Furman University Physics Department

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Special Projects

An Evaluation of Modified Newtonian Dynamics
by Simulation in a Parallel Computing Environment

Project Overview:

The use of computer modeling and simulation in both physics research and teaching is an increasingly important teaching methodology.  One reason for this is that there are relatively few problems that have exact solutions.  By applying computer modeling we are able to add many other situations that can be reasonably studied.  This approach is especially effective for small-scale problems. However, for very large-scale problems we are limited by the capability of the individual computers and the time available for the solution.  This is an unfortunate situation since many interesting physical problems lie in the realm of what are considered to be very large-scale problems.  One example is that of the dynamics of large collections of stars.  We find them in the form of star clusters and galaxies.  These objects become excellent test subjects for the study of gravitational interactions, since here we are able to see the effects of the force of gravity acting over a very long period of time. 

Our project involves the construction of a computer cluster with computers that are otherwise used for data collection in standard undergraduate physics laboratories. As a part of this we have implemented special software to connect the computers in a manner so that we can accomplish the solution of an astrophysics problem of interest in a parallel fashion.  The end result of this project is that we have developed the highly specialized computing environment that will allow both the students and faculty to perform studies of large-scale stellar dynamics.  Our first study involves the comparative simulation of the formation of a star cluster under normal Newtonian gravitational dynamics and in the case in which Newton's law of gravitation is modified along the lines suggested by the Modified Newtonian Dynamics (MOND) theory.   

Background Information on the Project:

Astronomical observations support the notion that as much as 95% of the matter in our universe is in the form of "dark matter ".  For example, there is very strong evidence in the form of the flat rotation curves for galaxies.  Dark matter has been given this designation because it is essentially invisible and produces no detectable radiation.  However, its existence has not been directly confirmed in any laboratory experiment.  This null result has led to an increasing interest in alternate solutions.  One possibility is that something may be incorrect in our formulation of the theory that describes the gravitational interaction of matter over great distances. 

An alternative theory that is gaining interest is modified Newtonian dynamics or simply MOND.  This is essentially an empirical theory and was originally proposed by M.  Milgrom in 1983.  It postulates that for accelerations below cm per second squared Newton's law of gravity would be expressed as .  Here is a new constant with units of acceleration.  In this form MOND can produce most of the effects thought to be due to dark matter, e.g. flat rotation curves for galaxies and by reproducing the Tully-Fisher law.  In our project one of the goals is then to examine the consequences of the MOND for the perihelion precession of planets and to study its effects on long-period comets. Consideration of these cases can, in principle, provide information that would be able to be verified by observation.  Another situation to be studied will be stellar dynamics in an open star clusters. 

Star clusters involve thousand of stars that are gravitationally bound together.  They are the consequence of gravity pulling together the stars over a very long period of time and over great distances.   This physical situation provides a clear case where MOND would be operative along with the normal Newtonian form of gravitation.  In order to see any consequences for MOND we are simulating the formation of a clusters composed of 1000 to 10000 stars.  This problem will take advantage of parallel computing through the use of a new computer cluster that we have developed within the Physics Department and in concert with Computing and Information Services at Furman University.  Again we would look for potentially testable predictions that would show a difference between a cluster formed under a purely Newtonian gravitational field and a scenario that includes MOND. 

MOND is an exciting idea that can eliminate the need for the postulation of dark matter.  It also poses a major test for General Relativity via the principle of equivalence and in finding a compatible formulation within Einstein's theory of gravity.  This is one aspect on which the project will concentrate.

Educational Perspective:

Astrophysics is a rapidly developing area of study and affords students the opportunity to participate in science that is regularly reported in the news media.  For this reason it is both exciting and highly motivational.  The project also gives the students an opportunity to work in a technical group setting with cutting edge technology and to participate in answering questions about the nature of the universe.  This project embodies Furman University's very strong commitment to engaged learning and currently involves two undergraduate physics students.  We expect to be able to present the results at conferences and in the form of publications.

This project makes use of computers that were acquired for our undergraduate introductory physics laboratories.  When the computers are not in use as part of experimental activities they are used as a cluster to perform the large-scale computations required by our project.  Hence we have saved significantly by funding these laboratories in a manner that is consistent with this dual utilization of these computers.

