Click here for DISCOVER Magazine's article on Roger Penrose and his alternate Interpretation (as opposed to Copenhagen and Many-Worlds) of Quantum Mechanics, and detail regarding the Dust-particle Tabletop Experiment he proposes to prove he is correct, as well as current work on same.

The article is also currently on the newsstands/bookstores (through late March 2010) in DISCOVER Magazine's "Extreme Universe" special issue.

There are actually people trying to make that happen. They claim they need 5 years given the engineering challenges. The article was written in 2005. It is now 2010, so .....

Well? Where are we?

I can help, because I'm aware of the challenges (making the apparatus as vibration-free as possible, with constant temperature in a wind-free environment). I've worked on a Mach-Zehnder Interferometer with the necessary lasers, mirrors, beam-splitters, concrete floor, balloon tires, in a quiet, dark air-tight lab in a high-vacuum environment, etc.

I'd like to put my Mechanical Engineering knowledge AND experience to good use. If anyone knows who is currently doing these tests, please forward this webpage to them. Thanks in advance.

Sincerely,

Steve Colyer

## Saturday, January 30, 2010

## Friday, January 29, 2010

### Hořava-Lifshitz gravity

Interestingly, if Hořava-Lifshitz gravity is combined with Kaluza-Colyer 5-D(4S+1T) Toroidal-Cylinder Theory, all the problems of particle physics are solved, including the masses of the fundamentals and Dark Energy and Dark Matter as well (they're geometrical, as expected).

LOL, no they're not. But only because I haven't fully explored Kaluza-Colyer theory! :-)

What's Horava Gravity? Read here:

Hořava-Lifshitz gravity

From Wikipedia, the free encyclopedia

Amazingly, Hořava-Lifshitz gravity seems tailor made for Bohmian Mechanics. Together, they give the first and only self-consistent philosophical framework to reconcile quantum mechanics' intrinsic non-locality with gravity's low-energy effective symmetry: Lorentz invariance.

LOL, no they're not. But only because I haven't fully explored Kaluza-Colyer theory! :-)

What's Horava Gravity? Read here:

Hořava-Lifshitz gravity

From Wikipedia, the free encyclopedia

**Hořava-Lifshitz gravity**(or**Hořava gravity**) is a theory of quantum gravity proposed by Petr Hořava in 2009.^{[1]}It solves the problem of different concepts of time in quantum field theory and general relativity by treating the quantum concept as the more fundamental so that space and time are not equivalent (anisotropic). The relativistic concept of time with its Lorentz invariance emerges at large distances. The theory relies on the theory of foliations to produce its causal structure. It is related to topologically massive gravity and the Cotton tensor. It is a possible UV completion of general relativity. The novelty of this approach, compared to previous approaches to quantum gravity such as Loop quantum gravity, is that it uses concepts from condensed matter physics such as quantum critical phenomena. Hořava's initial formulation was found to have side-effects such as predicting very different results for a spherical Sun compared to a slightly non-spherical Sun, so others have modified the theory. Inconsistencies remain.^{[2]}^{[3]}Amazingly, Hořava-Lifshitz gravity seems tailor made for Bohmian Mechanics. Together, they give the first and only self-consistent philosophical framework to reconcile quantum mechanics' intrinsic non-locality with gravity's low-energy effective symmetry: Lorentz invariance.

## References

**^**Quantum Gravity at a Lifshitz Point, Petr Hořava, 2009**^**On the Extra Mode and Inconsistency of Hořava Gravity, D. Blas, O. Pujolàs, S. Sibiryakov, 2009**^**Pathological behaviour of the scalar graviton in Hořava-Lifshitz gravity, Kazuya Koyama, Frederico Arroja, 2009

## See also

## External links

## Thursday, January 28, 2010

### The Post-Humous Nobel Prizes in Phyiscs

Note to The Royal Swedish Academy: Although the following people are quite dead, we your adoring public, and in great appreciation of the great man that Alfred Nobel was and in your good service thereof, and in great respect to his Last Will and Testament and the intent of his Prizes, humbly request you extend tradition and give post-humous Nobel Prizes in Physics to the following people for their extraordinary accomplishments in Physics:

- Nikola Tesla, of Austrian Empire/United States, and Thomas Edison, of United States, for multiple contributions in Electromagnetism

- Lise Meitner, of Austria/Sweden and Fritz Strassmann, of Germany, for contributions to nuclear fission

- Albert Einstein, of Germany/Switzerland/United States, for The Special Theory of Relativity

- Albert Einstein, of Germany/Switzerland/United States, for The General Theory of Relativity

- Albert Einstein, of Germany/Switzerland/United States, for Brownian Motion

- Satyendra Nath Bose, of India, for Bose-Einstein Condensates

- Albert Einstein, of Germany/Switzerland/United States, for Bose-Einstein Condensates

- Albert Einstein, of Germany/Switzerland/United States, for EPR

Einstein already won for the Photoelectric Effect, but most would agree he deserved 5 more.

