Lars Onsager's Regression Hypothesis

The goal of this section is to introduce Lars Onsager's regression hypothesis, a consequence of the fluctuation-dissipation theorem proved by Callen and Welton in 1951 (H. B Callen and T. A. Welton, Phys. Rev. 83, 34 (1951)). Here, we derive the regression hypothesis from the principles of statistical mechanics.

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 $ A$ for a system at thermal equilibrium. Such property fluctuates in time with spontaneous microscopic fluctuations

$\displaystyle \delta A(t) = A(t) - \langle A \rangle.$ (421)
Here, $ A(t)$ is the instantaneous value of the observable and $ \langle A \rangle$ is the equilibrium ensemble average
$\displaystyle \langle A \rangle = \frac{\text{Tr} \{ A(t) e^{-\beta H_0} \}}{\text{Tr}\{ e^{-\beta H_0} \}}.$ (422)
The average correlation between $ \delta A(t)$ and an instantaneous fluctuation at time zero $ \delta A(0)$ is described by the correlation function
$\displaystyle C(t) = \langle \delta A(t) \delta A(0) \rangle = \langle A(t) A(0)\rangle - \langle A \rangle^2,$ (423)

where $ \langle \rangle$ represents the equilibrium ensemble average. Note that at small times, instantaneous fluctuations are correlated and therefore,

$\displaystyle \lim_{t \rightarrow 0} C(t) = \langle (\delta A(0))^2 \rangle,$ (424)

has a finite value. However, at large times $ C(t)$ vanishes, i.e.,

$\displaystyle \lim_{t \rightarrow \infty } C(t) = 0,$ (425)

since $ \delta A(t)$ becomes uncorrelated to $ \delta A(0)$. 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

$\displaystyle \frac{\overline{A}(t) - \langle A \rangle}{\overline{A}(0) - \lan......frac{\langle \delta A(t) \delta A(0) \rangle}{\langle (\delta A(0))^2 \rangle}.$ (426)

where $ \overline{A}(t)$ describes the macroscopic relaxation of the observable towards its equilibrium value $ \langle A \rangle$ 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 $ f$ (e.g., an electric field) that couples to the dynamical variable $ A(t)$ (e.g., the instantaneous total dipole moment). The external field is assumed to be so weak that the perturbation Hamiltonian $ \Delta H$, written as an expansion in powers of $ f$, can be approximated to be first order in the field as follows,

$\displaystyle \Delta H = - f A(0).$ (427)

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

$\displaystyle H = H_0 + \Delta H,$ (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

$\displaystyle \overline{A}(t) = \frac{\text{Tr} \{ A(t) e^{-\beta (H_0 + \Delta H)} \}}{\text{Tr}\{ e^{-\beta (H_0 + \Delta H)} \}},$ (429)

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

$\displaystyle e^{-\beta (H_0 + \Delta H)} \approx e^{-\beta H_0} (1-\beta \Delta H).$ (430)

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

$\displaystyle \overline{A}(t) \approx \frac{\text{Tr} \{ e^{-\beta H_0 }(1-\bet......text{Tr} \{ e^{-\beta H_0} \} - \beta \text{Tr} \{ e^{-\beta H_0} \Delta H \}},$ (431)


$\displaystyle \overline{A}(t) \approx \frac{\text{Tr} \{ e^{-\beta H_0 } A(t) \......H A(t) \}}{\text{Tr} \{ e^{-\beta H_0 } \} (1-\beta \langle \Delta H \rangle)},$ (432)

which gives

$\displaystyle \overline{A}(t) \approx \frac{\langle A(t) \rangle}{1 - \beta \la...... \beta \frac{\langle \Delta H A(t) \rangle}{1 - \beta \langle \Delta H\rangle}.$ (433)

Introducing the approximation

$\displaystyle 1/(1-x) \approx 1+x,$ (434)

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

$\displaystyle \overline{A}(t) \approx \langle A \rangle - \beta \langle \Delta \Delta H \rangle (\langle A \rangle - \beta \langle \Delta H A(t) \rangle).$ (435)

Therefore, according to Eq. (435),

$\displaystyle \overline{A}(t) \approx \langle A \rangle - \beta (\langle \Delta H A(t) \rangle - \langle \Delta H \rangle \langle A \rangle) + O((\Delta H)^2).$ (436)

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

$\displaystyle \overline{A}(0) - \langle A \rangle \approx f \beta (\langle A(0)^2 \rangle - \langle A(0) \rangle^2),$ (437)


$\displaystyle \overline{A}(t) - \langle A \rangle \approx f \beta (\langle A(0) A(t) \rangle - \langle A(0) \rangle \langle A(t) \rangle).$ (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),

$\displaystyle \langle \delta A(t) \delta A(0) \rangle = \langle A(t) A(0) \rangle - \langle A \rangle^2.$ (439)