  Based on systematic analyses of several popular non-relativistic energy
density functionals with mean-field approximation, we estimate the value of the
fourth-order symmetry energy $E_{\text{sym,4}}(\rho)$ at nuclear normal density
$\rho_0$ and its density dependence, and explore the correlation between
$E_{\text{sym,4}}(\rho_0)$ and other macroscopic quantities of nuclear matter
properties. We use the empirical values of some nuclear macroscopic quantities
to construct model parameter sets by Monte Carlo method for the conventional
Skyrme-Hartree-Fock (SHF) model, the extended Skyrme-Hartree-Fock (eSHF) model,
the Gogny-Hartree-Fock (GHF) model, and the momentum-dependent interaction
(MDI) model. The value of $E_{\text{sym,4}}(\rho_0)$ is estimated to be
$1.02\pm0.49$ MeV for the SHF model, $1.02\pm0.50$ MeV for the eSHF model,
$0.70\pm0.60$ MeV for the GHF model, and $0.74\pm0.63$ MeV for the MDI model.
Moreover, our results indicate that the density dependence of
$E_{\text{sym,4}}(\rho)$ is model dependent, especially at higher densities.
Furthermore, we find that the $E_{\text{sym},4}(\rho_0)$ has strong positive
(negative) correlation with isoscalar (isovector) nucleon effective mass
$m_{s,0}^*$ ($m_{v,0}^*$) at $\rho_0$. In particular, for the SHF and eSHF
models, the $E_{\text{sym,4}}(\rho)$ is completely determined by the isoscalar
and isovector nucleon effective masses $m_{s}^*(\rho)$ and $m_{v}^*(\rho)$, and
the analytical expression is given. In the mean-field models, the magnitude of
$ E_{\text{sym,4}}(\rho_0)$ is generally less than $2$ MeV, and its density
dependence depends on models, especially at higher densities. $
E_{\text{sym,4}}(\rho_0)$ is strongly correlated with $m_{s,0}^*$ and
$m_{v,0}^*$.
