About Diffussion

Introduction

Fig.1 Concentration gradient.

Diffusion happens due to a gradient in a medium, for example if we add a drop of ink in water the ink will diffuse in the water to ultimately make an homogeneous mixture.

The Physical Process

When a inhomogeneity is set in a mix of two liquids the process of diffusion is set. Molecules of the more concentrated fluid diffuses until the concentration is uniform. Note that the process is not set by any kind of force but simply by a random process known as Brownian motion.

 
Brownian Motion

Fig.2 Brownian motion

A particle in a fluid is surrounded by many other particles which will interact with it though multiple collisions (see figure 2). Due to this constant bombardment the fluid particle will undergo an erratic movement throughout the space available. The size of every step is given by the mean free path (l) which is a measure of the distance covered between successive collisions, while the root-mean-squere speed is given by:

v(rms) ~ sqrt(kT/M)

where:

k: 1.3806503 × 10^-23 m² kg s^-2 K^-1 (Boltzmann constant)

T: Absolute temperature (K)
M: Mass of the particle

The proportionality constant will depend of the inner degrees of freedom of the molecule (vibrational, rotational, etc).

Diffusion

Fig. 3 Global flow to the right

Now lets consider a vessel  filled with two fluids A and B where fluid B is concentrated in a small region of the overall volume and fluid A is uniformly distributed in the vessel. You can think of fluid B as a drop of ink in a glass of water (fluid A).  


If c is the concentration of B at x (see Fig. 3) then, the concentrations at x - dx and x + dx will be given by:

c(-) = c(x)  + dx*dc/dx > c(x)                       (1)

c(+) = c(x) + dx*dc/dx < c(x)                      (2)


The amount of particles in the volume ΔV(-) contained between x - dx and x will be approximately N(-) = c(-) ΔV(-). Similarly, N(+) = c(+)ΔV(+) with N(-) > N(+). When a particle moves in ΔV(-) it can either move towards x or move away from x. In average, as many particles will be moving towards x as moving away from it in ΔV(-). The same arguments can be applied to ΔV(+). Therefore as many particles in ΔV(+) are moving away from x as moving towards it. But because ΔV(-) is more populated than ΔV(+)  as is inferred from the premise that c(-) > c(x), then more B particles are moving to the right of  x than to the left of x, i.e., there is a net flux of B particles moving towards the regions where the concentration of B is lower.

If the time between collisions is τ, and dx ~ l (mean-free-path),  then the flux towards the right would be approximately:

Φ ~ ΔcΔV/τ   (3)

Where Δc = dx*dc/dx, ΔV = ΔV(-) = ΔV(+) and equations (1) and (2) have been used.


Conclusion

Diffusion is the result of Brownian motion of the molecules of a fluid and not of any kind of forces appearing because of a concentration gradient. The speed of the diffusion process is given by the concentration difference between to regions, the amount of particles participating in the motion and the time between collisions. Note that higher collision rate (i.e., smaller τ) increases the speed of diffusion as collisions are at the core of the random nature of the movement of particles in a fluid.




 

   

   

CERN - OPERA Experiment Results

The Experiment

The CERN - OPERA (Oscillation Project with Emulsion-tRacking Apparatus) experiment was designed to measure in a straightforward way neutrino oscillations. The experiment was aimed to detect tau-neutrnos from the oscillation of mu-neutrinos during their travel from Geneva to Gran Sasso (L'aquila) in about 2.4ms. In OPERA, tau-leptons resulting from the interaction of tau-neutrinos will be observed in "bricks" of photographic emulsion films interleaved with lead plates.

Little has been discussed about neutrino transmutation so far, instead all of the attention has been focused on the apparently faster-than-light speed of the neutrinos (still to be confirmed by the scientific community by different experiments).

Relativistic Background

It is interesting to see how the laws of physics came under scrutiny after experiments involving the measurement of the space-time coordinates of a couple of related events, as happened with the Fizeau or Michaelson-Morley experiments.

