A century-old puzzle in statistical physics and its solution

New theories can not only give the results that have been established by experiments, but also predict new results, and the physically reasonable ingenuity of this result can only be attributed to the arrangement of nature.

Written by | Liu Quanhui (Ph.D. in Theoretical Physics, Professor, School of Physics and Microelectronics Science, Hunan University)

There is a big difference between physical and mathematical puzzles, and centuries of "advanced" mathematical puzzles are not uncommon, for example, Fermat's Last Theorem has been 358 years since it was proposed and proved; goldbach conjecture has been 280 years. Interestingly, for every mathematical puzzle solved, it is equivalent to killing a goose with a golden egg. There are also centuries-old problems in physics, and each solution is equivalent to finding a goose that lays a golden egg. These problems range from why the ice surface slips, to the structure of the universe and its evolution; complexities such as climate change, specifically such as the return of pigeons to their nests, and so on. The finite size effect puzzle in this paper, written in physics textbooks, has been published in a series of research papers throughout history, and is a small puzzle between mathematics and physics.

One

Limited size effect puzzle

Of all the statistical distributions, the most general one is an introduction to statistics. Statistics, however, cares about large sample sizes. Rare examples are not in the scope of statistics. Statistical physics also seeks the most general distribution, but a system often comes into contact with a heat library or particle library, and there may be only one particle in the system. In other words, in statistical physics, even if a particle appears, a distribution occurs. Single-atom heat engines have been a research hotspot in recent years, dealing with the statistical distribution of only one particle. Usually, thermodynamics deals with a system with a large number of particles, mathematically processed as an infinite number of particles, and the volume of the system should also be taken as infinite, and the density of the number of particles is unchanged, which is the so-called thermodynamic limit. At the other pole of the thermodynamic limit is a small system, or a system with few particles, or a system of limited size. The new effect that emerges here can be called the finite size effect, or the less particle effect.

However, for more than a century and a half, hundreds of thousands of scholars of statistical physics have been plagued by the problem that the applicability of the relevant theory requires a large number of particles, and only by satisfying this condition can they give statistical distributions with the help of mature mathematical tools. Of course, when the number of particles is large, these results are also correct and have been rigorously tested by experiments. So, when the number of particles is very small, is there a distribution? What is the form of the distribution? Mathematically, the problem comes down to how to deal with the logarithm lnx! of a variable x and its calculus, where the variable x can be understood as the number of particles, x= 0, 1, 2,...。 For example, x is between 1 and 10, which is the so-called less particle system.

In statistical physics, lnx! is often used as follows the Stirling formula, lnx! ≈xlnx-x。 This formula is very precise when the value of x is very large. But when x is smaller, for example when x is valued between 1 and 10, the precision of this formula is very poor. However, physically, it can only be taken in this way, and one more item and one less item cannot be done! The key reason is that only in this way can the entropy of a thermodynamic system be guaranteed. If a more precise approximation is used, the extensibility will inevitably be destroyed. In other words, the exact Stirling formula, if the final result given is correct, should be the so-called small particle effect or the finite size effect, but the probability is wrong.

When x is the number of particles, lnx! is an innate discrete function. Using the approximate formula lnx!≈ an important purpose of xlnx-x is to make lnx! a continuous function for calculus. However, when x is smaller, the precision of treating lnx! as a continuous function is very poor, and can only be treated as a discrete function, in this case, the differential dlnx of lnx! should be replaced by differential Δlnx!. However, there are many definitions of difference Δlnx!, for example, the previous differential Δlnx!= ln(x+1)!- lnx!=ln(x+1), the latter differential Δlnx!= ln(x)!- ln(x-1)!=ln(x), the first two steps difference, the last two steps difference,...,central difference, eccentric difference,... Wait a minute. With so many definitions, the physical results given are not the same from each other, and choosing any one of them is equivalent to introducing a new hypothesis, and even then, the result given is highly likely to be unreliable.

Thus, when statistical physics comes to lnx!, there are two places where strict processing is required, one is the precise expression of lnx!, and the other is strict discrete calculus.

Two

Find the most stable solution

Seeking the precise expression of the Raheem Formula for the function lnx!, physicists can't play mathematicians. Carry over the results found by mathematicians. So, the question is not whether to use the exact Stirling formula. The problem must arise on the function difference. After many explorations and failures, we finally found a narrow but ingenious way to solve this problem[1], and there was no need to use The Stirling formula at all.

