cold fusion, lenr, cmns, nuclear fusion

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241) Simulating experiments described in unit #240

Ludwik Kowalski (7/17/05)
Department of Mathematical Sciences
Montclair State University, Upper Montclair, NJ, 07043

Important clarification have been made in inut #242. It was added after Dr. Bass commented on what is shown below. Please consider the next unit to be the continuation of this unit.

1) Introduction:
In unit #240 I wrote about a paper of Robert Bass. Since then I found that the peper can also be downloaded from the library at . Here I am summarizing observations based on computer simulations of experiments described by Bass. My cathodes, like his idealized cathodes are assumed to be made from a CF-suitable material. I was motivated by desire to verify Robert’s statistical protocol. Instead of relying on his mathematical derivation I decided to use the brute-force approach -- the Monte Carlo method. Like Bass, I assumed that the only uncertainties result from random errors of measuring five quantities: A0, A1, A2, A3 and A4. These are amounts of helium produced in five experiments. The percentage errors. E0, E1, E2, E3 and E4, presumably independent of each other; are standard deviation for the corresponding value of Ak (k=0, 1, 2, 3, and 4). For example, if A2=1,000 units, and if the standard deviation associated with this result is 100 units then E2 =10%. I tested Robert’s protocol and tentatively found it to be not reliable.

Let me begin by describing basic assumptions. Each atom of helium, produced from fusion of two atoms of deuterim is known to liberate 23.6 MeV of nuclear energy. Assuming that this energy appears in the form heat (and not photons of gamma rays) I know that excess heat generated at the rate of 10 watts (6.24*10¹³ MeV per second) would produce 2.25*10¹⁷atoms of helium per hour. That quantity, helium generation rate, allowed me to calculate exact values of A_k for experiment durations of t_k= 0, 2, 4, 6 and 8 hours. The theoretical results were:

A0=0
A1=0.45*10¹⁸atoms
A2=0.90*10¹⁸atoms
A3=1.35*10¹⁸atomsA4=1.80*10¹⁸atoms

The experimentally reported results are expected to fluctuate around these numbers and my Monte Carlo program confirms this, as expected. Sizes of fluctuations are controlled by changing desired values of E0, E1, E2, E3 and E4. The program can rapidly simulate thousands of measurements. Parameters of Robert’s protocol are calculated, for each set of five measurements.

2) Bass’ protocol:
Robert claims that, as stated in the abstract, his protocol was designed to “instantly reject any doubt that the electrical pulse was creating ⁴He from some form of nuclear-chemistry process.” A pulse is simply one case of the Fleischmann-like dc electrolysis going on for a fixed length of time. In unit #240 I wrote that those who actually performed experiments described in the paper used more convincing graphical displays. But that might be a matter of taste. The main issue has to with validity of the proposed numerical protocol. Would it be confirmed or contradicted by simulations of experimental results?

Robert’s protocol consists of calculating three parameters (P1, P2 and P3) from available experimental data (from A0, A1, A2, A3 and A4). By definition:

P1 = 3*(A0 + 2 * A1 + A2 - A4) / 5

while

P2 = (A3 +2 * A4 -2 * A0 - A1)/10

Note that without random fluctuations of A1, A2 and A4, the value of P1 would be zero. This is simply a consequence that, under ideal conditions A2=2*A1 and A4=4*A1. My simulations, as expected, show that P1 approaches zero when bars of errors advance toward zero. This shows that P1 depends on precision of measurements of A1, A2 and A4. I find it strange that, according to the above definition, P1 is not affected by changes in A3. This conflicts with the idea that P1 is an indicator of the overall accuracy. Also note that P2, under ideal conditions, is positive. This too is a consequence of conditions imposed on durations of experiments. Once again I am puzzled by the absence of A2 in the definition of P2.

The third parameter is defined ambiguousely; it involves k and Ak

P3 = (1/3) * (Ak - P1 - k*P2)²

where k is the integer between 0 and 4. In what follows I will assume that (1/3) has to be multiplied by the sum of five quadratic terms, one for each k. In other words,

P3 = (1/3) * SUM (Ak - P1 - P2)² (Bass confirmed this on 7/20/05)

Perhaps there was a typing error and the symbol SUM was omitted. Note that I am avoiding Greek symbols because they are sometimes changed into nonsensical characters over the Internet.

3) My results:
Simulated sets of five Monte Carlo experiments are always successful, by definition. Knowing this I calculated P1, P2 and P3 from the simulated values of A0, A1, A2, A3 and A4. The results from five simulated sets are tabulated below. The first four sets refer to situations in which E0=E1=E2=E3=E4; the valus of assumed percentages are shown in the first column. In the last set I used a mixture (mx) of arbitrarily assumed percentages: E0=3%, E1=12%, E2=7%, E3=30% and E4=4%.

