Brownian Motion & Albert Einstein

EINSTEIN AND BROWNIAN MOTION

PRE-REQUISITE : A fundamental understanding of Brownian motion.

“No one before or since has widened the horizons of physics in so short a time as Einstein did in 1905”

--Abraham Pais in his biography of Einstein.

How has the analysis of Brownian motion & of its subsequent generalizations contributed to human knowledge?

            Even a minute part of the remarkably diverse list of applications that have emerged over the past 100 years is astonishing. The Brownian motion is relevant to varied fields of study like

n      Polymers in solution

n      Chemically reacting molecules

n      Neutrons in a nuclear reactor

n      Clouds

n      Sand piles

n      Quantum mechanics and quantum field theory

n      Fluctuations of the stock market

n      Avalanches

n      Fundamental issues in the theory of probability

n      Star clusters

n      Animal herds

n      Insect swarms

n      Techniques of computation

and so on. In 1900 Louis Bachelier had worked out the mathematical part that underlies the concept of Brownain motion in his doctoral thesis, in connection with an attempt to model the changing prices of shares in the stockmarket. Mathematicians such as A.N.Kolmogorov and Norbert Wiener has used his thesis as progenitor for many deep results in the theory of probability.

CONCLUSION:

Can we identify the most intense and sustained mental effort by a single person leading to the most profound results?

                        A unique answer cannot be given. Newton, Darwin and Einstein, when they scaled their respective heights, changed something forever. Their discoveries separate distinct eras in humankind’s understanding of the universe.

EINSTEIN’S THEORY OF RELATIVITY

EINSTEIN & SPACE - TIME:

1. He placed the physically measured notion of time on essentially the same footing as that of space.
2. Einstein’s radical advance, as we have said was to propose that time is also essentially, on the same footing as the three dimensions of space. Hence the notion of a 4-dimensional combined space – time.
3. Observers oriented differently in space-time would measure time intervals, by their respective clocks, differently from each other. This is known as relativistic time dilation.
4. There can be events in time whose ordering depends on the orientation of the observer in the space-time.
5. If one event can casually influence a second event, it actually turns out that the first will always be before the other for all observers however differently oriented they may be in space-time.

1. In any case, by viewing time on the same footing as space, Einstein’s proposal upset deep-seated notions of time such as the absolute nature of simultaneity.
2. It was Einstein’s insight that two observers who are moving with respect to each other with a uniform velocity should be thought of as oriented differently in space – time.
3. “Everything is Relative” is often the profoundly misleading conclusion drawn from the above observations on space and time.
4. Different observers in relative uniform motion may thus view events in space-time from different orientations , but nevertheless arrive at invariant conclusions about physical phenomena. Everything is not  relative.

CONCLUSION:

10. The year 1905 is thus quite a unique occasion, which is why Einstein’s discovery continues to be a lasting legacy to this day. Though there have been many new theories proposed in Physics ranging from the domain of sub-nuclear regime to inter-galactic scales, we have yet to see a modification to the kinematic framework proposed by Einstein.
11.  The postulates of the Special Theory of Relativity have become guiding principles in the formulation of new dynamical laws.
12. Einstein’s later discoveries in his study of gravitation further deepened the physicist’s conception of space and time.
13. It changed the idea of space-time as a passive arena for all events and rather made space-time itself a participant. The cosmological expansion of the universe is the most striking demonstration of this idea.

Einstein Albert Part - V

The origin and understanding of mass’ calculation

                        We let each segment of space-time keep track of how many Quarks and gluons it contains. We then treat these segments as an assembly of interacting subsystems.

                        In this context “we” means a collection of hard-working CPU’s of a large number of powerful computers in sync working at teraflop speeds for months at a time.They manage to calculate the mass of Quarks and Gluons.

