View source for Stability of matter

{{Short description|Problem in statistical physics}}{{Primary sources|date=November 2024}}

In [[physics]], the '''stability of matter''' refers to the ability of a large number of [[charged particle]]s, such as [[electron]]s and [[proton]]s, to form macroscopic objects without collapsing or blowing apart due to [[Electromagnetism|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 mechanics|quantum mechanical]] explanation. 

The first solution to this problem was provided by [[Freeman Dyson]] and Andrew Lenard in 1967–1968,<ref name="DysLen1">{{cite journal |last1=Dyson |first1=Freeman J. |last2=Lenard |first2=A. |title=Stability of Matter. I |journal=Journal of Mathematical Physics |date=March 1967 |volume=8 |issue=3 |pages=423–434 |doi=10.1063/1.1705209|bibcode=1967JMP.....8..423D }}</ref><ref name="DysLen2">{{cite journal |last1=Lenard |first1=A. |last2=Dyson |first2=Freeman J. |title=Stability of Matter. II |journal=Journal of Mathematical Physics |date=May 1968 |volume=9 |issue=5 |pages=698–711 |doi=10.1063/1.1664631|bibcode=1968JMP.....9..698L |doi-access=free }}</ref> but a shorter and more conceptual proof was found later by [[Elliott Lieb]] and [[Walter Thirring]] in 1975 using the [[Lieb–Thirring inequality]].<ref name="LT">{{cite journal |last1=Lieb |first1=Elliott H. |last2=Thirring |first2=Walter E. |title=Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter |journal=[[Physical Review Letters]] |date=15 September 1975 |volume=35 |issue=11 |pages=687–689 |doi=10.1103/PhysRevLett.35.687|bibcode=1975PhRvL..35..687L }}</ref> The stability of matter is partly due to the [[uncertainty principle]] and the [[Pauli exclusion principle]].<ref>{{Cite book |last=Marder |first=Michael P. |url=https://books.google.com/books?id=ijloadAt4BQC&q=marder+condensed+matter |title=Condensed Matter Physics |date=2010-11-17 |publisher=John Wiley & Sons |isbn=978-0-470-94994-8 |language=en}}</ref>

== Description of the problem ==

In [[statistical mechanics]], the existence of macroscopic objects is usually explained in terms of the behavior of the energy or the [[Thermodynamic free energy|free energy]] with respect to the total number <math>N</math> of particles. More precisely, the [[Ground state|ground-state]] energy should be a linear function of <math>N</math> for large values of <math>N</math>.
<ref>{{cite book |last1=Ruelle |first1=David |title=Statistical Mechanics: Rigorous Results |date=April 1999 |doi=10.1142/4090|isbn=978-981-02-3862-9| publisher=World Scientific|bibcode=1999smrr.book.....R }}</ref> 
In fact, if the  ground-state energy behaves proportional to <math>N^a</math> for some <math>a\neq1</math>, then pouring two glasses of water would provide an energy proportional to <math>(2N)^a-2N^a=(2^a-2)N^a</math>, which is enormous for large <math>N</math>. 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 <math>N</math>. 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 [[ion]]s), interacting solely via the [[Coulomb's law|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{{clarify|date=November 2024}} due to the long range of the Coulomb potential. Stability should be a consequence of [[Electric-field screening|screening effects]], but those are hard to quantify.

Let us denote by 
:<math>
H_{N,K}=-\sum_{i=1}^N\frac{\Delta_{x_i}}{2}-\sum_{k=1}^K\frac{\Delta_{R_k}}{2M_k}-\sum_{i=1}^N\sum_{k=1}^K\frac{z_k}{|x_i-R_k|}+\sum_{1\leq i<j\leq N}\frac{1}{|x_i-x_j|}+\sum_{1\leq k<m\leq K}\frac{z_kz_m}{|R_k-R_m|}
</math>
the [[Hamiltonian (quantum mechanics)|quantum Hamiltonian]] of <math>N</math> electrons and <math>K</math> nuclei of charges <math>z_1,...,z_K</math> and masses <math>M_1,...,M_K</math> in [[atomic units]]. Here <math>\Delta=\nabla^2=\sum_{j=1}^3\partial_{jj}</math> denotes the [[Laplace operator|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 <math>H_{N,K}</math>) is bounded from below by a constant times the total number of particles:
{{NumBlk|:|<math>
E_{N,K}=\min\sigma(H_{N,K})\geq -C(N+K).
</math>|{{EquationRef|1}}}}
The constant <math>C</math> can depend on the largest number of [[Spin (physics)|spin]] states for each particle as well as the largest value of the charges <math>z_k</math>. It should ideally not depend on the masses <math>M_1,...,M_K</math> so as to be able to consider the infinite mass limit, that is, classical nuclei.

