stability of matter

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 $N$ of particles. More precisely, the ground-state energy should be a linear function of $N$ for large values of $N$.

{{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 }}

In fact, if the ground-state energy behaves proportional to $N^a$ for some $a\backslash neq1$, then pouring two glasses of water would provide an energy proportional to $(2N)^a-2N^a=(2^a-2)N^a$, which is enormous for large $N$. 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 $N$. 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{{clarify|date=November 2024}} 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

:$$

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}

At the end of the 19th century it was understood that electromagnetic forces held matter together. However two problems co-existed.{{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|url-access=subscription }} Earnshaw's theorem from 1842, proved that no charged body can be in a stable equilibrium under the influence of electrostatic forces alone. The second problem was that James Clerk Maxwell had shown that accelerated charge produces electromagnetic radiation, which in turn reduces its motion. 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. 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.

Freeman Dyson showed{{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 }} 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 $N^\{7/5\}$ instead of being linear in $N$.{{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}}{{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}}

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 and David Ruelle formalized the conjecture in 1966{{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 }} According to Dyson, Fisher and Ruelled offered a bottle of Champagne to anybody who could prove it.{{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}} Dyson and Lenard found the proof of ({{EquationNote|1}}) a year later 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.{{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}}

According to Dyson himself, the Dyson–Lenard proof is "extraordinarily complicated and difficult" and relies on deep and tedious analytical bounds. The obtained constant $C$ 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.{{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 }}

They got a constant $C$ 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|:|$$

E_{N,K}\geq -0.231 q^{\frac23}N\left(1+2.16 Z(K/N)^{\frac13}\right)^2

|{{EquationRef|2}}}}

where $Z=\backslash max(z\_k)$ is the largest nuclear charge and $q$ is the number of electronic spin states which is 2. Since $N^\{1/3\}K^\{2/3\}\backslash leq\; N+K$, 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.{{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/ }}{{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}}{{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 }}

The Lieb–Thirring inequality was used to bound the quantum kinetic energy of the electrons in terms of the Thomas–Fermi kinetic energy $\backslash int\_\{\backslash mathbb\{R\}^3\}\backslash rho(x)^\{\backslash frac53\}d^3x$. 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{{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}}{{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}}

{{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.}}

The Lieb–Thirring approach has generated many subsequent works and extensions. (Pseudo-)Relativistic systems{{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}}{{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}}{{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 }}{{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}}

magnetic fields{{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}}{{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 }}

quantized fields{{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 }}{{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}}{{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}}

and two-dimensional fractional statistics (anyons){{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}}{{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}}

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 $K$ of nuclei.{{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 }}

• [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.

{{Reflist}}

Category:Mathematical physics

Category:Statistical mechanics theorems