Bohr–Van Leeuwen theorem: Difference between revisions
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an extension of the Bohr-van Leeuwen theorem through newly discovered magnetic field impacts on thermal bath's state. Tags: Reverted Visual edit: Switched |
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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. |
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. |
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==A classical extension of the Bohr-van Leeuwen theorem== |
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Recent research uncovered the effects of magnetic fieldson classical systems that were not anticipated by the original Bohr-van Leeuwen theorem<ref name=weak>{{Cite journal |last=Matevosyan |first=Ashot |last2=Allahverdyan |first2=Armen E. |date=2023-01-17 |title=Lasting effects of static magnetic field on classical Brownian motion |url=https://link.aps.org/doi/10.1103/PhysRevE.107.014125 |journal=Physical Review E |language=en |volume=107 |issue=1 |doi=10.1103/PhysRevE.107.014125 |issn=2470-0045}}</ref>. These effects have been demonstrated on a system composed of a classical charge coupled to an uncharged thermal bath, as described by the [http://www.scholarpedia.org/article/Caldeira-Leggett_model Caldeira-Leggett model] <ref> |
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{{cite journal |
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| last = Magalinskii |
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| first = V. B. |
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| title = Dynamical model in the theory of the Brownian motion |
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| journal = Soviet Physics JETP |
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| volume = 9 |
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| issue = 6 |
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| pages = 1381–1382 |
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| year = 1959 |
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| publisher = American Institute of Physics |
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}} |
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{{Cite journal |last=Zwanzig |first=Robert |date=1973-11-01 |title=Nonlinear generalized Langevin equations |url=https://link.springer.com/article/10.1007/BF01008729 |journal=Journal of Statistical Physics |language=en |volume=9 |issue=3 |pages=215–220 |doi=10.1007/BF01008729 |issn=1572-9613}} |
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{{Cite journal |last=Caldeira |first=A. O |last2=Leggett |first2=A. J |date=1983-09-01 |title=Quantum tunnelling in a dissipative system |url=https://linkinghub.elsevier.com/retrieve/pii/0003491683902026 |journal=Annals of Physics |volume=149 |issue=2 |pages=374–456 |doi=10.1016/0003-4916(83)90202-6 |issn=0003-4916}} |
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</ref>. The thermal bath is composed of independent oscillators with different frequencies weakly coupled to the charged particles. The Caldeira-Leggett model successfully reproduces the [[Boltzmann distribution|Gibbs distribution]] and [[Langevin equation]] for the charged ([[Brownian motion|Brownian]]) particle. |
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After the charged particle relaxes from a non-equilibrium state under a constant magnetic field, the final equilibrium state has zero [[magnetization]] (and zero angular momentum) following the Bohr-van Leeuwen theorem. However, the uncharged oscillators within the bath gain a substantial average angular momentum, meaning they start to rotate. Specifically, the bath oscillators are divided into two groups, each rotating in opposite directions, thereby respecting the conservation law of the angular momentum. |
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This effect is connected to the bath slightly falling out of equilibrium due to the influence of the charged (Brownian) particle and is persistent over arbitrarily long periods of time. Thus, the angular momentum of the bath is affected by the magnetic field over these periods <ref name=weak/>. |
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==References== |
==References== |
Revision as of 10:31, 15 August 2024
The Bohr–Van Leeuwen theorem states that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero.[1] This makes magnetism in solids solely a quantum mechanical effect and means that classical physics cannot account for paramagnetism, diamagnetism and ferromagnetism. Inability of classical physics to explain triboelectricity also stems from the Bohr–Van Leeuwen theorem.[2]
History
What is today known as the Bohr–Van Leeuwen theorem was discovered by Niels Bohr in 1911 in his doctoral dissertation[3] and was later rediscovered by Hendrika Johanna van Leeuwen in her doctoral thesis in 1919.[4] In 1932, J. H. Van Vleck formalized and expanded upon Bohr's initial theorem in a book he wrote on electric and magnetic susceptibilities.[5]
The significance of this discovery is that classical physics does not allow for such things as paramagnetism, diamagnetism and ferromagnetism and thus quantum physics is needed to explain the magnetic events.[6] This result, "perhaps the most deflationary publication of all time,"[7] may have contributed to Bohr's development of a quasi-classical theory of the hydrogen atom in 1913.