== History ==

=== 19th century physics ===
At the end of the 19th century it was understood that electromagnetic forces held matter together. However two problems co-existed.<ref name=":0">{{Cite journal |last=Jones |first=W |date=1980-04-01 |title=Earnshaw's theorem and the stability of matter |url=https://iopscience.iop.org/article/10.1088/0143-0807/1/2/004 |journal=European Journal of Physics |volume=1 |issue=2 |pages=85–88 |doi=10.1088/0143-0807/1/2/004 |bibcode=1980EJPh....1...85J |issn=0143-0807}}</ref> [[Earnshaw's theorem]] from 1842, proved that no charged body can be in a stable equilibrium under the influence of [[Electrostatics|electrostatic forces]] alone.<ref name=":0" /> The second problem was that [[James Clerk Maxwell]] had shown that accelerated charge produces [[electromagnetic radiation]], which in turn reduces its motion.<ref name=":0" /> 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.<ref name=":0" /> 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.<ref name=":0" />

=== Dyson–Lenard solution ===
[[Freeman Dyson]] showed<ref>{{cite journal |last1=Dyson |first1=Freeman J. |title=Ground-State Energy of a Finite System of Charged Particles |journal=Journal of Mathematical Physics |date=August 1967 |volume=8 |issue=8 |pages=1538–1545 |doi=10.1063/1.1705389|bibcode=1967JMP.....8.1538D }}</ref> in 1967 that if all the particles are [[bosons]], then the inequality ({{EquationNote|1}}) cannot be true and the system is thermodynamically unstable. It was in fact later proved that in this case the energy goes like <math>N^{7/5}</math> instead of being linear in <math>N</math>.<ref>{{cite journal |last1=Conlon |first1=Joseph G. |last2=Lieb |first2=Elliott H. |last3=Yau |first3=Horng-Tzer |authorlink3=Horng-Tzer Yau |title=TheN 7/5 law for charged bosons |journal=Communications in Mathematical Physics |date=September 1988 |volume=116 |issue=3 |pages=417–448 |doi=10.1007/BF01229202|bibcode=1988CMaPh.116..417C |doi-access=free}}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Solovej |first2=Jan Philip |title=Ground State Energy of the Two-Component Charged Bose Gas |journal=Communications in Mathematical Physics |date=December 2004 |volume=252 |issue=1–3 |pages=485–534 |doi=10.1007/s00220-004-1144-1|arxiv=math-ph/0311010 |bibcode=2004CMaPh.252..485L |doi-access=free}}</ref>
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 Fisher|Michael Fischer]] and [[David Ruelle]] formalized the conjecture in 1966<ref>{{cite journal |last1=Fisher |first1=Michael E. |last2=Ruelle |first2=David |title=The Stability of Many-Particle Systems |journal=Journal of Mathematical Physics |date=February 1966 |volume=7 |issue=2 |pages=260–270 |doi=10.1063/1.1704928|bibcode=1966JMP.....7..260F }}</ref> According to Dyson, Fischer and Ruelled offered a bottle of [[Champagne]] to anybody who could prove it.<ref name="youtube1">{{cite web |last1=Dyson |first1=Freeman |title=A bottle of champagne to prove the stability of matter |url=https://www.youtube.com/watch?v=KXzOdZmM7Zo |website=Youtube |date=5 September 2016 |access-date=22 June 2022}}</ref> Dyson and Lenard found the proof of ({{EquationNote|1}}) a year later<ref name="DysLen1" /><ref name="DysLen2" /> and therefore got the bottle.

=== Lieb–Thirring inequality ===
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.<ref>{{cite journal |last1=Lieb |first1=Elliott H |last2=Lebowitz |first2=Joel L |title=The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei |journal=[[Advances in Mathematics]] |date=December 1972 |volume=9 |issue=3 |pages=316–398 |doi=10.1016/0001-8708(72)90023-0 |doi-access=free}}</ref>

