# Bohr–Van Leeuwen theorem

Proof Statistical mechanics Thermodynamics Kinetic theory show Particle statistics show Thermodynamic ensembles show Models show Potentials show Scientists vte An intuitive proof The Bohr–Van Leeuwen theorem applies to an isolated system that cannot rotate. If the isolated system is allowed to rotate in response to an externally applied magnetic field, then this theorem does not apply.[8] If, in addition, there is only one state of thermal equilibrium in a given temperature and field, and the system is allowed time to return to equilibrium after a field is applied, then there will be no magnetization.

The probability that the system will be in a given state of motion is predicted by Maxwell–Boltzmann statistics to be proportional to {displaystyle exp(-U/k_{text{B}}T)} , where {displaystyle U} is the energy of the system, {displaystyle k_{text{B}}} is the Boltzmann constant, and {displaystyle T} is the absolute temperature. This energy is equal to the sum of the kinetic energy ( {displaystyle mv^{2}/2} for a particle with mass {displaystyle m} and speed {displaystyle v} ) and the potential energy.[8] The magnetic field does not contribute to the potential energy. The Lorentz force on a particle with charge {displaystyle q} and velocity {displaystyle mathbf {v} } is {displaystyle mathbf {F} =qleft(mathbf {E} +mathbf {v} times mathbf {B} right),} where {displaystyle mathbf {E} } is the electric field and {displaystyle mathbf {B} } is the magnetic flux density. The rate of work done is {displaystyle mathbf {F} cdot mathbf {v} =qmathbf {E} cdot mathbf {v} } and does not depend on {displaystyle mathbf {B} } . Therefore, the energy does not depend on the magnetic field, so the distribution of motions does not depend on the magnetic field.[8] In zero field, there will be no net motion of charged particles because the system is not able to rotate. There will therefore be an average magnetic moment of zero. Since the distribution of motions does not depend on the magnetic field, the moment in thermal equilibrium remains zero in any magnetic field.[8] A more formal proof So as to lower the complexity of the proof, a system with {displaystyle N} electrons will be used.

This is appropriate, since most of the magnetism in a solid is carried by electrons, and the proof is easily generalized to more than one type of charged particle.

Each electron has a negative charge {displaystyle e} and mass {displaystyle m_{text{e}}} .

If its position is {displaystyle mathbf {r} } and velocity is {displaystyle mathbf {v} } , it produces a current {displaystyle mathbf {j} =emathbf {v} } and a magnetic moment[6] {displaystyle mathbf {mu } ={frac {1}{2c}}mathbf {r} times mathbf {j} ={frac {e}{2c}}mathbf {r} times mathbf {v} .} The above equation shows that the magnetic moment is a linear function of the velocity coordinates, so the total magnetic moment in a given direction must be a linear function of the form {displaystyle mu =sum _{i=1}^{N}mathbf {a} _{i}cdot {dot {mathbf {r} }}_{i},} where the dot represents a time derivative and {displaystyle mathbf {a} _{i}} are vector coefficients depending on the position coordinates {displaystyle {mathbf {r} _{i},i=1ldots N}} .[6] Maxwell–Boltzmann statistics gives the probability that the nth particle has momentum {displaystyle mathbf {p} _{n}} and coordinate {displaystyle mathbf {r} _{n}} as {displaystyle dPpropto exp {left[-{frac {{mathcal {H}}(mathbf {p} _{1},ldots ,mathbf {p} _{N};mathbf {r} _{1},ldots ,mathbf {r} _{N})}{k_{text{B}}T}}right]}dmathbf {p} _{1},ldots ,dmathbf {p} _{N}dmathbf {r} _{1},ldots ,dmathbf {r} _{N},} where {displaystyle {mathcal {H}}} is the Hamiltonian, the total energy of the system.[6] The thermal average of any function {displaystyle f(mathbf {p} _{1},ldots ,mathbf {p} _{N};mathbf {r} _{1},ldots ,mathbf {r} _{N})} of these generalized coordinates is then {displaystyle langle frangle ={frac {int fdP}{int dP}}.} In the presence of a magnetic field, {displaystyle {mathcal {H}}={frac {1}{2m_{text{e}}}}sum _{i=1}^{N}left(mathbf {p} _{i}-{frac {e}{c}}mathbf {A} _{i}right)^{2}+ephi (mathbf {q} ),} where {displaystyle mathbf {A} _{i}} is the magnetic vector potential and {displaystyle phi (mathbf {q} )} is the electric scalar potential. For each particle the components of the momentum {displaystyle mathbf {p} _{i}} and position {displaystyle mathbf {r} _{i}} are related by the equations of Hamiltonian mechanics: {displaystyle {begin{aligned}{dot {mathbf {p} }}_{i}&=-partial {mathcal {H}}/partial mathbf {r} _{i}\{dot {mathbf {r} }}_{i}&=partial {mathcal {H}}/partial mathbf {p} _{i}.end{aligned}}} Therefore, {displaystyle {dot {mathbf {r} }}_{i}propto mathbf {p} _{i}-{frac {e}{c}}mathbf {A} _{i},} so the moment {displaystyle mu } is a linear function of the momenta {displaystyle mathbf {p} _{i}} .[6] The thermally averaged moment, {displaystyle langle mu rangle ={frac {int mu dP}{int dP}},} is the sum of terms proportional to integrals of the form {displaystyle int _{-infty }^{infty }(mathbf {p} _{i}-{frac {e}{c}}mathbf {A} _{i})dP,} where {displaystyle p} represents one of the momentum coordinates.