- Wolfgang Pauli, of Austria/United States, for Neutrinos

Pauli already won for his Exclusion Principle which launched Chemistry into the stratosphere, but he deserves another in our humble opinion

- Chung-Yao Chao, of China, for experiments leading to the recognition of the positron

- Boris Poldolsky, of Russia/United States, for EPR

- Nathan Rosen, of United States/Israel, for EPR

- Cesar Lattes, of Brazil, and Eugene Gardner, of United States, for discoveries regarding the pion for which their boss Cecil Powell won the award

- Julius Edgar Lilienfeld, of Austria/United States, and Oskar Heil of Germany/United States, for the groundwork that led to the invention of the transistor

- Chien-Shiung Wu, of China/United States, for disproving the law of conservation of parity and numerous contributions

- Ennackal Chandy George Sudarshan, of India/United States, and Robbert Marshak, of United States, for the V-A Theory of Weak Interactions (Sudershan is still alive)

- Ennackal Chandy George Sudarshan, of India/United States, for the Sudarshan diagonal representation (also known as Sudarshan-Glauber representation ) in quantum optics (Sudershan is still alive)

George Sudarshan currently holds the record of the most nominated Nobel Prize candidate alive who has yet to receive any Nobel Prize

- Yuval Ne'eman, of Israel, for his 1961 discovery of the classification of hadrons through the SU(3) flavour symmetry

- Jocelyn Bell Burnell, of United Kingdom, who discovered the first radio pulsars, with her thesis supervisor Antony Hewish, for which Hewish shared the NPP with Martin Ryle.

Burnell is still alive.

- George Gamow, of Russia/United States, for Quantum tunneling

- George Gamow, of Russia/United States, for Big Bang Cosmogony

- John Stewart Bell, of Northern Ireland, for Bell's Inequality Theorem

- Ralph Alpher, of United States, for predicting the cosmic microwave background radiation and working out in 1948 the underpinnings of the Big Bang theory

keep your eye on Horava gravity and therefore:

- David Bohm, of United States/Israel, for Hidden Variables Theory

We can think of dozens of others who are also deserving, but those are the ones who come first to mind.

We are open to discussion, and we humbly thank you Royals for your kind consideration.

## Wednesday, January 27, 2010

### Lisa Randall CAN'T SPELL "Supersymmetry"

**CONFLICT**A slide that Lisa Randall of Harvard used at a recent conference ( http://www.nytimes.com/2010/01/26/science/26essay.html ) to refer whimsically to issues and challenges facing physics.

From Wiki:

Did you mean:

*supersymmetry*

*?You may create the page "Supesymmetry", but consider checking the search results below to see whether it is already covered.*

- Supertrace

Martin, A Supesymmetry Primer, in Perspectives on supersymmetry, G. L. Kane, ed., p. 1-98 arXiv:hep-ph/9709356. S. Coleman and E.**...**5 KB (767 words) - 20:35, 10 October 2009

I'm just teasing Lisa, if you're reading this. I find your work fascinating, inspirational, and important. I'm sure it was a post-doc or grad student who made the error. ;-)

In more sobering news, I just found out that noted Cosmologist Andrew Lange of CalTech took his own life, at the far too young age of 53 (my age). Click on the following link for details of the sad news: Andrew Lange (1956-2010)

Rest in peace, Andrew.

### SOLITONS

**I'm zeroing in on that which I wish to specialize in for the next 22 years before the synapses in my brain begin to widen and dementia** begins. Since I've narrowed my search down to Nonlinear Dynamics, check, with N=Continuum, check, this is the first of several candidates:

### SOLITONS

### From Wikipedia, the free encyclopedia

For other uses, see Soliton (disambiguation).

In mathematics and physics, a **soliton**is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (The term "dispersive effects" refers to a property of certain systems where the speed of the waves varies according to frequency.) Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation".

## Definition

A single, consensus definition of a soliton is difficult to find. Drazin and Johnson (1989) ascribe 3 properties to solitons:^{[1]}

- They are of permanent form;
- They are localised within a region;
- They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift.

*soliton*for phenomena that do not quite have these three properties (for instance, the 'light bullets' of nonlinear optics are often called solitons despite losing energy during interaction).

## Explanation

Dispersion and non-linearity can interact to produce permanent and localized wave forms. Consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies. Since glass shows dispersion, these different frequencies will travel at different speeds and the shape of the pulse will therefore change over time. However, there is also the non-linear Kerr effect: the refractive index of a material at a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the dispersion effect, and the pulse's shape won't change over time: a soliton. See soliton (optics) for a more detailed description.Many exactly solvable models have soliton solutions, including the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine-Gordon equation. The soliton solutions are typically obtained by means of the inverse scattering transform and owe their stability to the integrability of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research.

Some types of tidal bore, a wave phenomenon of a few rivers including the River Severn, are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the undersea internal waves, initiated by seabed topography, that propagate on the oceanic pycnocline. Atmospheric solitons also exist, such as the Morning Glory Cloud of the Gulf of Carpentaria, where pressure solitons travelling in a temperature inversion layer produce vast linear roll clouds. The recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons.

