Bogoliubov inequality

The Bogoliubov inequality describes two inequalities that both make very general statements in statistical physics . The first so-called inequality is rather abstract and relates an expression formed with two operators, A or C , (an expected value of quantum mechanical operators in thermal equilibrium ) to a product of two correlation functions formed with the separate operators. The inequality was published in 1962 by the Russian physicist and mathematician Nikolai Nikolajewitsch Bogoljubow . Variant 2 is more concrete: it concerns the free energy of a thermodynamic system and its various approximations and is more generally known (see many standard textbooks on statistical physics ).

Content of variant 1

A physical system is considered , described by means of a Hamilton operator . Then the following applies for two operators and (for which the specified mean values exist, but which are otherwise arbitrary): ${\ displaystyle H}$ ${\ displaystyle A}$ ${\ displaystyle C}$

{\ displaystyle | \ langle [A, C] \ rangle | ^ {2} \ leq {\ frac {\ beta} {2}} \ langle \ {A, A ^ {\ dagger} \} \ rangle \ cdot \ langle [C ^ {\ dagger}, [H, C]] \ rangle \ qquad {\ text {with}} \ qquad \ beta = {\ frac {1} {k _ {\ mathrm {B}} T}}}

wherein as a commutator or as anti-commutator are to be understood as well as the expectation value of an operator as ${\ displaystyle [A, C]}$ ${\ displaystyle \ {A, A ^ {\ dagger} \}}$ ${\ displaystyle X}$

{\ displaystyle \ langle X \ rangle = {\ text {Sp}} (e ^ {- \ beta H} X) / {\ text {Sp}} (e ^ {- \ beta H})}

given is. is the Boltzmann constant . The (original) proof of the Mermin-Wagner theorem , a fundamental theorem about the impossibility of ordered two-dimensional ferromagnets (or superconductors or crystals) with isotropic interaction, is mainly based on this inequality. ${\ displaystyle k _ {\ mathrm {B}}}$

Proof idea

The proof of the Bogoliubov inequality is based on the fact that about

{\ displaystyle (A, B): = \ sum _ {nm} ^ {E_ {n} \ neq E_ {m}} \ langle n | A ^ {\ dagger} | m \ rangle \ langle m | B | n \ rangle {\ frac {e ^ {- \ beta E_ {m}} - e ^ {- \ beta E_ {n}}} {E_ {n} -E_ {m}}}}

a positive semi-definite scalar product can be defined. As a scalar product, it satisfies the Schwarz inequality :

{\ displaystyle | (A, B) | ^ {2} \ leq (A, A) \ cdot (B, B)}

If one now looks at the inequality, one obtains. ${\ displaystyle B = [C ^ {\ dagger}, H]}$

Variant 2

Another relationship is also known as Bogoliubov's inequality, but is more generally applicable, e.g. B. in the approximation of the so-called. Free energy of any thermodynamic system by approximation methods, z. B. by a molecular field approximation . This relationship, also known as "Bogoliubov's inequality", is based on the fact that in such cases the Hamiltonian of the system is replaced by an approximation . The relationship then applies ${\ displaystyle F}$ ${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle {\ mathcal {H}} _ {0}}$

{\ displaystyle F \ leq F_ {0} + \ langle {\ mathcal {H}} - {\ mathcal {H}} _ {0} \ rangle _ {0} \ ,,}

where on the right side of this inequality all expected values are to be calculated consistently with the approximation operator, e.g. B. The free energy is essentially the logarithm of the sum of states . The multiplication sign has now been replaced by the sum sign, +, which is appropriate because of the logarithmic character of the free energy ( ). ${\ displaystyle \ langle X \ rangle _ {0} = {\ frac {{\ rm {{track} \, \,}} (e ^ {- \ beta {\ mathcal {H}} _ {0}} X )} {{\ rm {trace}} \, \, e ^ {- \ beta {\ mathcal {H}} _ {0}}}} \ ,.}$ ${\ displaystyle F = - \ beta ^ {- 1} \ ln {\ rm {{track} \, \, e ^ {- \ beta {\ mathcal {H}}}}} \ ,.}$ ${\ displaystyle \ cdot \ ,,}$ ${\ displaystyle \ ln a \ cdot b = \ ln a + \ ln b}$

A proof of variant 2 can be found in the specified article. Both variants are based on similar ideas.

literature

Nolting: Quantum Theory of Magnetism , Teubner, Vol. 2

swell

↑ NN Bogoliubov, Physics. Treatise Soviet Union 6, 1, 113, 229 (1962).
↑ Mermin, Wagner Absence of ferromagnetism or antiferromagnetism in 1 or 2 dimensional isotropic Heisenberg models , Physical Review Letters, Vol. 17, 1966, p. 1133.
↑ see e.g. B. the English Wikipedia in the article Helmholtz_free_energy # Bogoliubov_inequality .

[1] NN Bogoliubov, Physics. Treatise Soviet Union 6, 1, 113, 229 (1962).

[2] Mermin, Wagner Absence of ferromagnetism or antiferromagnetism in 1 or 2 dimensional isotropic Heisenberg models , Physical Review Letters, Vol. 17, 1966, p. 1133.

[3] see e.g. B. the English Wikipedia in the article Helmholtz_free_energy # Bogoliubov_inequality .