The regression hypothesis states that *the regression of
microscopic
thermal fluctuations at equilibrium follows the macroscopic law of
relaxation
of small non-equilibrium disturbances* (L. Onsager, *Phys. Rev.***37**,
405 (1931); **38**, 2265 (1931)).

In order to understand this hypothesis, consider an observable
for a system at thermal equilibrium. Such property fluctuates in time
with
*spontaneous
microscopic fluctuations*

(421) |

(422) |

(423) |

where
represents the *equilibrium ensemble average*. Note that at small
times, instantaneous fluctuations are correlated and therefore,

(424) |

has a finite value. However, at large times vanishes, i.e.,

(425) |

since
becomes uncorrelated to .
This decay of correlations is the *regression of microscopic thermal
fluctuations* referred to in Onsager's hypothesis. Therefore,
Onsager's
regression hypothesis can be formulated as follows

(426) |

where
describes the *macroscopic relaxation* of the observable towards
its
equilibrium value
while the system evolves from an initial state that is not far from
equilibrium
to its final state in equilibrium with a thermal reservoir.

In order to derive Eq.(426) from the ensemble postulates of statistical mechanics, consider preparing the system in a state that is not far from equilibrium by applying to the system a weak perturbational field (e.g., an electric field) that couples to the dynamical variable (e.g., the instantaneous total dipole moment). The external field is assumed to be so weak that the perturbation Hamiltonian , written as an expansion in powers of , can be approximated to be first order in the field as follows,

(427) |

Assume that such perturbational field was applied until the system equilibrated according to total Hamiltonian

(428) |

The macroscopic relaxation of the system is analyzed by switching off the external perturbational field and computing the evolution of the non-equilibrium ensemble average

(429) |

as the system evolves towards equilibrium. Since the perturbation is small, we can introduce the approximation

(430) |

Substituting Eq. (430) into Eq. (429), we obtain

(431) |

or

(432) |

which gives

(433) |

Introducing the approximation

(434) |

for small , in Eq. (433), we obtain

(435) |

Therefore, according to Eq. (435),

(436) |

Substituting Eq. (427) into Eq. (436) and keeping only the terms that are first order in we obtain,

(437) |

and

(438) |

Finally, Eqs. (437) and (438) complete the derivation of Eq. (425) in terms of the ensemble average postulates of statistical mechanics, since according to Eq. (421),

(439) |