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. [3] The stability of matter is partly due to the uncertainty principle and the Pauli exclusion principle. [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 of particles. More precisely, the ground-state energy should be a linear function of for large values of . [5] In fact, if the ground-state energy behaves proportional to 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 scientists 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 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.
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:
(1) |
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.
At the end of the 19th century it was understood that electromagnetic forces held matter together. However two problems co-existed. [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] Classical physics was thus unable to explain the stability of matter and could only be explained with quantum mechanics which was developed at the beginning of the 20th century. [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. It was in fact later proved that in this case the energy goes like instead of being linear in . [8] [9] It is therefore important that either the positive or negative charges are fermions. 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.
As was mentioned before, stability is a necessary condition for the existence of macroscopic objects, 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 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. The obtained constant in ( 1 ) was also very large. In 1975, Elliott Lieb and Walter Thirring found a simpler and more conceptual proof, based on a spectral inequality, now called the Lieb–Thirring inequality. [3] [13] 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
(2) |
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 ( 1 ). The Lieb–Thirring idea 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 which states that atoms can never bind in Thomas–Fermi model. [14] [15] [16] The 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, Dyson wrote later [17] [18]
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 [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. The form of the bound ( 1 ) has also been improved over the years. For example, one can obtain a constant independent of the number of nuclei. [19] [30]
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.
Degenerate matter occurs when the Pauli exclusion principle significantly alters a state of matter at low temperature. The term is used in astrophysics to refer to dense stellar objects such as white dwarfs and neutron stars, where thermal pressure alone is not enough to prevent gravitational collapse. The term also applies to metals in the Fermi gas approximation.
A Fermi gas is an idealized model, an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.
Fermi liquid theory is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the behavior of many-body systems of particles in which the interactions between particles may be strong. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas, and why other properties differ.
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In astrophysics and condensed matter physics, electron degeneracy pressure is a quantum mechanical effect critical to understanding the stability of white dwarf stars and metal solids. It is a manifestation of the more general phenomenon of quantum degeneracy pressure.
A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting electrons in a one-dimensional conductor. Such a model is necessary as the commonly used Fermi liquid model breaks down for one dimension.
In particle physics, Fermi's interaction is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interacting with one another. This interaction explains beta decay of a neutron by direct coupling of a neutron with an electron, a neutrino and a proton.
Elliott Hershel Lieb is an American mathematical physicist. He is a professor of mathematics and physics at Princeton University. Lieb's works pertain to quantum and classical many-body problem, atomic structure, the stability of matter, functional inequalities, the theory of magnetism, and the Hubbard model.
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is
In theoretical condensed matter physics and quantum field theory, bosonization is a mathematical procedure by which a system of interacting fermions in (1+1) dimensions can be transformed to a system of massless, non-interacting bosons. The method of bosonization was conceived independently by particle physicists Sidney Coleman and Stanley Mandelstam; and condensed matter physicists Daniel C. Mattis and Alan Luther in 1975.
Walter Eduard Thirring was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.
The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The Thomas–Fermi model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory.
In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.
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The Lieb–Robinson bound is a theoretical upper limit on the speed at which information can propagate in non-relativistic quantum systems. It demonstrates that information cannot travel instantaneously in quantum theory, even when the relativity limits of the speed of light are ignored. The existence of such a finite speed was discovered mathematically by Elliott H. Lieb and Derek W. Robinson in 1972. It turns the locality properties of physical systems into the existence of, and upper bound for this speed. The bound is now known as the Lieb–Robinson bound and the speed is known as the Lieb–Robinson velocity. This velocity is always finite but not universal, depending on the details of the system under consideration. For finite-range, e.g. nearest-neighbor, interactions, this velocity is a constant independent of the distance travelled. In long-range interacting systems, this velocity remains finite, but it can increase with the distance travelled.
In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. They are named after E. H. Lieb and W. E. Thirring.
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