Two mathematicians from the University of Pennsylvania have found general solutions that were unknown for more than a hundred years to Boltzmann's conduction equation - an unsolvable equation that is 140 years old.
The research in question is partly a historical journey but mainly a mathematical proof that the mathematicians have been working on for two years. The research was carried out by Philip T. Grassman and Robert M. Strain from the Penn Department of Mathematics.
http://www.math.upenn.edu/~gressman/
http://www.math.upenn.edu/~strain/
http://www.math.upenn.edu/
During the 1860s and 1870s James Clerk Maxwell and Ludwig Boltzmann developed the conduction equation to predict the distribution of matter in a rare gas in space and how it responds to changes in quantities such as temperature, pressure, or velocity. At the same time that Boltzmann argued with Ernst Mach about the reality of the existence of atoms and molecules, he developed equations that had the power to lead to experiments that would reveal clues to the existence of molecules. This is how the Boltzen equation is kept a place of honor in the pantheon of science, if only because it is the equation that provides the model for gas behavior. The predictions that result from the equation regarding the behavior of the gas have indeed been backed up by heaps of experiments in many fields of physics.
The full confidence in the equation and the support it received from experiments - the assumption that gases are composed of molecules - led the scientific community to fully embrace the theory. What's more, the theory provided crucial and important predictions - the most important of which is that gases naturally tend to a state of equilibrium in the absence of external influences. One of the important tests of the equation is that even when the gas appears macroscopically at rest, there is molecular activity raging in the form of collisions. While these collisions cannot be observed, they explain the temperature of the gas.
And here mathematicians began to look for a proof and exact solutions to the equation. David Hilbert, the brilliant mathematician from Göttingen, who almost defeated Albert Einstein in the race to the theory of general relativity, tried for years to find formal solutions to Boltzmann's conduction equation, but in vain. This proves how difficult the equation is to prove rigorously.
Grassmann and Strain's interest was aroused in Boltzmann's mysterious equation due to the fact that it describes well the behavior of the physical world. But Magelia could still only find solutions for gases in perfect equilibrium.
Both mathematicians used techniques from modern mathematics in the field of partial differential equations (methods from pseudo-differential operators) and harmonic analysis. The mathematicians mainly relied on methods developed in this article:
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, "Uncertainty principle and kinetic equations", J. Funct. Anal. 255 (2008), no. 8, 2013–2066.
Thus, most of the mathematical techniques in general were developed during the last fifty to five years, so this is relatively new mathematics.
Grassmann and Strain proved with the help of these methods the existence of exact general classical solutions to the Boltzmann equation and a fast decay in time to an equilibrium state for the Boltzmann equation with long-range interactions. Existence of general classical solutions and rapid decay in time to equilibrium imply that the equation correctly predicts that the solutions will continue to conform to the behavior of the system and not some mathematical catastrophe. For example, the equation can suddenly lose its long-standing physical credibility as a result of a mathematical change that will occur in the equation. Rapid decay to equilibrium means that the effect of a small initial perturbation in the gas is short-lived and very quickly becomes imperceptible.
The research provides a renewed understanding of the effects due to contact collisions, when neighboring molecules but lightly touch each other instead of colliding head-on. It turns out that these contact collisions are the dominant type of collisions for the complete Boltzmann equation with the long-range interactions.
The researchers said, "It amazes us that an equation obtained by Boltzmann and Maxwell in 1867 and 1872 provides a fundamental example of geometric partial derivatives that appear in a physical model of the natural world." And Strain added, "the mathematical techniques designed to study these phenomena were only developed in the modern era." This means that Boltzmann and Maxwell were so genius to develop an equation almost two hundred years ago, one that would fit modern mathematical solutions only from five years ago...
Boltzmann's formulation from 1872 in the following source:
Ludwig Boltzmann, Lectures on gas theory, Translated by Stephen G. Brush, University of California Press, Berkeley, 1964.
And Maxwell's formulation from 1866:
J. Clerk Maxwell, On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society of London 157 (1867), 49–88.[
38 תגובות
Looking for an equation that links entropy to time and information...
Also in November 1957, the article by two scientists from the Department of Applied Mathematics of the Weizmann Institute was published
Solution of the Boltzmann-Hilbert integral equation II.
