-existence of movement at perpendicular axes to the magnetization axis, but seems that is non-probable.

The paper of C. Stoner and E. P. Wohlfarth, as well as its models are basic to start an understanding the major question, consider see:

C. Stoner and E. P. Wohlfarth Trans. Magn. 27, 1991(reprinted)

More recently, consider see:

Also:

Best regards

Marcos Nobre

Hi,

I would clip the raster to the layer of the islands, reclassify the raster with one classes (above 0.3 and the rest as nodata) and then do a zonal statistics for the island features with the reclassified raster as input value. Apart from the various statistics, the "zonal statistics as table" tool also produces the area of the raster inside the feature.

"My distribution is sometimes skewed, but not always...."

This is a nice property, which means that ht-index can capture the change or dynamics of the distribution; see the following figure, in which the shape of the trees is changing, implying that the distribution of all their branches is changing.

In general the Brownian motion is but sort of oscillations about the equillibrium point, or a steady path, so upon applying time averaging, this effect should zero out. However there are two things that  come into play, so the entire picture changes. Using the term "Brownian motion" you mean random movements of particles w/o referring to any equillibrium or steady path, as these are unknown to you. Secondary blood, if not all body fluids, is everything but a newtonian fluid. It has a memory. So such a fluid remembers (just to ilustrate this) that this particle was "going left", although it now "goes to the right".  I do not believe this would effect the action of gravity force per se, but in combination with "crowding out of heavier particles" under action of a cetrifugal forces in same body of the mixture, this could produce an apparent buoyant force. My insights may be very limited from this point on, because I have no experience with non-newtonian fluids. There is a group in the UK called FOAM, they deal in advanced comoutational fluid dynamic and may already have some software to lead you further on.  RGDS. Jerzy Klimkowski   

I worked with platelets and was never able to see Brownian motion. They are flat disks in the resting state, and about 3 um in diameter. 

Dear Francesco,

As the the entries in the array A are complex numbers, I assume A[][].imag = 0 means set the imaginary part of the number to zero.

Steve

Flo,  The best way to check your results is to plot them. Use log-log coordinates and plot the outputs of DFA and other methods you are using. You should be able to spot possible problems with the code and also you should be able to verify the scaling range of your time-series. Please remember that once you get close to sample size and to the sample resolution (max scale and min scale respectively) you will encounter finite size effects which affect your scaling estimate. Different methods respond differently to these finite size effects. Furthermore, there is no 'ideal' method which would recover the 'true' scaling exponent - all estimates suffer from finite size effects. Choosing the range of scaling (window sizes) is a bit of an 'art'. Generally, you need to choose the largest possible range, but sill stay away from gross finite size effects. Please also check earlier discussion of related matters with scaling estimates: https://www.researchgate.net/post/How_accurate_we_can_estimate_the_Fractal_Dimension/1

    i wrote to the author and he corrected me, the function is timestamps. 

Moein, it is very rare that you have to get the answer using just one pass estimation. In principle, you can always make a rough estimate first and then choose the method which works best in that range of H. In real life, I always used at least two methods and compared the results, often using various parameter settings such as # of vanishing moments with wavelets or degree of detrending using DFA. Integration prior to calculating H is another choice you need to make depending on where your H is. There is really no simple answer to your question from my experience. Once you see your results converging, you are likely working in the right direction using the right method (settings). 

In the absolute rate theory, which may be applicable to Brownian motion due to its stochastic nature,    the reaction rate constant  K⁺ is given by K⁺=  exp(-DelG/RT),  where Del G  is the Gibbs free energy of the activated complex particle at the saddle point configuration. One can also write  K⁺ = Gama⁺ C⁺/ Gama_R C_R,  where  +  denotes activated complex quantities.  C  is density of the nano particle at the reactant state. Then the rate of reaction is given by

  Rate= (RT/Nh) C⁺=  (RT/Nh)  K⁺ Gama_R / Gama⁺  C_R.  

Dear Humam,

You are right. There is a connection between quantum phenomena and the classical ones. But I should like to stress, that the quantum phenomena can be explained from the viewpoint of classical dynamics. It is very important, because in this case there is no necessity to quantize all in the world. In particular, one does not need to consider quantum gravitation, which is considered by some theorists as the main problem of  the elementary particle theory.

In most sources, the Brownian Motion and the Wienner Process are the same things. However, in some sources the Wiener process is the standard Brownian motion while a general Brownian Motion is of a form αW(t) + β.

A Brownian  Motion or Wienner process, is both a Markov process and a martingale. These two properties are very different. In fact, they have little in common. A random process X_t, adapted to a filtration F_t, is a martingale with respect to the filtration if conditional expectation of the increment E( X_t - X_s | F_s) =0 for all t>s (conditional increment equals 0).

The same process has Markov property if 

E( X_t  | F_s) = E( X_t | X_s) =0 for all t>s (future distribution is independent of past).

There are a variety of techniques that you could use to measure the Brownian diffusion effects, including particle tracking microrheology and dynamic light scattering.