The laws of electromagnetism predict the existence of a field which propagates at about 299,792,458 meters per second. H Lorentz discovered that the laws of electromagnetism where not invariant under Galilean transformations and investigated the type of transformations that would leave Maxwell's equations invariant. The final result was the Lorentz transformations.

The principle of relativity of Galileo states that:

       All laws are identical in all inertial systems of reference

From Maxwell equations it is deduced that the speed of light is finite and is the maximum speed at which electromagnetic interactions can propagate.

The combination of the principle of relativity with the finiteness of propagation of the electromagnetic field is called the principle of relativity of Einstein in contrast with the principle of relativity of Galileo, which was based on an infinite speed of propagation of interactions.

From the principle of relativity it follows in particular that the speed of light is the same in all inertial systems of reference.

Fundamental Forces

In nature, as far as we know there are four fundamental forces:

• Electromagnetic (responsible for electromagnetic phenomena)
• Nuclear Strong (responsible for binding nucleons together)
• Nuclear Weak (responsible for the radioactive decay of subatomic particles)
• Gravity (responsible for the gravitational interaction)

These four forces are independent in a deep sense. They can be studied independently because they are related to different types of processes.

Propagation Speed

When studying the electromagnetic field one can deduce that the field propagates at a velocity related to the electric and magnetic properties of the media. Special Relativity was developed while studying electromagnetic problems and it was accepted that the speed of light was the unreachable upper limit to the speed of all material particles as well as the speed of propagation of light signals.

The CERN - OPERA experiment has found (if the instrument where registering correctly) that the neutrino beam sent from Geneva to Gran Sasso travelled some 60ns faster than light signal.

It would really be surprising if:

1. The beam is proven to have travel faster than light (in the same medium)
2. The weak force can somehow be reduced to electromagnetic

This would mean a contradiction in the logic framework because in this case, we could produce interactions with a velocity superior than the maximum possible velocity for the propagation of interactions, which is absurd.

In any other case the discovery would only mean that the weak interaction travels faster than the electromagnetic interaction and the result of the experiment would be the first straightforward measurement for the speed of propagation of the weak interaction but logically does not create any conflict with the principle of relativity of Einstein. One just has to swap the electromagnetic field by the field under consideration (weak force in this case) and the framework is left unchanged.

Frames of Reference

A rather striking discovery would be if an interaction could somehow violate the principle of relativity which is the cornerstone of Special Relativity. This would open a door to a compete different type of physics where the inertial frame does play a role in the result of physical processes, but this has never been observed so far.

Conclusion

Whether the result of the experiment can be confirmed independently or not, the principle of relativity would not be challenged as far as the weak force cannot be reduced to the electromagnetic force.

The only reading from the result (if confirmed) would be that the weak interaction can travel faster than the electromagnetic interaction which by itself is not contradictory because they are different fields.

When studying problems of interactions, the speed of propagation for the considered field (electromagnetic, weak, strong or gravitational) should be the one used in the principle of relativity.

Food, Energy and Entropy.

Successful species

The success of species is related to it's capacity to extract and process information from it's environment in the most beneficial way and  maintaining this capacity through biological adaptation. Humans have adapted better than any other species, being able to modify the environment significantly in our own benefit. A key factor to this success is undoubtedly our highly developed brains which allows us to use language, reasoning, consciousness, etc.

The brain is connected to the external world (environment) through our senses (sight, touch, smell, taste and hearing). By processing signals coming from our senses our brain is capable to develop "consciousness" or "awareness" of what's around us and respond in a presumably coherent manner to these signals. Most of the processes executed by the brain such as recognizing a friend by looking at an image or grabbing the bus bar when it jumps over a big bump might seem trivial to most of us but they wouldn't be possible without a highly developed brain which is capable of processing great amounts of information and transforming it into "knowledge".

Knowledge and information are different things, the first is only reached after the processing of some amount of the latter. After this processing takes place, we became conscious or aware of some rule, fact, pattern, etc., being able to use this knowledge in a predictive way. Thus, for example, when we see grey clouds we know that it will possibly rain because we have experienced many times how grey clouds bring showers. The more information we can retain about something the better (more detailed) our knowledge about it could be. It might also happen that amounts of information reaching our senses never gets transformed in new knowledge due to noise, brain fatigue, distraction, etc. Redundant information never gets converted into knowledge but it might help to retain this knowledge thus avoiding erasure (to forget) from our conscious being.