The integrand is an odd function of {displaystyle p} , so it vanishes.

Therefore, {displaystyle langle mu rangle =0} .[6] Applications The Bohr–Van Leeuwen theorem is useful in several applications including plasma physics: "All these references base their discussion of the Bohr–Van Leeuwen theorem on Niels Bohr's physical model, in which perfectly reflecting walls are necessary to provide the currents that cancel the net contribution from the interior of an element of plasma, and result in zero net diamagnetism for the plasma element."[9] Diamagnetism of a purely classical nature occurs in plasmas but is a consequence of thermal disequilibrium, such as a gradient in plasma density. Electromechanics and electrical engineering also see practical benefit from the Bohr–Van Leeuwen theorem.

See also List of plasma (physics) articles References ^ John Hasbrouck van Vleck stated the Bohr–Van Leeuwen theorem as "At any finite temperature, and in all finite applied electrical or magnetical fields, the net magnetization of a collection of electrons in thermal equilibrium vanishes identically." (Van Vleck, 1932) ^ Alicki, Robert; Jenkins, Alejandro (2020-10-30). "Quantum Theory of Triboelectricity". Physical Review Letters. 125 (18): 186101. arXiv:1904.11997. Bibcode:2020PhRvL.125r6101A. doi:10.1103/PhysRevLett.125.186101. ISSN 0031-9007. PMID 33196235. S2CID 139102854. ^ Bohr, Niehls (1972) [originally published as "Studier over Metallernes Elektrontheori", Københavns Universitet (1911)]. "The Doctor's Dissertation (Text and Translation)". In Rosenfeld, L.; Nielsen, J. Rud (eds.). Early Works (1905-1911). Niels Bohr Collected Works. Vol. 1. Elsevier. pp. 163, 165–393. doi:10.1016/S1876-0503(08)70015-X. ISBN 978-0-7204-1801-9. ^ Van Leeuwen, Hendrika Johanna (1921). "Problèmes de la théorie électronique du magnétisme". Journal de Physique et le Radium. 2 (12): 361–377. doi:10.1051/jphysrad:01921002012036100. ^ Van Vleck, J. H. (1932). The theory of electric and magnetic susceptibilities. Clarendon Press. ISBN 0-19-851243-0. ^ Jump up to: a b c d e f Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism. Clarendon Press. pp. 6–7. ISBN 0-19-851791-2. ^ Van Vleck, J. H. (1992). "Quantum mechanics: The key to understanding magnetism (Nobel lecture, 8 December 1977)". In Lundqvist, Stig (ed.). Nobel Lectures in Physics 1971-1980. World Scientific. ISBN 981-02-0726-3. ^ Jump up to: a b c d Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006). The Feynman Lectures on Physics. Vol. 2. p. 34-8. ISBN 978-0465024940. ^ Roth, Reece (1967). "Plasma Stability and the Bohr–Van Leeuwen Theorem" (PDF). NASA. Retrieved 2008-10-27. External links The early 20th century: Relativity and quantum mechanics bring understanding at last Categories: Classical mechanicsElectric and magnetic fields in matterPhysics theoremsStatistical mechanicsStatistical mechanics theorems

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