The coefficients of viscosity and heat conduction
Solving the Boltzmann-Hilbert II integral equation for the coefficients of viscosity and thermal conductivity
CL Pekeris and Z. Alterman
Proceedings of the National Academy of Sciences USA, 43, 998-1007 (1957)
https://www.pnas.org/content/43/11/998
and in November 1962
Eigenvalues and Eigenfunctions of the linearized Boltzmann Collision Operator for a Maxwell Gas and for a Gas of Rigid Spheres
C. L Pekeris, and Alterman Z, Frankowski K
The Astrophysical Supplement Series 7, 291
https://bit.ly/32Hu6iY
Regarding the title "Mathematicians solve the 140-year-old Boltzmann equation this year"
I wanted to draw the attention of the editor and the readers to the article by Professor Chaim Pekris from the Weizmann Institute which was published in September 1955
SOLUTION OF THE BOLTZMANN-HILBERT INTEGRAL EQUATION
C. L Pekeris
Proceedings of the National Academy of Sciences of the United States of America. 41: 661-9.
https://bit.ly/38OWcgc
Pie was a delicious cake?
It is strange that Boltzmann committed suicide, felt human nature, stripped us of the laws of nature in a scientific way. A man like him was only inspired by scientific research. Boltzmann's workspace makes it possible to see the flowering of today's physics, which is called "non-classical". Everyone is retarded, not everyone commits suicide.
cedar,
It is possible to briefly explain what is happening in this discussion
The question is whether you are really interested or simply interested in expressing your displeasure from the discussion
And by Allah - I wish I knew what you are talking about …….
fresh,
Sometimes changing the physical constant or the coefficient can change the behavior of a linear to periodic to chaotic equation. See for example the famous predator-prey equation:
X(t+1)=cXt(xt-1) d
where t is the time point, C is the coefficient.
When c=0.5 the equation converges linearly. When c=1 it is cyclic and when it is 2 the equation is chaotic meaning it is sensitive to the initial conditions and does not repeat itself or converge to any value so that it is not possible without calculating the points to predict for example X at t=2000. Easy to check, take Excel and try.
sympathetic
Yes, it's me. And I agree more with LIZA's description of what I said.
I agree with you that he can be refreshed to answer the answers he was asked, and personally I do not agree with him about the 'butterfly principle' but I tried to express my opinion about what he is trying to explain.
Science does define chaos, and that is the linearity in science, that science chooses to define everything.
I would like to present the saying "the world is chaotic and science is linear" in a slightly different light.
In a certain sense, there is justice in things, and the meaning is this:
The scientists choose the scale in which they examine phenomena and the relevant parameters for examining each phenomenon. When examining the impact of any action, several aspects can be examined. If, for example, we look at the effect of swinging a ball in the air - at the atomic level, this action will cause a chain reaction that will cause the atoms of the ball to affect the atoms of the air which will affect other atoms (mine, of the table next to me, etc., etc.). If I don't swing the ball this action won't happen. That is, at the atomic level the world will be a completely different world if we lift the ball and if we don't.
We do not notice this because at the level of our description of the world, the thing has no material effect. But this is just an arbitrary choice of level of description for the phenomena.
In this sense it is science that linearizes.
It is very possible that for the phenomena that we currently call chaotic (such as the weather), in the future you will find an appropriate level of description (which looks at certain parameters that are currently unknown) where the phenomenon is not chaotic.
That is, the question of whether a phenomenon is chaotic or not is related to the way we choose to describe it
Anonymous (R*h Rafa*im?)
I cannot speak for Ra'anan, but we will examine the claim "the world is chaotic and science is linear".
First, science is not linear! Chaos theory is part of science and it defines what chaos is! Second, science describes the world and therefore must have the same properties of the world. The correct claim is that most of the phenomena occur very soon in a linear fashion. The fact that we can describe much of the world around us with Newton's laws, Boltzmann's equations, Maxwell's equations, Schrödinger's equations are all linear equations! The success of these equations in describing the world indicates that the world around us is pretty much linear. The linearity corrections that we often neglect can be neglected. In very certain systems, small disturbances to initial conditions cannot be neglected, usually these are complex systems such as the weather, but not exactly. These systems are sensitive to small disturbances.
We will look around us and try to estimate the number of systems we encounter every day that we cannot calculate.
According to Ra'anan's claim, any presence of a small disturbance creates the "butterfly effect", I assume that Ra'anan does not fly in airplanes
Because butterflies in Australia can cause the plane to crash. Raanan doesn't drive a car because the wind can divert it from its course and Raanan understands that the world can't have existed for billions of years because any small disturbance would have long ago thrown it into the sun or away from it. It may be that in the case of cars and airplanes the driver and the pilot oppose the butterfly effect and what about the orbit of the earth. There is no choice left to refresh but to assume that God is taking care of correcting the chaotic effects.
If I understood Raanan correctly, he means that the world is chaotic and science is linear. No?
fresh,
The following equation is not linear:
Y=AX^2+BX+C (general description of a parabolic trajectory)
is not a linear equation. Do you think she is chaotic?
