Mathematics of small doses

Single rare events are poorly amenable to statistical analysis, since each random implementation of the event is unique. If we observe unusual statistically reproducible patterns, then there is a desire to explain them to some specially organized structure of the object under study.

The absorption of solutions of rare random macromolecular clusters demonstrate a completely regular, stable hierarchical structure that can easily be taken as a special internal organization of the studied clusters in the study of a substance present in the field of ultra -low concentrations.

In fact, the observed statistical patterns with very bizarre regular behavior are due to the manifestation of the theoretical and malformation of rare random events and do not require the involvement of uncontrolled physical reasons to explain them.

I really like to read comments on popular articles on the discussion of such exciting minds of the topic of the topic as homeopathy: after the exchange of several trial opinions that resemble reconnaissance in battle before the artillery, the authors of the comments immediately move on to discussing each other, practically forgetting the plot. At the same time, by the depths of the opinions expressed, the discussion very quickly begins to resemble Ostap Bender with priests from the Golden Calf: “There is no God,” Ostap said. “God is,” the priests said ...

Among adherents of homeopathy, there is an opinion that the effect of a homeopathic remedy increases as the concentration of the current principle decreases. This statement, which, it seems to me, reflects the essence of homeopathy, is most convexly formulated in a joke: “... the patient stopped taking the drug and died of an overdose ...” The effect of treatment with a homeopathic drug is evaluated exclusively in terms of “sick X recovered” or “patient Y is better (worse).”

As a physicist, I always want to ask: “Gentlemen, is it possible to measure any active beginning of a homeopathic remedy in vitro?” The adherents of homeopathy may be known, but being sacred, it is hidden from the random gaze of outsiders. During the interrogation with the addiction, homeopaths are kept steadily, not giving out secrets, explaining the homeopathic effect “water memory”, “information field”, sometimes even appealing to quantum mechanics.

From the point of view of modern science, the phrase “Memory of Water”, “Information Field” is nothing more than spells that have no greater meaning than, say, “torsion fields”, “eternal nano -engine” or “cosmophysical factors”. The defect in the logical constructions of adherents of “non -traditional science” lies in the fact that they try to “explain” unexplained phenomena with the help of concepts that themselves turn out to be a “thing in themselves”. The bunch of “incomprehensible phenomenon is an explanation”, which claims to be scientific, must satisfy two conditions with the need. First: the phenomenon under discussion should be thoroughly verified in independent experiments. These experiments should give statistically reliable identical results, which exclude both the “human factor” (that is, the role of the experimenter himself in experience) and the role of all uncontrolled external conditions. Second: the theory that describes the phenomenon, no matter how new and revolutionary it may be, should go into the existing idea of ​​the structure of the world.

Let's get back to homeopathy. So, the concentration of the current beginning of the homeopathic remedy in the drug drug is such that it simply is not there, so there is nothing to measure in itself. But here, chemists and biologists come to the rescue of homeopaths who say: and we recorded the physiological effect of the action of ultra -small doses! In the solution, the substance is present, however, in insignificant quantities and nevertheless it acts on living organisms! We will not talk about physiological action, since we want to determine the physical effect precisely outside the biological organism, but since the substance is still present, you can try to study its molecular structure by setting the appropriate experiments, for example, by investigating the absorption of the solution of this substance at different frequencies.

I will be distracted by one paragraph from the main topic and try to briefly explain why the resonant absorption spectrum carries information about the structure of the dissolved substance. Classical work of M. Katsa 1966 Can One Hear The Shape of the Drum? (“Can I hear the shape of the drum?”) Was devoted to the study of such a question.

Let there be a circle of arbitrary shape on which the elastic film is stretched. Is it possible to find out the shape of the contour by the spectrum of the own fluctuations in the film? The answer to this question is this: by the spectrum you can find out the surface area, the length of the perimeter and topological connectivity (the number of holes in the surface), but the form cannot be found. Nevertheless, the information extracted from the spectrum of own vibrations of the elastic membrane is quite extensive. A similar question can be asked in relation to the molecule of a rather complex configuration, which with a certain stretch (in the so -called harmonic approximation) can be represented in the form of a graph made of springs, as shown in Fig. 1 on the left.