Finding the most general distribution in statistical physics can be understood as a process of finding variations, and if you walk on the right path, you will definitely encounter some "demons and ghosts". Let's illustrate this with a simple example.

Therefore, the post-differential solution is more stable. To give a reasonable statement, it is from mathematics or physics, as long as you can pick out the post-difference solution.

A very small number of researchers will go this far and adopt a new physical principle to treat the most stable solutions as real solutions and eliminate the unreal solutions.

However, even here, the physical problem has not been fundamentally solved.

Three

A little new mathematical trick: asynchronous difference

After a little operation, the results given by the new theory are divided into three parts. In the first part, when the particle count is large, the result fully returns to the traditional result. This part of the results has withstood the repeated test of experiments, and if the new theory does not give the same result, the theory must be wrong. In the second part, if there are only two particles in the system, such as bosons or fermions, the original distribution continues to work. This seems to suggest that the giant regular distribution in traditional statistical physics suggests this result. However, a hint is not the same as an establishment. Whether the results given by the giant regular distribution are suitable for a system with only one or two particles cannot be judged by statistical physics itself. In the third part, if there is only one particle in the system, the new theory holds that the particle's quantum nature suddenly disappears and all particles obey the same statistics, the Boltzmann statistics. This one result is that the theory first gives the result and then understands it later. A few more words may be said on this point.

Two particles are fermions or bosons, which quantum mechanics considers to be a consequence of the satisfaction of the state of a system with exchange symmetry, while quantum field theory holds that there is a simple one-to-one correspondence between spin and statistics. According to the new theory, if there were originally two bosons in the system, the boson distribution would be satisfied. Suddenly take one of them and ask, is the remaining one still a boson? Quantum mechanics argues that since the boson cannot be exchanged with other particles, it is said that the particle is a boson or not, and has lost its meaning. It is here that the new theory further predicts that the particle should satisfy Boltzmann's statistics. This result made me very anxious for a while, but then I suddenly understood that this was the ingenious and inevitable arrangement of nature. When there is only one particle, the whole system can be used as a localized region, and this particle is a localized particle, of course, it can only satisfy boltzmann statistics. Thus, the new theory predicts that as particles change from two to one, there will be a statistical transition between two statistics, at which time a new heat exchange will occur. Therefore, the article [1] envisions that the new theory can provide experimentally testable results.

Four

Summary

Obviously, the problem of finite size effects involved here is difficult because there are no proper mathematical tools. I was guided by the physical sense that there should probably be a physical result of what it looked like, and then pushed back the math and found that only the introduction of asynchronous differences could solve the problem. Perhaps, in mathematics, someone else has proposed the idea of asynchronous difference, but it has never been introduced into physics.

Asynchronous differences seem to break through mathematical stereotypes, but remain within the framework of mathematics. Asynchronous differentials are all around us, just a thin layer of paper. New theories can not only give the results that have been established by experiments, but also predict new results, and the physically reasonable ingenuity of this result can only be attributed to the arrangement of nature.

Postscript

I write articles all my life, and many articles are written and forgotten. After submitting the article, I don't know which publication I submitted. There is even a good article that was rejected because the first review had passed but could not be replied to in time. This article[1] is different, an account I gave myself after more than twenty years of teaching the course "Thermodynamics and Statistical Physics", and an assignment that I have decided to deliver since June 2020, when Professor Qian Hong of the University of Washington convened the International Seminar Nanothermodynamica series. The article was officially published on the "May Fourth" Youth Day, on the occasion of the second anniversary of the Nano thermodynamics Seminar, and I wish the seminar a permanent study. The editors of the Annals of Physics can't wait to publish the article, which also shows that the paper is original. However, I was a little angry with the continuous rejection of the manuscript, not only rewriting the first sentence of the paper into an advertisement: "This article aims to solve a fundamental problem with a long history in statistical physics", but also eagerly made a report at the 2020-2021 Chinese Physics Autumn Annual Conference for promotion. After receiving the paper, I actually couldn't believe it; after the proofreading, I bought a pot of flowers to commemorate myself; the article was published, and I had to write this article to brag about it.

Academics are the world's public instruments, and we sincerely look forward to the criticism and correction of experts, teachers and friends.

bibliography

[1] Q. H. Liu，Asynchronous finite differences in most probable distribution with finite numbers of particles，Annals of Physics，Vol. 441, June 2022, 168884. https://doi.org/10.1016/j.aop.2022.168884

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