  We study the statistically steady states of the forced dissipative
three-dimensional homogeneous isotropic turbulence at scales larger than the
forcing scale in real separation space. The probability density functions
(PDFs) of longitudinal velocity difference at large separations are close to
but deviate from Gaussian, measured by their non-zero odd parts. Under the
assumption that forcing controls the large-scale dynamics, we propose a
conjugate regime to Kolmogorov's inertial range, independent of the forcing
scale, to capture the odd parts of PDFs. The analytical expressions of the
third-order longitudinal structure functions derived from the
K\'arm\'an-Howarth-Monin equation prove that the odd-part PDFs of velocity
differences at large separations are small but non-zero, and show that the
odd-order longitudinal structure functions have a universal power-law decay
with exponent $-2$ as the separation tends to infinity regardless of the
particular forcing form, implying a significant coupling between large and
small scales. Thus, dynamics of large scales depart from the absolute
equilibrium, and we can partially recover small-scale information without
explicitly resolving small-scale dynamics. The departure from the statistical
equilibrium is quantified and found to be viscosity independent. Even though
this departure is small, it is significant and should be considered when
studying the large scales of the forced three-dimensional homogeneous isotropic
turbulence.