The project is an application of the recently released gridMathematica software by Wolfram Research.  This software complements the Mathematica environment that we are presently using in our courses and other computational activities.  The simulation model has been programmed in Mathematica and uses the Euler-Cromer method to solve the set of differential equations.  This algorithm has been chosen because of its potential to be distributed over the computer cluster.

II.     Project Technical Details:

The Furman University Physics Department recently purchased sixteen new computers for use in the Freshman Physics labs. The previous computers had been used in the lab for data acquisition and analysis using Pasco Scientific and Matlab software. In planning the new lab configuration we wished to make efficient use of the newly available computing power so it was decided to design the lab computer configuration to fulfill two tasks.

First the computers should be used in performing the standard physics experiments in the Introductory Mechanics and Electricity & Magnetism courses. We chose Pasco Scientific hardware and software for use in the introductory physics labs to maintain compatibility with the Pasco sensors already in use with the previous lab computer systems.

For the second task the computers should be easily configurable as a parallel computing system for theoretical exploration. We chose Mathematica with the gridMathematica toolkit as the parallel computing application since Mathematica is already heavily used in several upper level physics courses.

Model Details:

Our simulation requires the solution of the equations of motion for each star in the gravitational field produced by its nearest neighbors and those that are more remote from it.  The star cluster is divided into cells and the gravitational potential field due to each cell, up to dipole terms, is determined.  Then this is added to the potential field due to all of the stars that are in the same cells as the star of interest.  The acceleration field is found by computing the negative gradient of the total potential function. The MOND effects are included in a corresponding potential function.  This effect is turned ÒonÓ when the acceleration that the star experiences falls below the threshold value of cm per second squared.  This process in continued until the acceleration field for all of the stars is determined.  The Euler-Cramer method is then applied successively to first obtain velocity changes and then to determine the new position of the star.  Parallelization is achieved by dividing the star cluster at the cell level over the individual computers in the computer cluster.  As the cells are updated this information is distributed to all of the computers in the cluster.  Although we are using Mathematica throughout this problem this method is new in a Mathematica context because the standard NDSolve available in Mathematica can not be parallelized.  This represented a very significant technical obstacle which we were able to overcome. The end result of the simulation is a time step model of the evolving star cluster.

The Freshman Physics Lab Computer Systems:

The computers purchased for the freshman labs were Compaq Evo 510 systems with 2.4 GHz Pentium4 processors, 1.0GB of RAM, and 10/100 Mbs network cards. Each computer was also provided with a Pasco Scientific Workshop 750 USB interface to connect to appropriate experiment sensors. In addition to standard Windows XP operating system, Microsoft Office XP, Pasco DataStudio 1.8, Matlab 6.5 and Mathematica 4.2, including the gridMathmatica toolbox, the Cygwin 1.3.2 Linux-like application was installed to be used by Mathematica in parallel communications. 

The GridMathematica Toolkit:

We decided to implement the parallel computing capabilities of the gridMathematica toolkit in order to minimize the impact on the Windows XP operating system used in the introductory physics laboratories.  Creating a classic Beowulf cluster, without disrupting the current computing environment, would have been much more difficult than adding the parallel toolkit to the Mathematica program.  Also Mathematica is widely used in our curriculum so the students and faculty are already very familiar with it.

The Firewall:

We wanted to isolate the Physics Lab computers from the rest of a Furman University network to minimize the network traffic in the lab and thus maximize the efficiency of the system when used for parallel computing. A firewall was set up using an HP Vectra VL computer running RedHat Linux 9.0 and using the Firestarter program from Sourceforge.net (http://firestarter.sourceforge.net/). Firestarter provided a simple user interface with which to set up a workable firewall. After surveying the network traffic we decided to configure the firewall with the following host and port settings. This allowed the computers inside the lab computers to access the Internet and appropriate network services while still effectively isolating them from network traffic.

The Switch:

We wanted to provide fast computer communication between the computer nodes in the parallel system so all of the computers behind the firewall were networked using a 100base-T Hewlett Packard J4899a 48 port switch which allows us to expand the system by adding up to 32 more computers in the future.

The Private Network:

We have created a private network in the 192.168.0.0/8 domain. Each computer has been given a static IP address as shown below. Cygwin and gridMathematica are installed and configured on all of the computers. The parallel system was set up assuming that Phylab11-10 (or Phylab11-09) would be used as the master (client) and all other computers would be slaves (servers). This configuration may be easily changed by editing the slave list in the ÒLaunchSlaveÓ section of the relevant Mathematica notebook.