Friends:
This fresh one is just retarded and it's a shame for every letter that is given to it.
There is no doubt that after such a thoughtful, serious, persuasive and profound answer, I had no choice but to admit my mistake and retire to a Buddhist monastery where I could escape the shame that would be my share anywhere else...
Noam and Zvi
You're wrong.
refreshed
You are absolutely wrong in most of your statements. I would advise you to re-read the dialogue between Ehud and Yael (who understand what they are talking about) and also Ehud's responses to you because I suppose you can learn something from them.
However, every physical system has countless different effects. Usually most of them are unimportant (the gravitational effect of a certain star in Andromeda on a mathematical pendulum) and you can work with approximations that will give incredibly accurate results - in many cases, you can work with linear equations and still get fairly accurate results, sometimes you can't, and then you have to work with non-linear equations linearity.
Non-linear equations are not a guarantee for the creation of chaotic phenomena. There are many physical systems that are described by non-linear equations that are not chaotic at all and in fact many times their solution is based on the fact that over long enough times the solutions converge to solutions of a certain type (quite the opposite of chaos).
Another group of systems includes a very large sensitivity to the initial conditions, these are chaotic systems and in them the same butterfly effect takes place. This really does not mean that every physical system maintains this effect and no physicist will agree with you on your statement that there is a "principle of the butterfly effect" and every physical system is ultimately chaotic.
The butterfly effect is a simplistic and imprecise analogy that clarifies an idea that is only true sometimes (in this case, I kind of think not).
fresh,
You just don't understand!
There are linear phenomena, there are non-linear phenomena and there are chaotic phenomena.
Not every non-linear phenomenon is chaotic - Ehud also explained to you and you refuse to understand (or read).
Why don't you spend a little time, and try to understand the matter before your confusion confuses others?
On the face of it, it looks like some of the physics works linearly but in fact it is only almost linear and therefore it is actually chaotic.
refreshed
First of all, we are not talking about phenomena, the article talks about equations:
"Mathematicians solve the 140-year-old Boltzmann equation this year".
Beyond that, not every physical phenomenon is chaotic. Even if all the phenomena in the universe are described on
By non-linear equations this does not make them chaotic. There are areas in the phase space where they are
The linear approximation works well and there are areas where it doesn't. Look around you the universe is not chaotic we are
Able to calculate things forward and a lot of physics behaves linearly.
sympathetic
Every physical phenomenon in the universe is chaotic, there is nothing that is linear in reality, the linearity only makes the calculation easier to get an approximation, therefore the butterfly effect applies to everything in the universe not only weather.
refreshed
There is no such thing as the principle of the butterfly effect! The butterfly effect speaks of chaotic systems based on non-linear equations. In a system of linear equations, the reaction to a disturbance is proportional to the disturbance, hence the name linear. Boltzmann equations are linear equations!
As mentioned, the butterfly effect is not a principle but an effect that appears in a certain type of chaotic systems such as weather systems.
"The effect of a small initial disturbance in the gas is short-lived and very quickly it becomes imperceptible"? Doesn't this contradict the principle of the butterfly effect?
In addition, it is not clear what the invention and development of "new mathematics" by J. Funct is. Anal, this is how they invented new mathematics, and we were not informed? But seriously, what new developments are these?
Yael
In my opinion, you raise one of the most important questions regarding physics and it has many connections regarding the topic of the article.
Physics in my opinion is the theory of approximations. It is the approximations that allow us to solve complicated mathematical problems without a complete solution by using physical intuition. For example, mathematicians have not yet proven that there is always a solution to the Navier-Stokes equation in three dimensions or that the solution is not singular on the other hand, physicists and engineers know how to describe the flow of fluids in many cases. Physics allows us to separate the main from the bland. By the way, a reward of one million dollars is offered to those interested for the above proof regarding Navier-Stokes.
To compare physical theory with reality we use theory to neglect what is not important. For example, a ball falls under the influence of the earth's gravitation, should the influence of the gravitation of the moon, Mars, Jupiter be taken into account? The physical theory allows us to see what approximations can be made. In the case in question the influence of the Moon, Mars and Jupiter is negligible and this can be shown using Newton's laws. The greatness of physics is its ability to estimate how a solution of a mathematical equation will behave under certain conditions without having the solution in our hands. There are very few fiscal problems that can be solved analytically: a harmonic oscillator, a two-body problem, ... The greatness of physics is its ability to simplify complex problems by approximations and turn them into simple problems and the ability to estimate the corrections to the simple solution. In technical language, physics is based on the field of perturbation theory.