                       

                        Quarks are subject, in particular, to Heisenberg’s Uncertainty Principle, which tries to imply that if you try to pin down their position too precisely, their momentum will be wildly uncertain. To support the possibility of large momentum, they must acquire large energy. In other words, it takes work to pin Quarks down. Wavicles want to spread out. So there is a competition between two effects. To cancel the color change completely we would like to put together the quark and antiquark  at precisely the same place; but those wavicles resist localization, so the cancellation comes at a price.

                        A number of stable compromise solutions can be found, where the quark and the antiquark (or three quarks) are brought close together but not perfectly coincident. Their distribution is described by Quantum mechanical wave functions. Each possible stable wave-patterns corresponds to – indeed, in a profound sense  it is --  a different kind of particle that you can observe. There are patterns for protons and neutrons, and for each entry in our whole Greek and Latin smorgasbord. Each pattern has some characteristic energy, because the color fields are not entirely cancelled and because the wavicles are somewhat localized. And that energy, through Einstein’s m=E/c²,  is the origin of mass.

CONCLUSION: I again emphasize that our understanding of mass is not complete still. The value of the electron mass, in particular, remains deeply mysterious even in our most advanced speculations about the grand unification of fundamental forces and string theory.

Reference:

1. A Brief History of Time by Stephen Hawking.
2. “THE ORIGIN OF MASS” a Cover Story in Frontline  in WYP 2005.

                        

Einstein Albert Part - IV

How do we know QCD is right?

            Experiment is the ultimate arbiter of scientific truth. Since quarks and gluons cannot be seen through naked eyes, we have to rely on experiments.

Let us see what happens in a LEP

A funda (wich is very essential for making any sense at all of what happens) à “ACCORDING TO THE PRINCIPLES OF QUANTUM MECHANICS, THE RESULT OF AN INDIVIDUAL COLLISION IS UNPREDICTABLE”.

Thus different results emerge. By making many repetitions, we can determine the probabilities for different outcomes. These probabilities encode basic information about the underlying fundamental interactions; according to Quantum Mechanics, they contain all the meaningful information.

E1 à QED eventsàlepton + antilepton

E2 à QCD events à quark + antiquark

Final State : Particle* and its antiparticle** moving rapidly in opposite directions. Particle*                                Antiparticle** #electron                           #positron #muon                               #antimuon #tau                                   #antitau (1 unit of –ve charge)  (1 unit of +ve                                          charge) * & ** are similar in properties and are called as  leptons.

Final State: 10 or more particles selected from a menu of pions, rho mesons, protons, anti protons & many more. These particles are made up of quarks and gluons which in other circumstances strongly interact with each other. (Further explanation about E2 is given below)

E₁ : The final states are very simple states. Once produced, any of these particles could  -- in the language of elementary acts – attach a photon using a QED hub, or alternatively, in physical terms, radiate a photon. The basic coupling of photon to a unit charge is fairly weak. Hence, each additional attachment is predicted to decrease the probability of the process being described, and so the most usual case is no attachment. Infact, the final state that includes a photon does occur, with about 1% of the rate of the particles simply scattering off each other ( and similarly for the other leptons). By studying the details of these 3 particle events, such as the probability for the photon to be emitted in different directions (the “antenna pattern”) and with different energy, we can check all aspects of our hypothesis for the elementary act. Let us call this first class of outcomes “QED events”

E₂ : Here, they make a smorgasbord of the Greek and Latin alphabet. It is such a mess that physicists have given up on trying to describe all the possibilities and their probabilities in detail.

            Some simple patterns emerge if we focus on the overall flow of energy and momentum.

            90% of the cases ----- particles always emerge moving out in opposite directions to each other. We say these are back-to-back jets.

            9% of the cases ------ particles flow in three directions.

            0.9% of the cases ----- particles flow in four directions.

            The remaining broad class of outcomes are called as “QCD Events”.

The QCD events and the QED events begin to look familiar. Indeed, the pattern of energy flow is qualitatively the same in both cases, that is, heavily concentrated in a few narrow jets.

