In physics, the stability of matter refers to the ability of a large number of charged particles, such as electrons and protons, to form macroscopic objects without collapsing or blowing apart due to electromagnetic interactions.
Classical physics predicts that such systems should be inherently unstable due to attractive and repulsive electrostatic forces between charges, and thus the stability of matter was a theoretical problem that required a quantum mechanical explanation.
The first solution to this problem was provided by Freeman Dyson and Andrew Lenard in 1967–1968,[1][2] but a shorter and more conceptual proof was found later by Elliott Lieb and Walter Thirring in 1975 using the Lieb–Thirring inequality.
[4] In statistical mechanics, 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
, then pouring two glasses of water would provide an energy proportional to
Neglecting other forces, it is reasonable to assume that ordinary matter is composed of negative and positive non-relativistic charges (electrons and ions), interacting solely via the Coulomb's interaction.
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.
Proving that such a system is thermodynamically stable is called the stability of matter problem and it is very difficult[clarification needed] due to the long range of the Coulomb potential.
Stability should be a consequence of screening effects, but those are hard to quantify.
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
At the end of the 19th century it was understood that electromagnetic forces held matter together.
[6] Earnshaw's theorem from 1842, proved that no charged body can be in a stable equilibrium under the influence of electrostatic forces alone.
[6] The second problem was that James Clerk Maxwell had shown that accelerated charge produces electromagnetic radiation, which in turn reduces its motion.
[6] In 1900, Joseph Larmor suggested the possibility of an electromagnetic system with electrons in orbits inside matter.
He showed that if such system existed, it could be scaled down by scaling distances and vibrations times, however this suggested a modification to Coulomb's law at the level of molecules.
[6] Freeman Dyson showed[7] in 1967 that if all the particles are bosons, then the inequality (1) cannot be true and the system is thermodynamically unstable.
In other words, stability of matter is a consequence of the Pauli exclusion principle.
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 formalized the conjecture in 1966[10] According to Dyson, Fischer and Ruelled offered a bottle of Champagne to anybody who could prove it.
[11] Dyson and Lenard found the proof of (1) a year later[1][2] and therefore got the bottle.
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 and Joel Lebowitz in 1972.
[12] According to Dyson himself, the Dyson–Lenard proof is "extraordinarily complicated and difficult"[11] and relies on deep and tedious analytical bounds.
which was by several orders of magnitude smaller than the Dyson–Lenard constant and had a realistic value.
The latter is always stable due to a theorem of Edward Teller which states that atoms can never bind in Thomas–Fermi model.
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.
Lenard and I began with mathematical tricks and hacked our way through a forest of inequalities without any physical understanding.
Theirs was a gateway to the new world of ideas.The Lieb–Thirring approach has generated many subsequent works and extensions.
(Pseudo-)Relativistic systems[19][20][21][22] magnetic fields[23][24] quantized fields[25][26][27] and two-dimensional fractional statistics (anyons)[28][29] have for instance been studied since the Lieb–Thirring paper.