An important point is the justification of fiscal equations. You gave an example of the laws of relativity becoming Newtonian at low speeds. The physical laws in this case are the laws of special relativity, they allow us to see when Newton's laws will work well and physicists as practical people will work with Newton's laws instead of the full mathematical description of the problem. Without the theory of special relativity, Newton's laws gain their validity from matching the experiment, when it was possible to perform more complex experiments, it turned out that now the entire set of experiments is matched by more general laws, i.e. special relativity.
To summarize: physics is built on approximations (within a given physical theory) so that the problem becomes simple enough but not too simple! The proximates are the tool of physics to describe the world at the necessary level of complexity.
Make everything as simple as possible, but not simpler.
Albert Einstein
Liza
The motivation of researchers is indeed to try to answer questions of cause-effect relationships on the other hand the explanatory chain ends in my opinion with the physical law. For example, Kepler discovered the three laws of planetary motion, you can ask what causes the planets to move according to which laws and Newton's law of gravitation "answers" that. Therefore, the answer to the question of why planets move in elliptical orbits is Newton's laws.
Does the law better explain the reason and what is the reason for the law? In my opinion, laws of nature are a compact way to combine the results of many experiments. I will try to explain myself better using an analogy. Experimental results are like points on a set of axes, the law of nature is the curve (function) that describes the points, for example the curve closest to all the points. An infinite number of functions can be transferred to a finite collection of points, so that in order to confirm the selected function, the curve must be continued to areas where there are no points and add points (i.e. additional experiments) in this area and examine the fit.
What is the role of the physical "story" in this description? The story is what enables the connection between experience and mathematical functions and points. Example A ball falls under the influence of gravity. The physical story talks about a property of the mass ball,
Other properties of the ball can be ignored: its color, shape (soon first), the material it is made of, etc...
Now we can write an equation for a point mass on which a force acts. The power of the psychological theory is in finding the right parameters to describe the phenomenon. Example: experiments teach us that the color of the ball has no effect on how fast it falls. In conclusion, the story defines the mathematical space on which the questions are asked and the physical law binds the results.
sympathetic,
Thanks for your response. When I said that it is possible to bring a coin closer to a spherical counter I was trying to present a position. It's like arguing that if we neglect the speed of light and only look at slow objects, the equations of relativity become Newtonian mechanics. This does not mean that Newtonian mechanics is always correct, and the fact is that it is not. It is a good approximation to reality under specific conditions.
Experimental physics deals mainly with approximations and neglect. Or as we in the department usually say, "The attitude of physicists to mathematics is like the attitude of lawyers to the law book. You should know them and you should know how to bypass them."
And as for us, I remember from thermodynamics classes that there were some disturbing things. Clapperon approximation, ideal gas approximation, harmonic oscillator approximation, and the icing on the cake is a diagram of phase transitions of pressure as a function of volume, the lecturer literally drew this graph on the board and then he nonchalantly marked the middle line and said "this is not physical, and does not really exist in reality" and guess how Solve this problem? Just making an approximation!
sympathetic:
I would love to hear your opinion on cause and effect relationships in physics and the way they are expressed in equations.
It seems that these relations are not part of the mathematical formalism but are an integral part of the narrative side of physical theories. How do you explain that? Can causality be completely erased from physical theories?
It seems that on the one hand, one of the goals of a physical theory is to understand and reveal cause-effect relationships, on the other hand it seems that this is not at all necessary for an accurate formulation of a physical theory
Is there an experimental practice in physics that tries to reveal cause-effect relationships or is it just to confirm a fit to the equations?
Is there a parallel in physics for trying to quantitatively understand such relationships as they try to do, for example, in medicine (when trying to understand the cause of certain diseases)
Yael
First of all, thank you for bringing up the topic, which in my opinion is a very important point. Equations are ownership
The central role in exact science. Some of the site's commenters, and I don't mean you, don't understand this.
Exact science consists of a model that can contain a narrative side and words such as parallel universes, dark matter, extra dimensions and more. As long as words are involved, the model is not an exact science. Another step in becoming an exact science is the moment when the model is described in mathematical language, i.e. equations.
Here again it is pure mathematics, it is possible to prove that the equations have a solution, it is possible to prove units of the solution, and an analytical expression for the solution can be found. All these steps are mathematical and have almost nothing to do with science. The scientific part comes when you compare the solution of the equation with an experiment! Finding an analytical solution is not physics, it is pure mathematics again. The equations get their validity from comparison to experience only! As mentioned, an analytical solution has nothing and nothing to do with the confirmation of the equations.