QCD@	QED@@
Multiple jets are more common.	Multiple jets are less common.
The jets are sprays of several particles.	Here the jets are made up of a single particle.

Pre-requisites for understanding ^@ and ^@@  is “ASYMPTOTIC FREEDOM”.

ASYMPTOTIC FREEDOM:

Basic Concept: The probability for a fast-moving quark or gluon to radiate away some of its energy in the form of other quarks & gluons depends on whether this radiation is “hard” or “soft”.

Hard radiation is radiation that involves a substantial deflection of the particle doing the radiating, while soft radiation is radiation that does not cause such a deflection. Thus, hard radiation changes the flow of energy and momentum, while soft radiation merely distributes it among additional particles, all moving together. Asymptotic freedom says that hard radiation is rare, but soft soft radiation is not.

This distinction explains why on the one hand there are jets, and on the other hand why the jets are not single particles. A QCD event begins as the materialization of quarks and anti quarks, similar to how a QED event begins as the materialization of lepton-antilepton.

By studying the antenna patterns of the multi-jet QCD events we can check all aspects of our hypotheses for the underlying hubs. Just as for QED, such antenna patterns provide a wonderfully direct and incisive way to check the soundness of the elementary acts from which we construct QCD. “Testing QCD” is also known as “Calculating backgrounds”.

Result of Collisions at LEP:                                               

                                                              Event 1 (E₁)

                                                              Event 2 (E₂)

                                                            P(E₁) = P(E₂) = 0.5               

                                                

      

Fig (1) A three-jet event : The tracks of particles emerging from this high-energy collision at the LEP mark the direction set by the quark, an antiquark and a gluon. The probablility that a given jet pattern emerges depends on the relative angles between the jets and the total energies they carry in an intricate manner. QCD, the fundamental theory of these particles, allows us to predict this dependence precisely.

Einstein - Part III

What is Quantum Chromodynamics (QCD) ?

1.      It is the theory of quarks and gluons.

2.      It is the generalization of Quantum Electrodynamics (QED).

3.      BASIC CONCEPT OF QED à The response of photon to electric charge

à Emission of a photon by a charged particle

The elementary act combined with the electric and magnetic forces from atomic to cosmic scales determine the construction of various elements.

In this way all the content of Maxwell’s equations for radio waves and light, Schrodinger’s equation for atoms and chemistry & Dirac’s more refined version including spin – are encoded in QED.

QCD	QED
QCD  is a bigger domain than QED.
There are 3 kinds of charges in QCD. They are called as quarks. red green blue They are known as colours.	There is only one charge called as electric charge.

Every quark has one unit of one of the colour charges. Quarks also come in different flavours. The important two that play a major role in ordinary matter are ‘u’ and ‘d’.

u à up. ( u quarks à a unit of red charge)

d à down. (d quarks à a unit of green charge)

and so on, for a total of 6 different possibilities.

Instead of one phtoton that responds to electric charge, QCD has eight color gluons that can either respond to different colour charges or change one into another.

There exists a symmetry here which makes things less complicated.

e.g. If you interchange red with blue everywhere, you must still get the same rules.

                                          A two-jet event: The tracks of particles emerging from 

                                         the high-energy collision @ the large Electron Positron 

                                         Collider (LEP) @ the CERN laboratory near Geneva

                                         mark the directions set by an underlying quark & antiquark.

CONCLUSION:  QCD is faithfully encoded in a single elementary act & its symmetry cousins. Solving the equations of QCD mathematically can be very difficult, but if they are solved, the outcome is unambiguous.

Note: I welcome both positive and negative criticisms, since I think both can improve my writing a lot better. Pleased to rectify errors if found any.

See u all with the continuation of this in the next post.

Einstein - Part II

Mass has an origin?