A topological soliton, or topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution." Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise almost always because the differential equations must obey a set of boundary conditions, and the boundary has a non-trivial homotopy group, preserved by the differential equations. Thus, the differential equation solutions can be classified into homotopy classes. There is no continuous transformation that will map a solution in one homotopy class to another. The solutions are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the screw dislocation in a crystalline lattice, the Dirac string and the magnetic monopole in electromagnetism, the Skyrmion and the Wess-Zumino-Witten model in quantum field theory, and cosmic strings and domain walls in cosmology.

## History

In 1834, John Scott Russell describes his*wave of translation*.

^{[nb 1]}The discovery is described here in Scott Russell's own words:

^{[nb 2]}

*"I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation"*.

^{[2]}

Scott Russell spent some time making practical and theoretical investigations of these waves. He built wave tanks at his home and noticed some key properties:

- The waves are stable, and can travel over very large distances (normal waves would tend to either flatten out, or steepen and topple over)
- The speed depends on the size of the wave, and its width on the depth of water.
- Unlike normal waves they will never merge – so a small wave is overtaken by a large one, rather than the two combining.
- If a wave is too big for the depth of water, it splits into two, one big and one small.

^{[nb 3]}In 1895 Diederik Korteweg and Gustav de Vries provided what is now known as the Korteweg–de Vries equation, including solitary wave and periodic cnoidal wave solutions.

^{[3]}

^{[nb 4]}

In 1965 Norman Zabusky of Bell Labs and Martin Kruskal of Princeton University first demonstrated soliton behaviour in media subject to the Korteweg–de Vries equation (KdV equation) in a computational investigation using a finite difference approach. They also showed how this behavior explained the puzzling earlier work of Fermi, Pasta and Ulam.

In 1967, Gardner, Greene, Kruskal and Miura discovered an inverse scattering transform enabling analytical solution of the KdV equation. The work of Peter Lax on Lax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems.

## Solitons in fiber optics

*See also Soliton (optics)*

Lists of miscellaneous information should be avoided. Please relocate any relevant information into appropriate sections or articles. (August 2009) |

^{[4]}

In 1973, Akira Hasegawa of AT&T Bell Labs was the first to suggest that solitons could exist in optical fibers, due to a balance between self-phase modulation and anomalous dispersion. Also in 1973 Robin Bullough made the first mathematical report of the existence of optical solitons. He also proposed the idea of a soliton-based transmission system to increase performance of optical telecommunications.

Solitons in a fiber optic system are described by the Manakov equations.

In 1987, P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy, from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a dark soliton, in an optical fiber.

In 1988, Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the Raman effect, named for the Indian scientist Sir C. V. Raman who first described it in the 1920s, to provide optical gain in the fiber.

In 1991, a Bell Labs research team transmitted solitons error-free at 2.5 gigabits per second over more than 14,000 kilometers, using erbium optical fiber amplifiers (spliced-in segments of optical fiber containing the rare earth element erbium). Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses.

In 1998, Thierry Georges and his team at France Telecom R&D Center, combining optical solitons of different wavelengths (wavelength division multiplexing), demonstrated a data transmission of 1 terabit per second (1,000,000,000,000 units of information per second).

For some reasons, it is possible to observe both positive and negative solitons in optic fibre. However, usually only positive solitons are observed for water waves since any attempt to create a wave of depression results in a train of oscillatory waves. (A positive soliton is related to a positive sech

^{2}profile and a negative soliton is connected to a wave profile of the form −sech

^{2}.)

In 2000, Cundiff predicted the existence of a vector soliton in a birefringence fiber cavity passively mode locking through SESAM. The polarization state of such a vector soliton could either be rotating or locked depending on the cavity parameters.

^{[5]}

In 2008, D.Y.Tang

*et al.*observed a novel form of higher-order vector soliton from the perspect of experiments and numerical simulations. Different types of vector solitons and the polarization state of vector solitons have been investigated by his group.

^{[6]}

## Solitons in magnets

In magnets, there also exist different types of solitons and other nonlinear waves.^{[7]}These magnetic solitons are an exact solution of classical nonlinear differential equations — magnetic equations, e.g. the Landau-Lifshitz equation, continuum Heisenberg model, Ishimori equation, nonlinear Schrodinger equation and so on.

## Bions

This section requires expansion. |

*bion*.

In field theory Bion usually refers to the solution of the Born–Infeld model. The name appears to have been coined by G.W. Gibbons in order to distinguish this solution from the conventional soliton, understood as a

*regular*, finite-energy (and usually stable) solution of a differential equation describing some physical system.

^{[8]}The word

*regular*means a smooth solution carrying no sources at all. However, the solution of the Born-Infeld model still carries a source in the form of a Dirac-delta function at the origin. As a consequence it displays a singularity in this point (although the electric field is everywhere regular). In some physical contexts (for instance string theory) this feature can be important, which motivated the introduction of a special name for this class of solitons.

On the other hand, when gravity is added (i.e. when considering the coupling of the Born–Infeld model to General Relativity) the corresponding solution is called

*EBIon*, where "E" stands for "Einstein".