Proof
Statistical mechanics |
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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 , where is the energy of the system, is the Boltzmann constant, and is the absolute temperature. This energy is equal to the sum of the kinetic energy ( for a particle with mass and speed ) and the potential energy.[8]
The magnetic field does not contribute to the potential energy. The Lorentz force on a particle with charge and velocity is
where is the electric field and is the magnetic flux density. The rate of work done is and does not depend on . 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 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 and mass .
If its position is and velocity is , it produces a current and a magnetic moment[6]
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
where the dot represents a time derivative and are vector coefficients depending on the position coordinates .[6]
Maxwell–Boltzmann statistics gives the probability that the nth particle has momentum and coordinate as
where is the Hamiltonian, the total energy of the system.[6]
The thermal average of any function of these generalized coordinates is then
In the presence of a magnetic field,
where is the magnetic vector potential and is the electric scalar potential.
For each particle the components of the momentum and position are related by the equations of Hamiltonian mechanics:
Therefore,
so the moment is a linear function of the momenta .[6]
The thermally averaged moment,
is the sum of terms proportional to integrals of the form
where represents one of the momentum coordinates.
The integrand is an odd function of , so it vanishes.
Therefore, .[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.
A classical extension of the Bohr-van Leeuwen theorem
Recent research uncovered the effects of magnetic fieldson classical systems that were not anticipated by the original Bohr-van Leeuwen theorem[10]. These effects have been demonstrated on a system composed of a classical charge coupled to an uncharged thermal bath, as described by the Caldeira-Leggett model [11]. The thermal bath is composed of independent oscillators with different frequencies weakly coupled to the charged particles. The Caldeira-Leggett model successfully reproduces the Gibbs distribution and Langevin equation for the charged (Brownian) particle.
After the charged particle relaxes from a non-equilibrium state under a constant magnetic field, the final equilibrium state has zero magnetization (and zero angular momentum) following the Bohr-van Leeuwen theorem. However, the uncharged oscillators within the bath gain a substantial average angular momentum, meaning they start to rotate. Specifically, the bath oscillators are divided into two groups, each rotating in opposite directions, thereby respecting the conservation law of the angular momentum.
This effect is connected to the bath slightly falling out of equilibrium due to the influence of the charged (Brownian) particle and is persistent over arbitrarily long periods of time. Thus, the angular momentum of the bath is affected by the magnetic field over these periods [10].
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. hdl:10669/82347. 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" (PDF). Journal de Physique et le Radium. 2 (12): 361–377. doi:10.1051/jphysrad:01921002012036100. S2CID 97259591.
- ^ Van Vleck, J. H. (1932). The theory of electric and magnetic susceptibilities. Clarendon Press. ISBN 0-19-851243-0.
- ^ 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.
- ^ 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.
- ^ a b Matevosyan, Ashot; Allahverdyan, Armen E. (2023-01-17). "Lasting effects of static magnetic field on classical Brownian motion". Physical Review E. 107 (1). doi:10.1103/PhysRevE.107.014125. ISSN 2470-0045.
- ^ Magalinskii, V. B. (1959). "Dynamical model in the theory of the Brownian motion". Soviet Physics JETP. 9 (6). American Institute of Physics: 1381–1382. Zwanzig, Robert (1973-11-01). "Nonlinear generalized Langevin equations". Journal of Statistical Physics. 9 (3): 215–220. doi:10.1007/BF01008729. ISSN 1572-9613. Caldeira, A. O; Leggett, A. J (1983-09-01). "Quantum tunnelling in a dissipative system". Annals of Physics. 149 (2): 374–456. doi:10.1016/0003-4916(83)90202-6. ISSN 0003-4916.