For example: finding an analytical solution to Schrödinger's equation does not prove its correctness as above for Maxwell's equation and other physical equations. Their confirmation comes from comparison with experience!
The approximation you are talking about and you brought as an example the five shekel coin that is approximated as a count is an approximation that is done within the framework of a given equation to solve them. It is more convenient to solve a problem with spherical symmetry. The success of the approximation does not increase or decrease the strength of the equation, but rather confirms or undermines the assumptions of the approximation. In a naive way, the physical equations can be thought of as the laws of nature. Under the assumption of certain laws of nature it is possible to use their various approximations to compare with experience or to calculate practical systems for example nuclear reactors that I have already mentioned.
By the way, if anything, something that happens many times is that there are solutions to the equations that are disqualified because they are not physical.
There are several more important points in the context of an exact solution to equations that I hope to write about later.
I did not diminish the value of the Boltzmann equation, who like me knows how widely this equation is used. I just said that an analytical or mathematical solution or anything that contributes to the proof of a theory should not be underestimated.
Even a NIS 5 coin can be considered a spherical count, when calculating its capacity.
There are branches of thermodynamics that are at best a wonderful approximation to reality, and at worst, equations that may yield unphysical solutions.
Liel Petar
1]. This is not a "mathematical proof" but an "analytical solution" to the equation.
2]. The fact that the equation has an analytical solution is not enough to confirm its physical "correctness".
3]. An equation, or more precisely, a law of nature formulated as an equation, is considered "correct" from a physical point of view, if and only if,
It describes with reasonable statistical accuracy, the physical observations and measurements.
4]. The analytical solution has its advantage, that it facilitates and facilitates the calculations.
5]. There can be, and there are, additional mathematical equations, which have an analytical solution but which do not have an intelligible or physical legality and, on the other hand, many physical laws described by equations that do not have analytical solutions but only computational approximations.
6]. The Boltzmann equation is not a completely arbitrary equation, but is based on certain physical assumptions. The above assumptions are also not completely arbitrary but are based on previous experiments and physical knowledge.
This is indeed explicitly stated in the article.
Yael:
I share the enthusiasm but not the reason.
The fact that the equation describes reality is not a conclusion from the mathematical proof in question here (which is nothing more than a mathematical proof of solving the equation).
This is a conclusion from the experiments.
These experiments were carried out a long time ago and everyone knows that the equation works (it is also written in the article).
What they didn't know was how to calculate an exact solution for it and that's what the mathematicians have now discovered.
It is important to understand that finding a mathematical solution to an equation does not say anything about its suitability to describe any physical thing.
Surprising and exciting. Thanks for the news Gali.
sympathetic,
If an arbitrary equation is sufficient to describe a phenomenon in our world, it does not mean that it is correct, it can just be a good approximation under specific conditions. Mathematical proof is also important.
The Boltzmann equation is not an isotropic equation used only to describe the dynamics of gas particles. The Boltzmann equation is used in calculations of electron transport in a conductor, heat conduction and neutron transport in reactors. There is an exaggeration in the claim that the Boltzmann equation is an unsolvable equation. The Boltzmann equation allows flux calculations in nuclear reactors and thanks to the ability to solve it, nuclear power reactors can be operated safely. Unlike mathematics, in physics there is not much interest in an exact solution to a problem (it's nice that it exists, but not necessary). The Boltzmann equation has countless close ones, for example the diffusion equation. The following give satisfactory solutions for practical level problems. With the development of computers, it is now possible to solve complicated mathematical problems in a reasonable amount of time, and this fact also lowers the value of exact solutions.
To summarize: the Boltzmann equation is a very important equation in physics with many practical implications.
The Boltzmann equation has many physical relatives (for decades) that allow its solution under certain conditions.
Probably because the exact mathematical solution does not have many practical consequences, it is nice that the solution is found, but it is more in the direction of mathematical curiosity.
I don't think I mentioned that the Boltzmann equation has 7 dimensions. But it is true in the article describing the discovery in the university's announcement that it is a 7-dimensional equation. In mathematics, it is not about spatial dimensions, but the intention is that to define something you need to have 7 parameters. Therefore, it is not a solution that will give us some mysterious gas that has the ability to open a hidden gate to parallel universes with seven dimensions. This is something that is obviously much more formal, but also greatly complicates the equation as you see here.
Dr. Gali Weinstein
It says there that this is a 7-dimensional equation.
Can you explain why?
Thanks.
Interesting article.
You guess right.
Penn math department? I assume this is the math department of the University of Pennsylvania
A fresh wind started blowing on the site
Thanks to Gali.