To determine the spectrum of own vibrations of such molecules, it is necessary to design the Laplace matrix A = { aij }, which corresponds to the ensemble of the graphs and is built according to the following rules: if particles with numbers I and J ( i ≠ j ) are connected by the spring, then aij = –1, otherwise AIJ = 0 . The sum of values ​​in a line with a reverse sign, as shown in Fig. 1 right.

Calculating our own values ​​of the matrix A, we can extract information about the internal device of the corresponding ensemble of the molecules. Such an experiment can be implemented by taking, for example, a laser with a rebuilt frequency and irradiate a solution of molecules at different frequencies. By measuring the spectrum of absorption of the solution at the frequency of λ , we will get a histogram of resonant frequencies ρ (λ), which indicates how many times in the molecule spectrum there was the frequency of vibration λ .

So, the definition of absorption spectrum is a convenient way to test the molecular structure of aggregates of a substance placed in a low molecular weight solvent. Imagine now that someone, working with ultra -low concentrations of a substance in water, has received a stable hierarchical self -like structure ρ (λ) with periodically repeated maximums such as depicted in Fig. 2.

What conclusion about the structure of molecules I want to make, looking at this relief? The first assumption that comes to the head is as follows: the complex structure of resonances is due to the complex intra-molecular organization of the sample molecules, and the frequency is associated with the characteristic structural features of macro-molecular aggregates.

In fact, it turns out that this is not at all the case. The complex structure of the histogram ρ (λ) with enviable constancy is repeated for any rare macro-molecular aggregates and thereby in no way connected with their internal organization. To verify this statement, we generated many matrices and the type depicted in Fig. 1 with a random very rare arrangement of non -zero elements a II , a set of own frequencies was determined for each matrix, averaged by a large number of different matrices and the corresponding histograms ρ (λ) were built.

Each time we received the same picture with the same arrangement of peaks. The meticulous reader may have a question: is this repeatability to defect the generator of random numbers, which is used to build different matrices A? Answer: No, not related. Dependence depicted in Fig. 2, can be calculated theoretically, without resorting to computer calculations. As a result, it can be shown that the sequences of maximums, most slowly decreasing in height (such as, for example, a sequence of 1, 2, 3, 4, ... in Fig. 2), form the so -called plywood sequences, well known in the theory of numbers.

So, figuratively speaking, any macro-molecular unstructured dirt in fairly small amounts, which is in a large volume of a low molecular weight solvent, will give a repeated clear signal shown in Fig. 2. With a decrease in the concentration of mud, the maximum will gradually become lower and lower, but their position will remain unchanged. In the end, one central peak will remain in a clean solution, which meets the monomolecular environment (all its own values ​​λ of the Laplace matrix are 0).

I would like to note that the opportunity to get “something from nothing” is not at all crossing out studies regarding the biological activity of the ultra -low concentrations of macromolecular substances. An example given above is the warning of mathematicians for physicists, chemists, and primarily biologists: be careful when working with small concentrations of substances, since under these conditions ordinary noise looks completely unusual and the desired can be easily accepted for the actual deep analysis of physical measurements. So, let's summarize. The collective manifestation of random non-proper events can be very complex and have a very bizarre distribution function, consisting of intermittent maximums and minimums, the position of which on the horizontal axis is determined exclusively by theoretical and visual patterns that have nothing to do with the specific internal organization of macromolecular clusters. Thus, working with systems in the field of very low concentrations, the collective statistics of rare events should be carefully taken into account in order to clearly separate the useful signal from noise.

I would like to finish this text by recalling the razor of Occam: "There is no need to multiply entities without the need." The search for an explanation of complex patterns should be started with the simplest and most transparent considerations.