## See also

- compacton, a soliton with compact support
- freak waves may be a related phenomenon.
- oscillons
- peakon, a soliton with a non-differentiable peak
- soliton (topological)
- non-topological soliton,in Quantum Field Theory
- Q-ball a non-topological soliton
- soliton (optics)
- soliton model of nerve impulse propagation
- topological quantum number
- sine-Gordon equation
- nonlinear Schrödinger equation
- vector soliton
- soliton distribution

## Notes

**^**"Translation" here means that there is real mass transport, although it is not the same water which is transported from one end of the canal to the other end by this "Wave of Translation". Rather, a fluid parcel acquires momentum during the passage of the solitary wave, and comes to rest again after the passage of the wave. But the fluid parcel has been displaced substantially forward during the process – by Stokes drift in the wave propagation direction. And a net mass transport is the result. Usually there is little mass transport from one side to another side for ordinary waves.**^**This passage has been repeated in many papers and books on soliton theory.**^**Lord Rayleigh published a paper in Philosophical Magazine in 1876 to support John Scott Russell's experimental observation with his mathematical theory. In his 1876 paper, Lord Rayleigh mentioned Scott Russell's name and also admitted that the first theoretical treatment was by Joseph Valentin Boussinesq in 1871. Joseph Boussinesq mentioned Russell's name in his 1871 paper. Thus Scott Russell's observations on solitons were accepted as true by some prominent scientists within his own life time of 1808–1882.**^**Korteweg and de Vries did not mention John Scott Russell's name at all in their 1895 paper but they did quote Boussinesq's paper of 1871 and Lord Rayleigh's paper of 1876. The paper by Korteweg and de Vries in 1895 was not the first theoretical treatment of this subject but it was a very important milestone in the history of the development of soliton theory.

## References

### Inline

**^**Drazin & Johnson (1989) p. 15.**^**J. Scott Russell. Report on waves,*Fourteenth meeting of the British Association for the Advancement of Science*, 1844.**^**Korteweg, D.J.; de Vries, G. (1895), "On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Waves",*Philosophical Magazine***39**: 422–443**^**"Photons advance on two fronts",*EETimes.com*, October 24, 2005.**^**Cundiff, S.T.; Collings, B.C.; Akhmediev, N.N.; Soto-Crespo, J.M.; Bergman, K.; Knox, W.H. (1999), "Observation of Polarization-Locked Vector Solitons in an Optical Fiber",*Physical Review Letters***82**(20): 3988, doi:10.1103/PhysRevLett.82.3988**^**Tang, D.Y.; Zhang, H.; Zhao, L.M.; Wu, X. (2008), "Observation of high-order polarization-locked vector solitons in a fiber laser",*Physical Review Letters***101**(15): 153904, doi:10.1103/PhysRevLett.101.153904**^**A.M., Kosevich; Gann, V.V.; Zhukov, A.I.; Voronov, V.P. (1998), "Magnetic soliton motion in a nonuniform magnetic field",*Journal of Experimental and Theoretical Physics***87**(2): 401–407, doi:10.1134/1.558674**^**Gibbons, G.W. (1998),*Born-Infeld particles and Dirichlet*p*-branes*,**514**, pp. 603–639, doi:10.1016/S0550-3213(97)00795-5

### General

- N. J. Zabusky and M. D. Kruskal (1965).
*Interaction of 'Solitons' in a Collisionless Plasma and the Recurrence of Initial States.*Phys Rev Lett 15, 240 - A. Hasegawa and F. Tappert (1973).
*Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion.*Appl. Phys. Lett. Volume 23, Issue 3, pp. 142–144. - P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy (1987)
*Picosecond steps and dark pulses through nonlinear single mode fibers.*Optics. Comm. 62, 374 - P. G. Drazin and R. S. Johnson (1989).
*Solitons: an introduction.*Cambridge University Press, 2nd ed., ISBN 0521336554 - N. Manton and P. Sutcliffe (2004).
*Topological solitons.*Cambridge University Press. - Linn F. Mollenauer and James P. Gordon (2006).
*Solitons in optical fibers.*Elsevier Academic Press. - R. Rajaraman (1982).
*Solitons and instantons.*North-Holland.

## External links

Wikimedia Commons has media related to: Solitons |

- Solitons, solitary waves and secondary or baby solitary waves in discrete media
- Heriot-Watt University soliton page
- The many faces of solitons
- Klaus Brauer's soliton page
- Helmholtz Solitons, Salford University
- Solitons and Soliton Collisions
- John Scott Russell and the solitary wave
- Severn Bore web site
- John Scott Russell biography
- Soliton in Electrical Engineering
- Miura's home page
- Short didactic review on optical solitons
- Photograph of Soliton on the Scott Russell Aqueduct
- Solitons possible agent of nerve transmission (PDF)(pnas.org)
- Three Solitons Solution of KdV Equation
- Three Solitons (unstable) Solution of KdV Equation
- Solitons & Nonlinear Wave Equations
- Periodically Amplified Soliton Systems
- Solitary waves in discrete media
- Envelope Soliton Oscillator

## Sunday, January 24, 2010

### LOVE AFFAIRS and DIFFERENTIAL EQUATIONS

From Albert Einstein's notebooks, on a sweet spring day in the early days of The Institute for Advanced Study at Fine Hall in Princeton once upon a time, on his attempt to solve The Three-body Problem:*

Modeling of Love-Hate relationships (from here)

See: Strogatz S.H., Love Affairs and differential equations, Math. Magazine, 61,35,1988.