The Cygwin/OpenSSH and GridMathematica:

A local account with the username ÒeinsteinÓ exists on each computer. Einstein is the only account authorized to run the Mathematica/gridMathematica parallel computing operations. No other user has the necessary access to the encrypted SSH communication of Cygwin used by Mathematica to launch the slave processes on the remote computers.

A Mathematica Test Notebook:

The following listing, copied from a Mathematica Notebook, contains the commands needed to launch slaves processes on remote computers.  These slave processes are then controlled by gridMathematica toolbox commands from the master computer. In this case the master computer is Phylab12-10. The current configuration may be run with either Phylab12-10 or Phylab11-09 as a master computer controlling the rest of the cluster. With minor editing of the $AvailableMachines parameter the cluster may be split and each Master computer may control a subset of the large cluster so that two experiments may be run simultaneously.

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<< Parallel`

$MachineName

$RemoteCommand =
    "ssh `1` \"/usr/local/bin/math -mathlink -linkprotocol TCP -linkmode
    Connect -linkname `2` < /dev/null >& /dev/null &\"";

SetOptions[LaunchSlave, ConnectionType -> LinkCreate];

$AvailableMachines = {

      RemoteMachine["192.168.0.101"],
      RemoteMachine["192.168.0.102"],
      RemoteMachine["192.168.0.103"],
      RemoteMachine["192.168.0.104"],
      RemoteMachine["192.168.0.105"],
      RemoteMachine["192.168.0.106"],
      RemoteMachine["192.168.0.107"],
      RemoteMachine["192.168.0.108"],
      RemoteMachine["192.168.0.109"],     

      RemoteMachine["192.168.0.111"],
      RemoteMachine["192.168.0.112"],
      RemoteMachine["192.168.0.113"],
      RemoteMachine["192.168.0.114"],
      RemoteMachine["192.168.0.115"],
      RemoteMachine["192.168.0.116"]
};

LaunchSlaves[$AvailableMachines]//TableForm

TableForm[
  RemoteEvaluate[{$ProcessorID,$MachineName, $SystemID, $ProcessID, $Version}],
TableHeadings \[Rule] {None, {"ID", "host", "OS", "process",  "Mathematica Version"}}]

$Slaves

CloseSlaves[]

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Equipment Costs:

The items listed below provide a working system for both freshman lab experiments and the parallel computing project.  The items in the first table were needed to upgrade the freshman physics labs without regard to the parallel computing project. The primary cost of the parallel computing addition is the $1030 license fee for adding gridMathematica to the current Mathematica license. We also could have saved about $400 on the HP fast ethernet switch if we had chosen a model with less expansion capability. In any event adding the parallel computing capability cost about $1500 or less than 5% of the total project cost.

Items purchased specifically to enable the parallel computing capability on all computers

Items needed to extend the fast ethernet from PH203 to PH201

 We now have a working system, but there is one more step needed to make the installation permanent. This is part of the general upgrading of the networking infrastructure in Plyler Hall. The expenditure has been approved, but the work has not yet been performed.

Summary:

We have created a powerful 16-node parallel computing cluster for a very small additional cost, approximately 5% over the cost to upgrade our introductory physics laboratory computers for normal experimental work. This allows us to have, in essence, two different types of physics labs for the price of one.

The parallel cluster can be utilized to study large-scale physical systems by the use of models that can simulate the dynamics of systems.  This capability can be used to address cutting edge scientific issues.  For example, our current project investigates an alternative to Newton's theory of gravitation, MOND.  We can contribute to the development of scientific evidence that can either validate or invalidate this alternative theory of gravity.

Two students are currently working with this system.  We plan to involve many more students in projects that will utilize this system.  Because it uses Mathematica, our standard math tool for our students, the system is readily available for student driven projects.

The parallel cluster gives the Physics Department and Furman University the capability to carry out very large scale simulations in many areas such as classical mechanics, quantum mechanics, structural and dynamic modeling, in materials science and astrophysics.

The technology that we have developed is machine and operating system independent and will be propagated onto new computers as we upgrade our laboratory system over time.  We can expand the present system to a 48 computer cluster, and provisions have been made so that each Master computer may control a subset of the large cluster so that two simulations may be run concurrently. Through this computer cluster we have gained parity with institutions that have large mainframe systems.  We have done this for a very small cost.