According to Dyson himself, the Dyson–Lenard proof is "extraordinarily complicated and difficult"<ref name="youtube1" /> and relies on deep and tedious analytical bounds. The obtained constant <math>C</math> in ({{EquationNote|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]].<ref name="LT" /><ref name="LT2">{{cite journal |last1=Lieb |first1=Elliott H. |last2=Thirring |first2=Walter E. |title=Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities |journal=Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann |date=31 December 2015 |pages=269–304 |doi=10.1515/9781400868940-014|isbn=978-1-4008-6894-0 }}</ref> 
They got a constant <math>C</math> which was by several orders of magnitude smaller than the Dyson–Lenard constant and had a realistic value.
They arrived at the final inequality
{{NumBlk|:|<math>
E_{N,K}\geq -0.231 q^{\frac23}N\left(1+2.16 Z(K/N)^{\frac13}\right)^2
</math>|{{EquationRef|2}}}}
where <math>Z=\max(z_k)</math> is the largest nuclear charge and <math>q</math> is the number of electronic spin states which is 2. Since <math> N^{1/3}K^{2/3}\leq N+K</math>, this yields the desired linear lower bound ({{EquationNote|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]].<ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Simon |first2=Barry |title=Thomas-Fermi Theory Revisited |journal=[[Physical Review Letters]] |date=10 September 1973 |volume=31 |issue=11 |pages=681–683 |doi=10.1103/PhysRevLett.31.681|bibcode=1973PhRvL..31..681L |url=https://authors.library.caltech.edu/85914/ }}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H |last2=Simon |first2=Barry |title=The Thomas-Fermi theory of atoms, molecules and solids |journal=[[Advances in Mathematics]] |date=January 1977 |volume=23 |issue=1 |pages=22–116 |doi=10.1016/0001-8708(77)90108-6|doi-access=free}}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H. |title=Thomas-fermi and related theories of atoms and molecules |journal=Reviews of Modern Physics |date=1 October 1981 |volume=53 |issue=4 |pages=603–641 |doi=10.1103/RevModPhys.53.603|bibcode=1981RvMP...53..603L }}</ref> 
The Lieb–Thirring inequality was used to bound the quantum kinetic energy of the electrons in terms of the Thomas–Fermi kinetic energy <math>\int_{\mathbb{R}^3}\rho(x)^{\frac53}d^3x</math>. Teller's no-binding theorem was in fact also used to bound from below the total Coulomb interaction in terms of the simpler [[Hartree|Hartree energy]] appearing in Thomas–Fermi theory. Speaking about the Lieb–Thirring proof, Dyson wrote later<ref name="selecta">{{cite book |last1=Lieb |first1=Elliott H. |url=http://projecteuclid.org/euclid.bams/1183555452 |title=The Stability of Matter: From Atoms to Stars: Selecta of Elliott H. Lieb |date=2005 |publisher=Springer |isbn=978-3-540-22212-5 |editor-last1=Thirring |editor-first1=Walter |doi=10.1007/b138553}}</ref><ref name="youtube2">{{cite web |last1=Dyson |first1=Freeman |date=5 September 2016 |title=Lieb and Thirring clean up my matter stability proof |url=https://www.youtube.com/watch?v=8nngG6ipyVs |access-date=22 June 2022 |website=youtube}}</ref>