Lets imagine a Romeo (R) and Juliet (J) "troubled" romance, where:

R(t)=Romeo's Love/Hate for Juliet at time t

J(t)= Juliet's Love/Hate for Romeo at time t

with positive values signifying love and negative values hate.

A first order system of equations to model the evolution in time of the relationship can be written as (Rdot = dR/dt = rate of change of R, and similarly for Jdot):

where a,b,c,d are parameters which can be positive, negative or zero, with the following "meanings":

For instance a case where Romeo has both a>0 and b>0 can be called an "

But if a<0 and b>0 ("

If a<0 and b<0 ("

If a>0 and b<0 ("

Typical issues of these "dynamical love systems" is which relationships are "viable"...

Notice that the "fixed points", that is where the system will stabilize would be given by

Rdot=0

Jdot=0

that is :

a R + b J =0

c R + d J =0

which is a system of two algebraic equations with two unknowns (R and J).

Let's analyze some special cases:

1) Two identical cautious lovers: a=d<0 , b=c>0

Then det=ad-bc=a2 -b2 , and the solutions behave in the following way:

i) If a2>b2 the lovers are more cautious than "responsive" and the relationship "fizzles out " to mutual indifference R=J=0 (caution leads to apathy)

ii) If a2

2) Out of touch with their own feelings: a=0, d=0 (lovers only react to the others feelings)

Then det=ad-bc=-bc, and the equations are:

Rdot = b J

Jdot = c R

Find out what happens!

3) Do opposites attract? Analyze d=-a, c=-b.

4) Do identical lovers make for good couples? d=a, c=b

5) Analyze your own "made up" case of interest!

Here's what interests me, the Eurorock band "T'Pau" performing their 1987 hit, "Heart and Soul":

* - OK, I lied, sorry. That is NOT from Einstein's notebooks. I don't know where it's from, but it's funny. :-)

No need to explore three bodies for the moment though. The two body problem is easier, is linear, and here's a sweet application by Strogatz:

First, click here to see a New York Times guest columnist piece by Stephen Strogatz on Romeo-and-Juliet Mathematics. The replies are pretty funny. :-)

Click here to see Strogatz' original 1988 piece on the topic.

See: Strogatz S.H., Love Affairs and differential equations, Math. Magazine, 61,35,1988.

Lets imagine a Romeo (R) and Juliet (J) "troubled" romance, where:

R(t)=Romeo's Love/Hate for Juliet at time t

J(t)= Juliet's Love/Hate for Romeo at time t

with positive values signifying love and negative values hate.

A first order system of equations to model the evolution in time of the relationship can be written as (Rdot = dR/dt = rate of change of R, and similarly for Jdot):

**Rdot = a R + b J****Jdot = c R + d J**where a,b,c,d are parameters which can be positive, negative or zero, with the following "meanings":

**a**and**d**: "**cautiousness**" (throw towards (if a,d>0) the other or avoid (if a,d<0) the other)**b**and**c**: "**responsiveness**" (degree at which they react to the other's advances)For instance a case where Romeo has both a>0 and b>0 can be called an "

**eager beaver**" (he gets excited by Juliet's love and is further excited by his own feelings into a "snowball of affection").But if a<0 and b>0 ("

**cautious lover**"), it means that the more Romeo loves Juliet (R>0), the more he wants to "run away" from her (Rdot more negative, particularly acute near marriage decisions...); and the more he hates her (R<0) the more he increases his love (Rdot more positive, nothing like distance to inflame his fellings).If a<0 and b<0 ("

**cautious and unresponsive**") usually not a good chance for romance, "lets just be friends" type...If a>0 and b<0 ("

**daring but unresponsive**") is more the "narcisist" type...Typical issues of these "dynamical love systems" is which relationships are "viable"...

Notice that the "fixed points", that is where the system will stabilize would be given by

Rdot=0

Jdot=0

that is :

a R + b J =0

c R + d J =0

which is a system of two algebraic equations with two unknowns (R and J).

Let's analyze some special cases:

1) Two identical cautious lovers: a=d<0 , b=c>0

Then det=ad-bc=a2 -b2 , and the solutions behave in the following way:

i) If a2>b2 the lovers are more cautious than "responsive" and the relationship "fizzles out " to mutual indifference R=J=0 (caution leads to apathy)

ii) If a2

2) Out of touch with their own feelings: a=0, d=0 (lovers only react to the others feelings)

Then det=ad-bc=-bc, and the equations are:

Rdot = b J

Jdot = c R

Find out what happens!

3) Do opposites attract? Analyze d=-a, c=-b.

4) Do identical lovers make for good couples? d=a, c=b

5) Analyze your own "made up" case of interest!

