The most likely origin of this is the competition between exchanges of even numbers of particles (giving antiferromagnetic interactions) and exchanges of odd numbers of particles (giving ferromagnetic interactions).3 The magnetic tendencies of an exchange can be derived from the fact that n particles exchange is equivalent to n-1 two-particle exchanges. Thus, a direct independent calculation of the Gruneisen constant is a useful complement to the work of Ceperley and Jacucci.
CHAPTER 2
Experimental Information
Fitting these experimentally measured quantities allows the calculation of some leading coefficients in the series. This effective Hamiltonian can be described as shown below in terms of nuclear spins.
Theoretical Calculations
In light of the data, the idea that steric (hardcore) impedance would have a very different effect on two-particle exchange than some higher exchanges, such as four-particle, nearest neighbor, was advanced by Roger, Hetherington, and Delrieu.5 greater effect of hard -core repulsion on two-particle exchange was thought to reduce its predominance and perhaps to suppress it to the point of insignificance. Three quantities are known from experiments with sufficient accuracy to fit the parameters of this model: .the leading term in the high-temperature expansion of the specific heat e2.
CHAPTER 3
The Path Integral
The Many Particle Density Matrix to First Order
U is often referred to as an effective potential because of its similarity to its role in the short-time density matrix. A constant term corresponding to the normalization factors in the free particle density matrices of Eq.
The Monte Carlo Method
The ratio between the probability of accepting a move [x]i --t [x]j divided by the probability of accepting a move [x]j - [x]i is then. The first term in Eq. 3.64) gives the intermediate positions of the particles at time T if they followed the classical trajectory of free particles.
Summary
One update of the system consists of this procedure applied to each time slice (excluding the endpoints) of each particle in the system. The new particle positions are generated one at a time, starting with the middle time slice between the two fixed endpoints of the new wire (Fig 14).
Measuring the Observable
Two different techniques were used to reduce the statistical errors in the Monte Carlo observables. One other interesting piece of information can be extracted from the way the observable scales with the particle number: a measure of the number of particles that play a significant role in the exchange processes.
The above procedure shows the moment for one particle, but can make mistakes in identifying the location of the exchange. Therefore, the definition used for the instanton is the average of the individual particle instantons for all particles participating in the exchange.
CHAPTER 5
Hard Spheres
The Two-Particle Density Matrix
To ensure that the density matrix is zero at u = O, the free particle density matrix:i for the image particle that jumps from (-u, v) to (u', v') in a time E is subtracted from the free particle density matrix:i for the image particle that jumps from (-u, v) to (u', v') in a time E. particle density matrix for the particle that jumps from (u, v) to (u', v') in a time E. Multiplying the first term in Eq. 5.7) by the expression for Pcm from Eq. 5.4) gives the free particle density matrix for the two particles in the original system.
The Observable
Use this and eliminate f /f' between Eq. 5.10) and (5.11) give the Grunesian parameter in terms of the derivative of J p with respect to the hard core diameter,. 5.12) if the three is replaced by two, the equation is valid in two dimensions. Taking the derivative of the path integral given in Eq. 5.8), can be used to calculate the derivative of U with respect to D. An identical argument to that used in Section 3.4b is used to relate the derivative of the logarithm of the exchange rate with respect to the hard-core diameter to the Monte Carlo observable in the large /3 limit .
It is interesting to note that the magnitude of the Gruneisen parameters is very similar for the three types of exchange, despite their significantly different geometries.
Solid 3 He
- Testing the Program
- a Free Particle
- b Harmonic Oscillator in One Dimension
- c Energy Check
- Determination of E and f3
The basic element used for all the 3He calculations is the two-particle density matrix. The fit is of the form. 6.19) Uo(r) is referred to as an endpoint approximation and is calculated from the new values of p1(r,r';/3) after a matrix squaring using These coefficients are calculated for values of r corresponding to each of the non-linear mesh points.
25 shows all the data for NT,meas=81, including the smallest system data, and Fig.
CHAPTER 7
The Theoretical Calculation of Roger
The potential along the exchange path is estimated from a one-dimensional sinusoidal potential, which is essentially the first fourth component of the potential along the exchange path. 7.8) where L is half the length of the exchange path, and VM is the value of the potential for the classical path at the point rrrid (and presumably the maximum). For the two-dimensional triangular lattice with the potential of Eq. 7.10) this leads to the following expression for the three-particle exchange energy J3 ,. where a is the grid spacing. 7.11) is not an accurate estimate of J3 because the prefactors are calculated too roughly, but the Gruneisen parameter for exchange should be given accurately within the validity of the high-density approximation.
