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Physics updates 2014 (Jan-Jun)...

ecterrab 14540
June 19 2014
8 0

Hi,
An interesting sequence of enhancements and new developments happened in the Physics package during this first half of the year. During the last month, improvements happened in the handling of Vectorial expressions and quantum mechanics using Dirac’s notation. During April and part of May it was the turn of general relativity enhancements.

Some of the developments are also interesting beyond Physics. For example: it is now possible to multiply equations. Suppose you have A = B   (1), and C = D   (2), multiplying as in (1) (2) now results in lhs((1)) lhs((2)) = rhs((1)) rhs((2)), saving a lot of typing. You can also perform (1)/(2) or (1)^2. Some enhancements in Physics related simplification, integration, `assuming`, and typesetting - e.g. the simplification and integration of spherical harmonics (SphericalY function) are also part of the update.

These developments are available to everybody as usual in the Maplesoft R&D Physics webpage. Below there is a list of the developments for the last month as seen in the worksheet that comes in the zip with the Physics update.

 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

DEs and Mathematical Functions Updates...

ecterrab 14540
differential-equations
May 06 2014
2 0

This is the first presentation of updates for the DE and Mathematical Functions programs of Maple 18. It includes several improvements, all in the Mathematical Functions sector, as well as some fixes. The update and instructions for its installation are available on the Maplesoft R&D webpage for DEs and mathematical functions. Some of the items below were mentioned here in Mapleprimes - you are welcome to present suggestions or issues; if possible they will be addressed right away in the next update.

  • Filling gaps in the FunctionAdvisor regarding all the 6 complex components: abs, argument, conjugate, Im, Re, signum, as well as regarding Heaviside (step function), Dirac, min and max.
  • Fix the simplification and differentation rule for doublefactorial
  • Make convert(..., hypergeometric) work the same way as convert(blabla, hypergeom)
  • Implement integral forms for Heaviside(z) and JacobiAM(z, k) via convert(..., Int)
  • Implement appropriate display for the inert %intat function as well as its conversion to the inert Int
  • Make the FunctionAdvisor/DE return not just the PDE system satisfied by f(z, k) = JacobiAM(z, k)and also (new) the ODE satisfied by f(z) = JacobiAM(z, k)
  • Fix conversion rule from Heaviside(z) to Sum
  • Fix unexpected error interruption when differentiating min(...) and max(...) containing more than three arguments
  • Fix issue in simplify/conjugate
  • Improvement in expand/int: factors in disguise are put outside the integration sign
  • Various improvements in the case of multiple integrals involving the Dirac function
  • Make Intat fully inert (before it was evaluating its arguments)
  • Make value of inert indexed objects work

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Superfluidity in Quantum Mechanics...

ecterrab 14540
physics
April 13 2014
7 0

The attached presentation is the last one of a sequence of three on Quantum Mechanics using Computer Algebra, covering the field equation for a quantum system of identical particles, its stationary solutions and the equations for small perturbations around them and, in this third presentation, the conditions for superfluidity of such a system of identical particles at low temperature. The novelty is again in how to tackle these problems in a computer algebra worksheet.

The Landau criterion for Superfluidity
  

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft, Canada

 

A Bose-Einstein Condensate (BEC) is a medium constituted by identical bosonic particles at very low temperature that all share the same quantum wave function. Let's consider an impurity of mass M, moving inside a BEC, its interaction with the condensate being weak. At some point the impurity might create an excitation of energy `&hbar;`*omega[k] and momentum `&hbar;` `#mover(mi("k"),mo("&rarr;"))`. We assume that this excitation is well described by Bogoliubov's equations for small perturbations `&delta;&varphi;` around the stationary solutions `&varphi;```of the field equations for the system. In that case, the Landau criterion for superfluidity states that if the impurity velocityLinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) is lower than a critical velocity v[c] (equal to the BEC sound velocity), no excitation can be created (or destroyed) by the impurity. Otherwise, it would violate conservation of energy and momentum. So that, if LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) < v[c] the impurity will move within the condensate without dissipation or momentum exchange, the condensate is superfluid (Phys. Rev. Lett. 85, 483 (2000)). Note: low temperature liquid 4He is a well known example of superfluid that can, for instance, flow through narrow capillaries with no dissipation. However, for superfluid helium, the critical velocity is lower than the sound velocity. This is explained by the fact that liquid 4He is a strongly interacting medium. We are here rather considering the case of weakly interacting cold atomic gases.

Landau criterion for superfluidity
 

 

Background: For a BEC close to its ground state (at temperature T = 0 K), its excitations are well described by small perturbations around the stationary state of the BEC. The energy of an excitation is then given by the Bogoliubov dispersion relation (derived previously in Mapleprimes "Quantum Mechanics using computer algebra II").

 

epsilon[k] = `&hbar;`*omega[k] and `&hbar;`*omega[k] = `&+-`(sqrt(k^4*`&hbar;`^4/(4*m^2)+k^2*`&hbar;`^2*G*n/m))

 

where G is the atom-atom interaction constant, n is the density of particles, m is the mass of the condensed particles, k is the wave-vector of the excitations and omega[k] their pulsation (2*Pi time the frequency). Typically, there are two possible types of excitations, depending on the wave-vector k:

• 

In the limit: proc (k) options operator, arrow; 0 end proc, "epsilon[k]&sim;`&hbar;`*k*"v[c] with v[c] = sqrt(G*n/m), this relation is linear in k and is typical of a massless quasi-particle, i.e. a phonon excitation.
• 

In the limit: proc (k) options operator, arrow; infinity end proc, `&sim;`(epsilon[k], `&hbar;`^2*k^2/(2*m)) which is the dispersion relation of a free particle of mass "m,"i.e. one single atom of the BEC.

