# Average value, theorem on variations of the

*in statistical mechanics*

A statement on the variation of the average value of a dynamic magnitude in the Gibbs statistical aggregate as a result of an infinitesimal change in the Hamiltonian. The variation of the average depends, generally speaking, on the way of "inclusion" of the change in the Hamiltonian and on the initial conditions. Let a system of many interacting particles (a quantum system or a classical system), that can explicitly be described by a time-independent Hamiltonian $ H $, be in thermodynamic equilibrium at the initial moment of time $ t \rightarrow - \infty $. As a result of an adiabatic inclusion of an infinitesimal, time-dependent perturbation of the Hamiltonian

$$ H \rightarrow H + \delta V (t), $$

where

$$ \delta V (t) = \ \sum _ {(E) } e ^ {\epsilon t - iE t } \delta V _ {E} ,\ \ \epsilon > 0,\ \ \epsilon \rightarrow +0, $$

the Gibbs equilibrium average $ \langle A\rangle $ of the explicitly time-independent dynamic variable $ A $ will change (in a linear approximation with respect to the perturbation) by the magnitude

$$ \delta \langle A (t) \rangle = \ - 2 \pi i \sum _ { (E) } e ^ {\epsilon t - iEt } \ll A \mid \delta V _ {E} \gg _ {E} ^ {( \mathop{\rm ret} ) } , $$

where $ {\ll A \mid \delta {V _ {E} } \gg } _ {E} ^ {( \mathop{\rm ret} ) } $ is the Fourier transform of the retarded commutator Green function (cf. Green function in statistical mechanics).

The main application of the theorem is in the theory of non-equilibrium irreversible processes (in which one formulation of the theorem is also known as the fluctuation-dissipation theorem), and in deducing the chains and systems of equations for the Green functions from the chains and systems of equations for the correlation functions (cf. Correlation function in statistical mechanics).

#### References

[1] | R. Kubo, J. Phys. Soc. Japan , 12 (1957) pp. 570 |

[2] | N.N. Bogolyubov, jr., B.I. Sadovnikov, Zh. Eksper. Teoret. Fiz. : 43 (1962) pp. 677 |

[3] | N.N. Bogolyubov jr., B.I. Sadovnikov, "Some questions in statistical mechanics" , Moscow (1975) (In Russian) |

[4] | S.V. Tyablikov, "Methods of the quantum theory of magnetism" , Plenum (1967) (Translated from Russian) |

#### Comments

#### References

[a1] | N.N. Bogolyubov, N.N. Bogolyubov jr., "Introduction to quantum statistical mechanics" , World Sci. (1982) |

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Average value, theorem on variations of the.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Average_value,_theorem_on_variations_of_the&oldid=45535