Here's what interests me, the Eurorock band "T'Pau" performing their 1987 hit, "Heart and Soul":

## Tuesday, January 19, 2010

### NONLINEAR DYNAMICS

**NON-LINEAR DYNAMICS and APPLICATIONS for PHYSICS**

(work in progress, stay tuned)

Much of this blog article is my interpretation of things learned from Cornell University Mathematics professor Dr. Stephen Strogatz's 1994 book: Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering

Outline

**I. Dynamics - HISTORY**

**II. Dynamics - Mathematics**

**- forthcoming, see Calculus IV in the meantime**

**III. Linear Dynamics - Applications in Physics**

**IV. Nonlinear Dynamics - Applications in Physics**

**V. Nonlinear Dynamics - Applications in Other Fields**

**VI. Personal History**

**I. Dynamics - HISTORY**

A.

**Issac Newton**- Differential Equations and the Two-Body Problem

B.

**Henri Poincaré**- Qualitative Geometric Approach; Chaos

C. Early-Mid 20th Century Mathematicians (

**Birkhoff, Kolmogorov, Arnold, Moser**)- Nonlinear oscillators; Complex behavior in Hamiltonian mechanics

D.

**Edward Lorenz**- 1963 Discovery of Chaotic Motion on a Strange Attractor

E. 1970's Mathematicians (

**Ruelle & Takens, May, Mandelbrot**) - Turbulence in Fluids; Population Biology; Fractals

F. 1970's Physics -

**Mitchell Feigenbaum**: Universality and renormalization; Chaos and Phase transitions linked

G. 1970's Biology -

**Arthur Winfree**: Nonlinear oscillators in biology

H. 1980's to present - Widespread interest in chaos, fractals, oscillators and their applications

**II. Dynamics - Mathematics - forthcoming**

**III. Linear Dynamics - Applications in Physics**

**A. N= 1 variable (Growth, Decay, or Equilibrium)**

**1. Exponential Growth**

**2. RC Circuit**

**3. Radioactive decay**

**B. N=2 variables (Oscillations)**

**1. Linear oscillator**

**2. Mass and spring**

**3. RLC Circuit**

**4. 2-body problem (Kepler, Newton)**

**C. N>=3 variables**

**1. Civil engineering, structures**

**2. Electrical Engineering**

**D. N >>> 1 variables (Collective phenomena)**

**1. Coupled harmonic oscillators**

**2. Solid-state physics**

**3. Molecular dynamics**

**4. Equilibrium statistical dynamics**

**E. Continuum (Waves and patterns)**

**1. Elasticity**

**2. Wave equations**

**3. Electromagnetism (Maxwell)**

**4. Quantum mechanics (Schrodinger, Heisenberg, Dirac)**

**5. Heat and diffusion**

**6. Acoustics**

**7. Viscous fluids**

**IV. Nonlinear Dynamics - Applications in Physics**

**A. N=1 variable**

**1. Fixed points**

**2. Bifurcations**

**3. Overdamped systems, relaxation mechanics**

**4. Logistic equation for single species**

**B. N=2 variables**

**1. Pendulum**

**2. Anharmonic oscillators**

**3. Limit cycles**

**4. Biological oscillators (neurons, heart cells)**

**C. N>=3 variables (Chaos)**

**1. Strange attractors (Lorenz)**

**2. 3-body problem (Poincare)**

**3. Chemical kinetics**

**4. Iterated maps (Feigenbaum)**

**5. Fractals (Mandelbrot)**

**6. Forced nonlinear oscillators (Levinson, Smale)**

**7. Practical uses of chaos**

**8. Quantum chaos ?**

**D. N >>> 1 variables**

**1. Coupled nonlinear oscillators**

**2. Lasers, nonlinear optics**

**3. Nonequilibrium statistical mechanics**

**4. Nonlinear solid-state physics (semiconductors)**

**5. Josephson arrays**

**6. Heart cell synchronization**

**7. Neural networks**

**8. Immune systems**

**9. Ecosystems**

**10. Economics**

**E. Continuum (Spatio-temporal complexity)**

**1. Nonlinear waves (shocks, solitons)**

**2. Plasmas**

**3. Earthquakes**

**4. General Relativity (Einstein)**

**5. Quantum field theory**

**6. Reaction-diffusion, biological and chemical waves**

**7. Fibrillation**

**8. Epilepsy**

**9. Turbulent fluids (Navier-Stokes)**

**42. Life**

**V. Nonlinear Dynamics - Applications in Other Fields - forthcoming**

Click here to see a highly interactive version of Brian Castellani's Complexity Map, as shown below:

**VI. Personal History**

I'm as happy as a Philosophy Grad Student at the beginning of his first lecture of his first teaching job teaching "Introduction to Plato". *

Why?

Because, I've finally found my specialty, thanks to United Parcel Service delivering the following book from Amazon to my doorstep yesterday:

By the way, I really hate, loathe and despise the term "Chaos Theory." The proper description would be "Structure-in-Chaos" Theory. It's young, it's happening, it's taking off in many different fields, and the high-speed computers of Computer Scientists and their wonderful Algorithms are its very best friend. We have miles to go before we sleep. Time to get cracking! :-)

From Wikipedia, at which an input of "Nonlinear Dynamics" directs to "Nonlinear differential equations" under "Nonlinear systems." Under that section it states:

A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics, the Lotka–Volterra equations in biology, and the Black–Scholes PDE in finance.