{{Quote|text=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.}}

=== Further work ===
The Lieb–Thirring approach has generated many subsequent works and extensions. (Pseudo-)Relativistic systems<ref name="LieYau">{{cite journal |last1=Lieb |first1=Elliott H. |last2=Yau |first2=Horng-Tzer |authorlink2=Horng-Tzer Yau |title=The stability and instability of relativistic matter |journal=Communications in Mathematical Physics |date=June 1988 |volume=118 |issue=2 |pages=177–213 |doi=10.1007/BF01218577|bibcode=1988CMaPh.118..177L |doi-access=free}}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Siedentop |first2=Heinz |last3=Solovej |first3=Jan Philip |title=Stability and instability of relativistic electrons in classical electromagnetic fields |journal=Journal of Statistical Physics |date=October 1997 |volume=89 |issue=1–2 |pages=37–59 |doi=10.1007/BF02770753|arxiv=cond-mat/9610195 |bibcode=1997JSP....89...37L |doi-access=free}}</ref><ref>{{cite journal |last1=Frank |first1=Rupert L. |last2=Lieb |first2=Elliott H. |last3=Seiringer |first3=Robert |title=Stability of Relativistic Matter with Magnetic Fields for Nuclear Charges up to the Critical Value |journal=Communications in Mathematical Physics |date=20 August 2007 |volume=275 |issue=2 |pages=479–489 |doi=10.1007/s00220-007-0307-2|bibcode=2007CMaPh.275..479F |doi-access=free|arxiv=math-ph/0610062 }}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Loss |first2=Michael |last3=Siedentop |first3=Heinz |title=Stability of relativistic matter via Thomas-Fermi theory |journal=Helvetica Physica Acta |date=1 December 1996 |volume=69 |issue=5–6 |pages=974–984 |arxiv=cond-mat/9608060 |bibcode=1996cond.mat..8060L |url=https://collaborate.princeton.edu/en/publications/stability-of-relativistic-matter-via-thomas-Fermi-theory |issn=0018-0238}}</ref> 
magnetic fields<ref>{{cite journal |last1=Fefferman |first1=C |authorlink1=Charles Fefferman |title=Stability of Coulomb systems in a magnetic field. |journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |date=23 May 1995 |volume=92 |issue=11 |pages=5006–5007 |doi=10.1073/pnas.92.11.5006|pmid=11607547 |pmc=41836 |bibcode=1995PNAS...92.5006F |doi-access=free}}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Loss |first2=Michael |last3=Solovej |first3=Jan Philip |title=Stability of Matter in Magnetic Fields |journal=[[Physical Review Letters]] |date=7 August 1995 |volume=75 |issue=6 |pages=985–989 |doi=10.1103/PhysRevLett.75.985|pmid=10060179 |arxiv=cond-mat/9506047 |bibcode=1995PhRvL..75..985L |s2cid=2794188 }}</ref> 
quantized fields<ref>{{cite journal |last1=Bugliaro |first1=Luca |last2=Fröhlich |first2=Jürg |last3=Graf |first3=Gian Michele |title=Stability of Quantum Electrodynamics with Nonrelativistic Matter |journal=[[Physical Review Letters]] |date=21 October 1996 |volume=77 |issue=17 |pages=3494–3497 |doi=10.1103/PhysRevLett.77.3494|pmid=10062234 |bibcode=1996PhRvL..77.3494B }}</ref><ref>{{cite journal |last1=Fefferman |first1=Charles |authorlink1=Charles Fefferman |last2=Fröhlich |first2=Jürg |last3=Graf |first3=Gian Michele |title=Stability of Ultraviolet-Cutoff Quantum Electrodynamics with Non-Relativistic Matter |journal=Communications in Mathematical Physics |date=1 December 1997 |volume=190 |issue=2 |pages=309–330 |doi=10.1007/s002200050243|bibcode=1997CMaPh.190..309F |doi-access=free}}</ref><ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Loss |first2=Michael |title=Stability of a Model of Relativistic Quantum Electrodynamics |journal=Communications in Mathematical Physics |date=1 July 2002 |volume=228 |issue=3 |pages=561–588 |doi=10.1007/s002200200665|arxiv=math-ph/0109002 |bibcode=2002CMaPh.228..561L |doi-access=free}}</ref>
and two-dimensional [[fractional statistics]] ([[anyons]])<ref>{{cite journal |last1=Lundholm |first1=Douglas |last2=Solovej |first2=Jan Philip |title=Local Exclusion and Lieb–Thirring Inequalities for Intermediate and Fractional Statistics |journal=Annales Henri Poincaré |date=June 2014 |volume=15 |issue=6 |pages=1061–1107 |doi=10.1007/s00023-013-0273-5 |arxiv=1301.3436 |bibcode=2014AnHP...15.1061L |doi-access=free}}</ref><ref>{{cite journal |last1=Lundholm |first1=Douglas |last2=Solovej |first2=Jan Philip |title=Hardy and Lieb-Thirring Inequalities for Anyons |journal=Communications in Mathematical Physics |date=September 2013 |volume=322 |issue=3 |pages=883–908 |doi=10.1007/s00220-013-1748-4|arxiv=1108.5129 |bibcode=2013CMaPh.322..883L |doi-access=free}}</ref>
have for instance been studied since the Lieb–Thirring paper. 
The form of the bound ({{EquationNote|1}}) has also been improved over the years. For example, one can obtain a constant independent of the number <math>K</math> of nuclei.<ref name="LieYau" /><ref>{{cite journal |last1=Hainzl |first1=Christian |last2=Lewin |first2=Mathieu |last3=Solovej |first3=Jan Philip |title=The thermodynamic limit of quantum Coulomb systems Part II. Applications |journal=[[Advances in Mathematics]] |date=June 2009 |volume=221 |issue=2 |pages=488–546 |doi=10.1016/j.aim.2008.12.011|doi-access=free|arxiv=0806.1709 }}</ref>

== Bibliography ==
* ''[https://link.springer.com/book/10.1007/b138553 The Stability of Matter: From Atoms to Stars]''. Selecta of [[Elliott H. Lieb]]. Edited by W. Thirring, and with a preface by F. Dyson. Fourth edition. Springer, Berlin, 2005.
* [[Elliott H. Lieb]] and [[Robert Seiringer]], ''[https://www.cambridge.org/core/books/stability-of-matter-in-quantum-mechanics/BC90EBAF135B745979EA749076A2F931 The Stability of Matter in Quantum Mechanics]''. Cambridge Univ. Press, 2010.
* [[Elliott H. Lieb]], [https://doi.org/10.1090/S0273-0979-1990-15831-8 The stability of matter: from atoms to stars]. ''Bull. Amer. Math. Soc. (N.S.)'' '''22''' (1990), no. 1, 1-49.

== References ==
{{Reflist}}

[[Category:Mathematical physics]]
[[Category:Statistical mechanics theorems]]