 

Problem: An impurity of mass M moves with velocity `#mover(mi("v"),mo("&rarr;"))` within such a condensate and creates an excitation with wave-vector `#mover(mi("k"),mo("&rarr;"))`. After the interaction process, the impurity is scattered with velocity `#mover(mi("w"),mo("&rarr;"))`.

 

a) Departing from Bogoliubov's dispersion relation, plus energy and momentum conservation, show that, in order to create an excitation, the impurity must move with an initial velocity

 

LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) >= v[c] and v[c] = sqrt(G*n/m)

 

  

When LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) < v[c] , no excitation can be created and the impurity moves through the medium without dissipation, as if the viscosity is 0, characterizing a superfluid. This is the Landau criterion for superfluidity.

 

b) Show that when the atom-atom interaction constant G >= 0 (repulsive interactions), this value v[c] is equal to the group velocity of the excitation (speed of sound in a condensate).

Solution
   

 

References

NULL

[1] Suppression and enhancement of impurity scattering in a Bose-Einstein condensate

[2] Superfluidity versus Bose-Einstein condensation
[3] Bose–Einstein condensate (wiki)

[4] Dispersion relations (wiki)

 


Download QuantumMechanics3.mw   QuantumMechanics3.pdf

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Mini-Course: Computer Algebra for Physic...

ecterrab 14540
physics manual
March 31 2014
13 3

This is a 5-days mini-course I gave in Brazil last week, at the CBPF (Brazilian Center for Physics Research). The material will still receive polishment and improvements, towards evolving into a sort of manual, but it is also interesting to see it exactly as it was presented to people during the course. This material uses the update of Physics available at the Maplesoft Physics R&D webpage.

Mini-Course: Computer Algebra for Physicists

 

Edgardo S. Cheb-Terrab

Maplesoft

 

 

This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do in physics with paper and pencil. It is oriented to people not familiar with computer algebra (sections 1-5), as well as to people who are familiar but want to learn more about how to use it in Physics.

 

Motivation
 

 

Among other things, with computer algebra:

 

• 

You can concentrate more on the ideas (the model and its formulation) instead of on the algebraic manipulations
• 

You can extend your results with ease
• 

You can explore the mathematics surrounding your problem
• 

You can share your results in a reproducible way - and with that exchange about a problem in more productive ways
• 

After you learn the basics, the speed at which algebraic results are obtained with the computer compensates with dramatic advantage the extra time invested to formulate the problem in the computer.

 

All this doesn't mean that we need computer algebra, at all, but does mean computer algebra can enrich our working experience in significant ways.

What is computer algebra - how do you learn to use it?
   

What is this mini-course about?
   

What can you expect from this mini-course?
   

 

Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the algebraic computation part of physics into interesting discoveries and fun.

1. Arithmetic operations and elementary functions
   

2. Algebraic Expressions, Equations and Functions
   

3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations
   

4. Algebraic manipulation: simplify, factor, expand, combine, collect and convert
   

5. Matrices (Linear Algebra)
   

6. Vector Analysis
   

7. Tensors and Special Relativity
   

8. Quantum Mechanics
   

9. General Relativity
   

10. Field Theory
   

BrasilComputacaoAlgebrica.mw.zip

BrasilComputacaoAlgebrica.pdf 

Edgardo S. Cheb-Terrab
Physics, Maplesoft

General Relativity using Computer Algebr...

ecterrab 14540 Maple
physics relativity
February 20 2014
10 6

I was recently asked about performing some General Relativity computations from a paper by Plamen Fiziev, posted in the arXiv in 2013. It crossed my mind that this question is also instrumental to illustrate how these General Relativity algebraic computations can be performed using the Physics package. The pdf and mw links at the end show the same contents but with the Sections expanded.

 

General Relativity using Computer Algebra

 

Problem: for the spacetime metric,

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))

 

a) Compute the trace of

 

"Z[alpha]^(beta)=Phi R[alpha]^(beta)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"

 

where `&equiv;`(Phi, Phi(r)) is some function of the radial coordinate, R[alpha, `~beta`] is the Ricci tensor, `&Dscr;`[alpha] is the covariant derivative operator and T[alpha, `~beta`] is the stress-energy tensor

 

T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))

b) Compute the components of "W[alpha]^(beta)"" &equiv;"the traceless part of  "Z[alpha]^(beta)" of item a)

 

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)" obtained in b)

 

Background: The equations of items a) and b) appear in a paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by Plamen Fiziev, a Maple user.  These equations model a problem in the context of a Branse-Dicke theory with vanishing parameter "omega." The Brans–Dicke theory is in many respects similar to Einstein's theory, but the gravitational "constant" is not actually presumed to be constant - it can vary from place to place and with time - and the gravitational interaction is mediated by a scalar field. Both Brans–Dicke's and Einstein's theory of general relativity are generally held to be in agreement with observation.

 

The computations below aim at illustrating how this type of computation can be performed using computer algebra, and so they focus only on the algebraic aspects, not the physical interpretation of the results.

a) The trace of " Z[alpha]^(beta)=Phi R[alpha]^(beta)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"
   

b) The components of "W[alpha]^(beta)"" &equiv;"the traceless part of " Z[alpha]^(beta)"
   

c) An exact solution for the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)"
   

 

GeneralRelativit.pdf    GeneralRelativity.mw

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

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