One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.

Going to the top of the page:

In mathematics, a

**nonlinear system**is a system which is not linear, that is, a system which does not satisfy the superposition principle, or whose output is not proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables.

Nonlinear problems are of interest to physicists and mathematicians because most physical systems are inherently nonlinear in nature. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos. The weather is famously nonlinear, where simple changes in one part of the system produce complex effects throughout.

And that's it for today. For all my regular readers (all 4 of you ... it would be 5 but Mom passed away in 2008) I'm afraid I will spend less time on-line and at this blog, as I have much to read. I won't go away completely, but for the most part I'll be incognito. Cheers and farewell, and here's hoping I do Mom proud, wherever she is, when I publish my first paper in 6 months to 3 years, or so.

* - most of whom start off with: "I am SO ENVIOUS of you people! You are about to hear about Plato for the FIRST time!" They have their point.

Sincerely,

S'Colyer

P.S. For your viewing and listening pleasure (subjective), a VERY non-linear song:

Here is the Number One most popular song in America today, Empire State of Mind by Jay-Z and Alicia Keys. Keys' bits are beautifully linear, Jay-Z's nonlinear. Somehow, they blend well:

## Monday, January 18, 2010

### Has Koelman Successfully Eliminated Dark Energy Thanks to the Recent Work of Verlinde?

"Hammock Physicist" Johannes Koelman has made a bold statement at his blog regarding the title of this blarticle, which I strongly recommend we read then examine by clicking here. Read the replies as well, as they're also interesting and no less important.

Essentially, the bulk of Koelman's work is as follows, and in his words from his article:

I previously posed the question “can dark energy, just like gravity, be understood as an entropic effect?”.

To my astonishment, it appears that a quick 10 minute exercise in determining the entropic force exerted on the entire observable universe indeed yields an effect with the right order of magnitude to explain the cosmic dark energy (or cosmic acceleration). It seems that a dark energy effect emerges from Verlinde's holographic description. All 123 orders of magnitude of the dark energy mismatch evaporate when considering the cosmic acceleration as a result of a holographic entropic force.

The line of reasoning is as follows (for ease of notation I will work in natural units and leave out factors c, G and h-bar):

1. Consider the cosmic horizon (the edge of the observable universe: a sphere with radius R approximately equal to 2.7 10^61 in natural units)

2. According to the holographic principle the observable universe is encoded in N = pi R^2 bits located at the cosmic horizon,

3. A finite temperature T is associated with this horizon. This temperature is determined by an equipartition of the energy Mc^2 contained within the horizon over the bits (degrees of freedom) associated with the horizon. Here M = 1.4 10^60 is the observable mass of the universe. Using the equipartition expression ½kT = M/piR^2, it follows that kT = 3 10^-64.

4. An entropic force F = kT grad(N) = kT grad(pi R^2) =2 pi kT R is associated with the horizon. This results in a cosmic expansion (1/R) d^2 R/dt^2 = F/MR = 2 pi kT / M = 1.3 10^-123. Presto!

--- Postscript ---

Two critical notes to the above speculative derivation need to be mentioned here:

1. As I set out to explain more than a hundred orders of magnitude mismatch, I have not bothered myself with numerical factors of order unity. As a result,the end result could be off by a factor of two or so.

2. More importantly, a full evaluation of the above simple expansion model yields an expansion (1/R)d^2R/dt^2 = (2c/R)^2. Whilst this expression yields the right order of magnitude for the expansion, it is not constant and therefore not in line with the full Lambda-CDM model cosmological model. Is it a coincidence that the current value of the dark energy density comes out right? Or does the Lambda-CDM model need a modification?

Essentially, the bulk of Koelman's work is as follows, and in his words from his article:

I previously posed the question “can dark energy, just like gravity, be understood as an entropic effect?”.

To my astonishment, it appears that a quick 10 minute exercise in determining the entropic force exerted on the entire observable universe indeed yields an effect with the right order of magnitude to explain the cosmic dark energy (or cosmic acceleration). It seems that a dark energy effect emerges from Verlinde's holographic description. All 123 orders of magnitude of the dark energy mismatch evaporate when considering the cosmic acceleration as a result of a holographic entropic force.

The line of reasoning is as follows (for ease of notation I will work in natural units and leave out factors c, G and h-bar):

1. Consider the cosmic horizon (the edge of the observable universe: a sphere with radius R approximately equal to 2.7 10^61 in natural units)

2. According to the holographic principle the observable universe is encoded in N = pi R^2 bits located at the cosmic horizon,

3. A finite temperature T is associated with this horizon. This temperature is determined by an equipartition of the energy Mc^2 contained within the horizon over the bits (degrees of freedom) associated with the horizon. Here M = 1.4 10^60 is the observable mass of the universe. Using the equipartition expression ½kT = M/piR^2, it follows that kT = 3 10^-64.

4. An entropic force F = kT grad(N) = kT grad(pi R^2) =2 pi kT R is associated with the horizon. This results in a cosmic expansion (1/R) d^2 R/dt^2 = F/MR = 2 pi kT / M = 1.3 10^-123. Presto!