We all know that food is important to us because we use it as main source of the energy needed to perform our daily physical and intellectual activities. If we are depraved of it for a long enough period time we would weaken and eventually die. Similarly (although not equal) to what happens to a car engine once all available fuel has been used. We see how both living creatures and machines need an external source of energy to function.

Entropy and Negative Entropy

At the absolute zero, the entropy of any substance is zero. This is the state of maximum order. In this state there is only one possible structure so

S = k_Bln1 = 0 

Where  k_B = 1.381 x 10^-23 Joules/Kelvin is the Boltzmann constant.

Therefore the state of the substance is certain. As we raise the temperature there will be more states compatible with the observed macroscopic state (not just one) and therefore for these states of temperature T > 0 K:

S = k_BlnW > 0 (W > 1)

Note that W represents the number of micro states compatible with the observed macroscopic state. Now the uncertainty of the state for the substance has been increased (higher disorder). Therefore, some of the information we had about the substance has been lost (we've gone from one possible state or outcome to several).

An isolated system or a system always evolves in a way that increases its entropy and after more or less time reaches a estate of maximum entropy (greatest disorder or maximum number of possible outcomes). Thus,  in the state of equilibrium there is a maximum number of micro states (outcomes) compatible with the observed macroscopic state. This natural tendency to chaotic states is inherent to all things.

Lets return to the example of of a substance at a temperature T > 0 K. As discussed before, in this state there are more than one micro states compatible with the observed macroscopic state. If we increase the temperature by heating up the substance then the number of micro states will also increase further resulting in an increase of entropy and also there will be a reduction of our knowledge about the state of the substance.  We can see how increasing the entropy of a substance increases its structural disorder.

Lets now consider the case of cooling down a substance by sucking heat out from it. As the temperature approaches zero kelvin, the number of possible micro states (possible outcomes) are reduced and so is the entropy. The process of cooling is equivalent to the absorption of negative entropy, which increases the information associated to the state substance (lower entropy means more certainty about the state of the substance). We can see a strong bond between negative entropy and information.

To quantify this information content we can consider the process of cooling a substance from a state where W micro states are compatible with the actual macro state at temperature T > 0 K to a state of temperature T' < T where W' micro states can be realized, with W' < W. The entropy of these states are:

S = k_BlnW

S' = k_BlnW'

and the information content associated to the process is:

I = S - S' = - ΔS

which is equal to the decrease in entropy of the substance (note that information is always positive or zero). We will see now that feeding upon negative entropy is how living organisms can maintain complex and highly organized states overcoming the natural tendency of things to reach equilibrium.

Feeding upon Negative Entropy

From thermodynamics we know that an isolated system tends to reach a structure of maximum entropy which is the most disordered structure. How is it possible from an statistical point of view the the faculty of a living organism of delaying this 'decay'? The answer is: by eating, drinking, breathing, etc. The technical term is metabolism. Originally the idea of metabolism was an exchange of materials but this is absurd. Any atom is as good as any other of the same kind. What would be gained by exchanging them? In the same manner the energy content of an adult organism is stationary (same as the material content). Since any calorie is as good as any other, a mere exchange of calories cannot be of great help.

Then what is contained in our food that keep us from death?

Every process that take place in nature results in an increase of entropy. Therefore a living organism constantly increases it's entropy and tend to approach states closer to thermal equilibrium (death). Therefore, living organisms can only stay alive by sucking negative entropy from it's environment. It is easy to see that what really fuels an organism is the negative entropy obtained from food, air, sun rays, etc. This negative entropy keep us from dying by driving entropy out of our systems and thus by keeping us in an fairly orderly state.

The key question from a nutritional point of view is how to quantify the negative entropy content available  in  different foods. Presumably highly processed foods that had gone through high heating and some chemical reactions (i.e., irreversible processes) will  increase their entropy contents and therefore decrease the negative entropy content according to the second principle of thermodynamics.