%    A0    A1       A2        A3        A4        P1         P2        P3      sqr(P3)

1    0	 4.59e17  9.06e19  1.36e18   1.81e18  -2.04e15    4.54e17    1.52e31   3.90e15
10   0	 4.53e17  9.61e17  1.39e18   1.91e18  -2.04e16    4.88e17    1.52e33   3.90e16
30   0   4.61e17  1.09e18  1.49e18   2.31e18  -6.14e16    5.65e17    1.36e34   1.17e17
60   0   4.71e17  1.27e18  1.62e18   2.82e18  -1.22e17    6.81e17    5.48e34   2.34e17
mx   0   5.51e17  9.82e17  1.36e18   1.85e18  -6.14e15    4.61e17    1.36e32   1.17e16

I see that P1 is negative in all sets. This might be due to the suspected typing error (ommision of A2 from the definition of P1). Journal of New Energy, as far as I know, does not filter publications through the peer review process. The last column shows square roots of P3. In the published paper that parameter is labeled as sigma; it does not differ from P2 by as much as P3 does. According to Bass, the experiment is successful when P1<<P2 and when sigma<<P2/2. I cannot comment on these inequalities without knowing that published definitions do not suffer from typing errors. For the time being I simply observe that, in the 60% set, sigma is NOT MUCH smoller than P2. I am confident that the trivial Monte Carlo code (shown below) has no errors.

If I were asked to peer-review Robert’s paper I would first insist on the removal of the ambiguity from the definition of P3. Then I would ask for the confirmation that printed definitions of parameters P1 and P2 contain no typing errors. If I were assured that definitions of P1, P2 and P3, in the Monte Carlo code below, are consistent with the protocol then I would suspect that the protocol is not reliable. My recommendation (to the editor of the journal) would be to send the manuscript to a statistician able to examine the mathematical derivation of the protocol.

4) Final comments
a) My guess about the definition of P3, as used in the code, might not be valid. If this is true then I would like to receive the correct definition. It will not be difficult to modify my code and to generate a new table of simulated results. I would be glad to do this.

b) Note that Robert’s protocol does not depend of explicit values of E0, E1, E2, E3 and E4. My results, on the other hand, show that relations between P1, P2 and P3 do depend on the imposed random fluctuations. This troubles me. But I cannot take anything seriously without being certain that definitions of P1, P2 and P3, in the Monte Carlo code, have no errors.

c) My Monte Carlo code, in True Basic, should be readable by a person familiar with programming in any procedural language, such as Fortran or Pascal. That is why I am listing the code below. Be aware that in True Basic exclamation signs identify comments. The tiny code was written to emphasize readability, not compactness. Comments concerning simulations would be appreciated.

Program robert_bass                             ! Coded by Ludwik Kowalski (July of 2005)
! *********************** 
dim A(0 to 4)                                   ! place holders for A0, A1, A2, A3 and A4
dim D(0 to 4)                                   ! for durations of five experiments in hours
let E0=0.1                                      !  this means 10%
let E1=0.1
let E2=0.1
let E3=0.1
let E4=0.1
let wattage=10
let D(0)=0                                      ! five durations (in hours)
let D(1)=2
let D(2)=4
let D(3)=6
let D(4)=8
let rate=2.25e16*wattage                        ! expected atoms/hour
for j=0 to 4
     let r=rnd                                  ! random number between -1 and +1
     let A(j)=(rate+r*E0*rate)*D(j)             ! simulated A0, A1, A2, A3 and A4
next j
let P1=(3*A(0)+2*A(1)+A(2)-A(4))/5              ! P1 was dA in Robert’s paper
let P2=(A(3)+2*A(4)-2*A(0)-A(1))/10             ! P2 was deltaA in his paper     
         let P30 = (A(0) - P1 - 0*P2)^2                                                      
         let P31 = (A(1) - P1 - 1*P2)^2 
         let P32 = (A(2) - P1 - 2*P2)^2         ! five components of the sum (to get P3)
         let p33 = (A(3) - P1 - 3*P2)^2 
         let P34 = (A(4) - P1 - 4*P2)^2
let P3=(P30+P31+P32+P33+P34)/3                  ! P3 was sigma squared in the paper

print "A0=";A(0);"  A1=";A(1);"  A2=";A(2);"  A3=";A(3);"   A4=";A(4)
print "P1=";P1;"   P2=";P2;"   P3=";P3

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