Whilst gravity will always be acting on your particles, the Brownian diffusion in the horizontal plane will not be effected by this, and similarly if you take measurements for a dilute suspension, inter particle interactions could also be ignored. 

Calculus ultimately relies on infinite precision, the differential being infinitessemal, i.e. the next number to zero. In many instances this causes no problems even if it is mathematical garbage. Some authors now state than their differential is a small number not an infinitessimal then procced using the normal methods of calculus. Where Nature disagrees then we end up with contradictions like the UV catastrophy. Since we do not know when Nature will not bowl us a googly we should always check that taking the differential to be arbtrary small actually works. This is seldom done so there is no literature on this. There is no a priori method of introducing indeterminacy to calculs but if indeterminacy exists in Nature, she will let us know if we ask the right questions. Planck set up his differential of action and increased the pecission of his calculations  he found that when da = h he got the correct spectrum but lost it again when da<h Thus he found the indetrminacy in his calculus. I have never seen any other method than try it and see.

Why don't you upload the article for easy visibility?

LAMMPS and GROMACS are popular simulation packages for molecular dynamics (MD) simulations. Both can be used to simulate Brownian Dynamics by using Langevin dynamics. See "fix langevin" in LAMMPS or "bd integrator" in GROMACS.

LAMMPS: lammps.sandia.gov

GROMACS: www.gromacs.org

There is a Gibbs energy barrier between two stable positions of a ion. To go from one stable position to another, the ion must overcome this barrier. This is possible because the ion is not frozen - it vibrates around its equilibrium position and each time it vibrates towards the barrier, there is a possibility of surpassing it. Let f₀ be the frequency with which the ion tries to overcome the barrier by vibrating towards the barrier (the so-called attempt frequency). The jump frequency f is then given by

in which G is the Gibbs energy of migration from one stable position to another (height of the barrier), k is the Boltzmann's constant and T is temperature. The exponential term corresponds to the probability of surpassing the barrier at each try.

The form of the equation above shows clearly that the jump frequency increases with temperature. And this is reinforced by the fact that G usually decreases as the temperature increases.

The effect of ion size and concentration on the jump frequency is via G. Usually, the smaller and lighter the ion, the lower is the barrier, resulting in greater jump frequencies. The effect of concentration of defects on G, on the other hand, is less obvious. High concentration of defects can lead to structural distortions and/or defect interactions, for instance, that can increase or reduce G.

One way to investigate the effect of concentration on G, and consequently on f, is by computational modeling. I recommend the GULP program, by Julian Gale.

PS: Even for a given ion with a given concentration in a fixed temperature, there may be more than one possible migration routes, with different Gs (leading to different fs), one for each route. Again, this can be investigated by computer simulations, as in the paper attached.

Wouldn't singular gold atoms react with the water to form some hydrated Au + or Au 2+ ions?

I guess if you do gold (nano) particles in water that potential for gold surfaces is already quite good.

I do not see any analytic papers on fractional ARIMA calculation. I agree with you that this is a gap. That is PhD work-to-be done. In my queueing paper, we conquered the M/M/r/K/K calculation. It may inspire your work.

I see. Thank you very much!

Different statistical distribution base :-)

There are, however, conservative force traps which also use laser beams - the dipole traps. A blue-detuned laser beam when focued on laser-cooled atoms can trap the atoms. This kind of trap is 'conservative' in the sense that the the flow of energy is both ways between the atoms and the laser beams. The optical force in such traps arises from the intensity gradient of the laser beam at the focus...Good Luck!

Dear Hamid,

I'd like to add some points about your question from a brief literature review point of view. A number of possible mechanisms have been proposed for the enhancement of effective thermal conductivity, which included Brownian motion effect, interfacial liquid layering, clustering and networking. We are facing 2 different ways: For nanofluids application under forced convective conditions, possible contributions to the complex behavior of the nanofluids may include the Brownian motion, particle rotation and its associated micro convection, particle migration and the resulting non-uniform property profile, and suppression or interruption of the boundary layer. The mechanisms associated with nanofluids under natural convective heat transfer conditions are expected to be quite different including particle-fluid slip and sedimentation of nanoparticles.

For more info about your no. 3 question, please look at our publication (http://dx.doi.org/10.1007/s10973-014-4048-0)

Good luck

In a celebrated paper published in 1905  by A.Einstein, the motion of pollen grains in water (i.e. Brownian motion) was calculated on the basis of molecular nature of water and random collisions of these molecules with the grains that results in a mean squared displacement proportional to time and diffusion constant, this paper historically proved the molecular nature of water and relation of their mean velocity with  temperature. On the other hand the definition of "temperature"  based on the motion (kinetic) of molecules and famous works conducted by Boltzmann and Maxwell had been shown that the assumption of atomic or molecular nature plus the random motion of them could obtain a consistent theory for definite description of  macroscopic thermodynamical parameters such as temperature, pressure, heat conduction and internal energies of systems compatible with thermodynamic laws.Moreover this assumptive motion has been observed directly by electron microscopy and STM etc.