Stability of matter refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
and Andrew Lenard in 1967–1968,
but a shorter and more conceptual proof was found later by
Elliott Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.
Lieb is ...
and
Walter Thirring
Walter Thirring (29 April 1927 – 19 August 2014) was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.Thirring, H. Über die Wirkung rotierender ferner Massen ...
in 1975.
Background and history
In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, the existence of macroscopic objects is usually explained in terms of the behavior of the energy or the free energy with respect to the total number
of particles. More precisely, it should behave linearly in
for large values of
.
In fact, if the free energy behaves like
for some
, then pouring two glasses of water would provide an energy proportional to
, which is enormous for large
. A system is called ''stable of the second kind'' or ''thermodynamically stable'' when the (free) energy is bounded from below by a linear function of
. Upper bounds are usually easy to show in applications, and this is why people have worked more on proving lower bounds.
Neglecting other forces, it is reasonable to assume that ordinary matter is composed of negative and positive non-relativistic charges (
electrons
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
and
nuclei), interacting solely via the
Coulomb force
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
. A finite number of such particles always collapses in classical mechanics, due to the infinite depth of the electron-nucleus attraction, but it can exist in quantum mechanics thanks to
Heisenberg's uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. Proving that such a system is thermodynamically stable is called the stability of matter problem and it is very difficult due to the long range of the Coulomb potential. Stability should be a consequence of screening effects, but those are hard to quantify.
Let us denote by
:
the quantum Hamiltonian of
electrons and
nuclei of charges
and masses
in atomic units. Here
denotes the Laplacian, which is the quantum kinetic energy operator. At zero temperature, the question is whether the ground state energy (the minimum of the spectrum of
) is bounded from below by a constant times the total number of particles:
The constant
can depend on the largest number of spin states for each particle as well as the largest value of the charges
. It should ideally not depend on the masses
so as to be able to consider the infinite mass limit, that is, classical nuclei.
Dyson Dyson may refer to:
* Dyson (surname), people with the surname Dyson
* Dyson (company), a Singaporean multinational home appliances company founded by James Dyson
* Dyson (crater), a crater on the Moon
* Dyson (operating system), a Unix general-pur ...
showed in 1967 that if all the particles are
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
, then the inequality () cannot be true and the system is thermodynamically unstable. It was in fact later proved that in this case the energy goes like
instead of being linear in
.
It is therefore important that either the positive or negative charges are
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
. In other words, stability of matter is a consequence of the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
. In real life electrons are indeed fermions, but finding the right way to use Pauli's principle and prove stability turned out to be remarkably difficult. Michael Fischer and
David Ruelle
David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term ''strange attractor'', and developed a n ...
formalized the conjecture in 1966 and offered a bottle of
Champagne
Champagne (, ) is a sparkling wine originated and produced in the Champagne wine region of France under the rules of the appellation, that demand specific vineyard practices, sourcing of grapes exclusively from designated places within it, spe ...
to anybody who could prove it.
Dyson and Lenard found the proof of () a year later
and therefore got the bottle.
As was mentioned before, stability is a necessary condition for the existence of macroscopic objets, but it does not immediately imply the existence of thermodynamic functions. One should really show that the energy really behaves linearly in the number of particles. Based on the Dyson-Lenard result, this was solved in an ingenious way by
Elliott Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.
Lieb is ...
and
Joel Lebowitz
Joel Louis Lebowitz (born May 10, 1930) is a mathematical physicist widely acknowledged for his outstanding contributions to statistical physics, statistical mechanics and many other fields of Mathematics and Physics.
Lebowitz has published mor ...
in 1972.
The Dyson-Lenard proof is ''″extraordinarily complicated and difficult″''
and relies on deep and tedious analytical bounds. The obtained constant
in () was also very large. In 1975,
Elliott Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.
Lieb is ...
and
Walter Thirring
Walter Thirring (29 April 1927 – 19 August 2014) was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.Thirring, H. Über die Wirkung rotierender ferner Massen ...
found a simpler and more conceptual proof, based on a spectral inequality, now called the
Lieb-Thirring inequality.
They got a constant
which was by several orders of magnitude smaller than the Dyson-Lenard constant and had a realistic value.
They arrived at the final inequality
where
is the largest nuclear charge and
is the number of electronic spin states which is 2. Since
, this yields the desired linear lower bound ().
The idea of Lieb-Thirring was to bound the quantum energy from below in terms of the
Thomas-Fermi energy. The latter is always stable due to a theorem of
Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care fo ...
which states that atoms can never bind in Thomas-Fermi theory.
The new
Lieb-Thirring inequality was used to bound the quantum kinetic energy of the electrons in terms of the Thomas-Fermi kinetic energy
. Teller's ''No-Binding Theorem'' was in fact also used to bound from below the total Coulomb interaction in terms of the simpler Hartree energy appearing in Thomas-Fermi theory. Speaking about the Lieb-Thirring proof, Freeman Dyson wrote later
: ''″Lenard and I found a proof of the stability of matter in 1967. Our proof was so complicated and so unilluminating that it stimulated Lieb and Thirring to find the first decent proof. (...) Why was our proof so bad and why was theirs so good? The reason is simple. Lenard and I began with mathematical tricks and hacked our way through a forest of inequalities without any physical understanding. Lieb and Thirring began with physical understanding and went on to find the appropriate mathematical language to make their understanding rigorous. Our proof was a dead end. Theirs was a gateway to the new world of ideas.″''
The Lieb-Thirring approach has generated many subsequent works and extensions.
(Pseudo-)Relativistic systems
,
magnetic fields
quantized fields
and two-dimensional
fractional statistics
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
(
anyons
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchanging ...
)
have for instance been studied since the Lieb-Thirring paper.
The form of the bound () has also been improved over the years. For example, one can obtain a constant independent of the number
of nuclei.
Bibliography
*
The Stability of Matter: From Atoms to Stars'. Selecta of
Elliott H. Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physics#Mathematically rigorous physics, mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, Cond ...
. Edited by W. Thirring, and with a preface by F. Dyson. Fourth edition. Springer, Berlin, 2005.
*
Elliott H. Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physics#Mathematically rigorous physics, mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, Cond ...
and
Robert Seiringer
Robert Seiringer (1 September 1976, Vöcklabruck) is an Austrian mathematical physicist.
Life and work
Seiringer studied physics at the University of Vienna, where in 1999 he acquired his diploma and in 2000 with Jakob Yngvason as thesis advi ...
,
The Stability of Matter in Quantum Mechanics'. Cambridge Univ. Press, 2010.
*
Elliott H. Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physics#Mathematically rigorous physics, mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, Cond ...
The stability of matter: from atoms to stars ''Bull. Amer. Math. Soc. (N.S.)'' 22 (1990), no. 1, 1-49.
References
{{Reflist
Mathematical physics
Statistical mechanics theorems