--- Postscript ---

Two critical notes to the above speculative derivation need to be mentioned here:

1. As I set out to explain more than a hundred orders of magnitude mismatch, I have not bothered myself with numerical factors of order unity. As a result,the end result could be off by a factor of two or so.

2. More importantly, a full evaluation of the above simple expansion model yields an expansion (1/R)d^2R/dt^2 = (2c/R)^2. Whilst this expression yields the right order of magnitude for the expansion, it is not constant and therefore not in line with the full Lambda-CDM model cosmological model. Is it a coincidence that the current value of the dark energy density comes out right? Or does the Lambda-CDM model need a modification?

Erik Verlinde (b. 1962)

## Friday, January 15, 2010

### Milgrom, MOND & TeVeS vs Dark Matter

Those interested in Dark Matter should find the following Dark Matter computer simulation of a Dwarf Galaxy interesting:

The simulated dwarf from Science News on Vimeo.

You can read about it in the Science News article I came across today: here.

DARK MATTER has yet to be proven, and the competing theory of MOND/TeVeS in which Newton's equations (particularly F=ma) are modified and "Dark Matter" "particles" are unnecessary, has yet to be falsified, yet it gets little to no attention, in comparison.

Mordehai Milgrom is THE guy in MOND. Here is his picture:

Read all about Mordehai Milgrom and the MOND/TeVeS vs Dark Matter controversy: here.

The simulated dwarf from Science News on Vimeo.

You can read about it in the Science News article I came across today: here.

DARK MATTER has yet to be proven, and the competing theory of MOND/TeVeS in which Newton's equations (particularly F=ma) are modified and "Dark Matter" "particles" are unnecessary, has yet to be falsified, yet it gets little to no attention, in comparison.

Mordehai Milgrom is THE guy in MOND. Here is his picture:

Read all about Mordehai Milgrom and the MOND/TeVeS vs Dark Matter controversy: here.

## Wednesday, January 6, 2010

### Philosophers Wanted: Long Hours, Low Pay, Important Work (Time, Gravity, Verlinde)

**LATEST**

**UPDATE (JAN. 24) -**Lubos is no longer indecisive and attacks Verlinde with a vengeance, here:

The Reference Frame: Why gravity can't be entropic

**"Historically, the greatest difficulty in scientific revolutions is usually not the missing piece but the extraneous one - the assumption that we've all taken for granted but is actually unnecessary. Philosophers are trained to smoke out these mental interlopers. Many of the problems that scientists now face are simply the latest guise of deep questions that have troubled thinkers for thousands of years. Philosophers bring this depth of experience with them. Many have backgrounds in physics as well."**

.....

*George Musser, Scientific American Senior Editor***I. Time**

II. Gravity

III. In Conclusion

I. TIME

II. Gravity

III. In Conclusion

I. TIME

**Time**. It's a Dimension. It's the 4th of the 4 dimensions we know of. It's also the strangest of them all, due to its apparent uni-directionality. Entropy and The Second Law of Thermodynamics seem to be involved.

First up, Richard Feynman's lecture at Cornell:

And then there's this, by Science Comedian Brian Malow :

**"If we are considering the fundamental level of reality, and asking the most fundamental questions about dynamics, we come up against the question “What decides how things change?” At this fundamental level, the physical laws can seem somewhat arbitrary (for example, the amount of charge on an electron). In fact, at this most fundamental level, the only principle which seems likely to describe dynamics seems to come from mathematics not physics: a system will have many more possible disordered states than ordered states, so a system which changes state randomly will most likely move to a more disordered state.**

"While the second “law” of thermodynamics is “just” a statistical principle, it is a mightily powerful statistical principle! This is because the basis of the second law – that “disorder will increase” – seems so obvious, and seems to appeal to a fundamental, platonic principle of mathematics. For this reason, the second law manages to appear even more fundamental and unbreakable than the other physical laws, which seem rather arbitrary in comparison. Hence Arthur Eddington’s famous quote: “If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can offer you no hope; there is nothing for it but to collapse in deepest humiliation.”

"At the most fundamental level, I would just imagine physical dynamics are described by change of entropy – I can’t imagine any more fundamental principle which could possibly describe change."

"While the second “law” of thermodynamics is “just” a statistical principle, it is a mightily powerful statistical principle! This is because the basis of the second law – that “disorder will increase” – seems so obvious, and seems to appeal to a fundamental, platonic principle of mathematics. For this reason, the second law manages to appear even more fundamental and unbreakable than the other physical laws, which seem rather arbitrary in comparison. Hence Arthur Eddington’s famous quote: “If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can offer you no hope; there is nothing for it but to collapse in deepest humiliation.”

"At the most fundamental level, I would just imagine physical dynamics are described by change of entropy – I can’t imagine any more fundamental principle which could possibly describe change."

*..... Andrew Thomas of "What is Reality?" fame, on "The Arrow of Time"***II. GRAVITY**

**Gravity**. It's a Force. It's the force we've known about the longest, yet, the one we seem to know the least about. It's the strangest of them all due to its weakness, and its range.

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