The calculation of the Gruneisen parameter for the three-particle exchange was performed in the same way as that calculated in Chapter 6 (using the same program), with the dimension set to 2 and with a two-dimensional, triangular grid.
Conclusions
The results, together with Roger's predictions, and the values from the experiment are shown below in Table 7.1. The fit line drawn through the experimental data (Figure 26) is in fact an exponential fit and not a power law fit, but the range of a is so small that the variation of the Gruneisen parameter is not significant compared to the variation of the data , especially at the larger values of a. Finally, this Monte Carlo calculation could be modified to control the effect of outside-me exchange by operating in three dimensions with the addition of a substrate potential varying in the direction perpendicular to the plane.
The effect of interlayer exchange can also be measured, although exchange with the liquid layer would necessarily involve some kind of averaging over different starting points for the particles whose path begins in the liquid.
Conclusion
Estimation of Computer Runtimes for Larger Systems
To estimate the computer time required to obtain the value of I with first-order error bars for 54- and 128-particle systems, consider observational Monte Carlo data measured in the no-exchange state for several system sizes as listed in Table 8.1 . Since the error is measured by the total number of measurements, N, as 1/JIV, this error reduction would require an increase in run time by a factor of 36. Second, the factor comes from the increase in time required to calculate the interactions between all particles in the system during the update and measurement phases of the calculation .
The next generation of parallel computers, which will be available in the next two years, will have nodes running at approximately 10 Mflops (million floating-point instructions per second), compared to a Cray maximum program speed of approximately 40 Mflops. if it is vectorized.
The Future
Scaling the statistical error with the system size, a factor will be required to maintain the error bars calculated for the 54-particle system. One way to achieve a speedup of about 3-4 is to vectorize the code for the Cray. The new position of the middle time slice must be generated first, but once that is completed, the generation of the two three-particle subthreads can proceed simultaneously.
It is interesting to note that for the small systems consisting of the exchanging group of particles being updated in a system of 16 particles, the Gruneisen parameters (see Table 5.2) are of the same order of magnitude as for 3He.
Appendix A
Parallel Programming Issues
A Powerful Parallel Programming Model for Regular Problems
Individual array nodes are typically much simpler than a host, often with no memory protection and less memory than a typical multi-user computer. The second part, which runs on the matrix, contains only the computationally intensive parts of the program and the communication required to read the data into the cube and print it out. The first routines described are used to provide a transparent mapping of the topology of the field connections to the grid, suitable for a parallel decomposition of the problem being solved.
One of the most useful routines (and especially used in the program listed in Appendix C) is .
Implementing a Portable Parallel Program
If the next thread is on the next node in the time direction, two copies of that fixed time are kept, one on each node. As mentioned above, t.Lrange[l] at one node is equal to ts_range[O] at the other node in the time direction. Therefore, for the value nP = 7, at most 16 nodes can be used in the time direction.
This is achieved by using the spatial coordinates of the processors in the two-dimensional communication mesh ((pcoord[O],pcoord[l]) = (time, space)).
Appendix B
Random Numbers
Linear Congruential
The way this algorithm is used on a parallel computer is by noting that a relation can be written that gives then+kth random number in terms of n,. The forms for Ak and Bk are easily derived by repeatedly substituting the expression for Rj, in terms of Rj_1 , into the recursion relation of Eqn.
The Polar Method of Producing Normal Random Numbers
Appendix C
Source Code
Set up shells of particles that are within distances in the array layer [][] of the exchange region. Calculate the part of the action that is different for the old and the new thread. If lpfl==O, the particle's interactions are calculated with all other particles in the system.
If mode==!, only the nearest periodic expansion is included in the calculation of the interactions, if mode==O, several periodic expansions are included.
Appendix D
The He-He Interaction Potential
The Aziz Form
In the words of Aziz et al., "Despite some remaining discrepancies, when all the different macroscopic properties are taken into account, the potential produces the best representation of the helium interaction potential available at this time." See Fig 29 for a graph of this potential.
The Ceperley-Partridge Form
Appendix E
Density Matrix Coefficients
Lattice Sites