            That a question makes grammatical sense does not guarantee that it is answerable, or even coherent. The concept of mass is one of the first things we discuss in our freshman mechanics class. i.e., Engineering Mechanics. Classical mechanics, is , literally, unthinkable without it. Newton’s second law of motion says that the acceleration of a body is given by dividing the force acting upon it by its mass. So a body without mass would not know how to move, because you would be dividing by zero. Also, in Newton’s Law of Gravity, the mass of an object governs the strength of the force it exerts. One cannot build up an object that gravitates, out of material that does not , so you cannot get rid of mass without getting rid of gravity.

            Finally, the most basic feature of mass in classical mechanics is that it is conserved. For example, when you bring together two bodies, the total mass is just the sum of the individual masses. This assumption is so deeply ingrained that it was not even explicitly formulated as a law. Altogether, in the Newtonian framework it is difficult to imagine what would constitute an “origin of mass” , or even what this phrase could possibly mean. In that framework mass just is what it is – a primary concept.

            Later developments in Physics make the concept of mass seem less irreducible. Einstein’s famous equation for the inter convertibility of mass and energy, already mentioned was the watershed. In modern particle accelerators, this possibility comes to life. For example, in a large Electron Positron Collider (LEP), at the CERN laboratory near Geneva, beams of electrons and anti electrons (positrons) were accelerated to enormous energies. Powerful, specially designed magnets controlled the paths of the particles, and caused them to circulate in opposite directions around a big storage ring. The path of these beams intersected at a few interaction regions, where collisions could occur. When a collision between a high – energy electron and a high – energy positron occurs, we often observe that many particles emerge from the event. The total mass of these particles can be 1000’s of times the mass of the original electron and positron. Thus mass has been created, physically, from energy.

            Having convinced ourselves that the question of the origin of mass might make sense, let us now come to grips with it, in the concrete form that it takes for ordinary matter. Ordinary matter is made from atoms. The mass of atoms is overwhelmingly concentrated in their nuclei. The surrounding electron are of course crucial for discussing for discussing how atoms interact with each other – and thus for chemistry, biology and electronics. But they provide less than a part in a thousand of the mass! Nuclei, which provide the lion’s share of mass, are made up of protons and neutrons. All these are familiar, well established story dating back to 70 years or more. Newer and perhaps less familiar, but by now no less well- established, is the next step: protons and neutrons are made from quarks and gluons. So most of the mass of matter can be traced , ultimately, back to quarks and gluons.

Einstein - Part I

EVERYDAY work on the frontiers of modern physics usually involves complex concepts and extreme conditions. We speak of quantum fields, entanglement or super symmetry, and analyze the ridiculously small or conceptualize the incomprehensibly large. Just as Willie Sutton famously explained that he robbed banks because “that’s where the money is”, so we do these things because “that’s where the Unknown is”. It is an amazing and delightful fact, however, that occasionally this sophisticated work gives answers to childlike questions about familiar things. The world of quarks and gluons, casts brilliant new light on one such childlike question: What is the origin of mass?

            This is an especially appropriate topic for the World Year of Physics, because it relates so closely to the circle of ideas around Albert Einstein’s most famous equation , E = mc^2 . That equation, written in that form, immediately suggests the possibility of converting small quantities of mass into large quantities of energy – a suggestion that was realized, of course, with the development of nuclear reactors and nuclear weapons. It is worth noting, however, that this is not the way the equation appears in Einstein’s original paper. In that paper you do not find E = mc^2 , but rather m = E/c^2. The difference is trivial algebraically, but profound conceptually, for the second (original) form of the equation suggests something quite different : the possibility to derive mass from energy. For a modern physicist , and even for Einstein in 1905, this sounds a deeper resonance. Energy appears a pervasive, primary concept in modern physics , and there is no real prospect of explaining it in terms of something more basic. For mass the situation is quite different. The title of Einstein’s paper is “Does the Inertia of a Body Depend Upon Its Energy Content?”. It shows that from the beginning, Einstein was thinking about questioning the foundations of fundamental physics, not making bombs. Modern physics, as I shall now explain in my next post, answers his question with a resounding “Yes!”