  We study the entanglement and the Bell nonlocality of a coupled two-qubit
system, in which each qubit is coupled with one individual environment. We
study how the nonequilibrium environments (with different temperatures or
chemical potentials) influence the entanglement and the Bell nonlocality. The
nonequilibrium environments can have constructive effects on the entanglement
and the Bell nonlocality. Nonequilibrium thermodynamic cost can sustain the
thermal energy or particle current and enhance the entanglement and the Bell
nonlocality. However, the nonequilibrium conditions (characterized by the
temperature differences or the thermodynamic cost quantified by the entropy
production rates) which give the maximal violation of the Bell inequalities are
different from the nonequilibrium conditions which give the maximal
entanglement. When the Bell inequality has asymmetric observables (between
Alice and Bob), for example the $I_{3322}$ inequality, such asymmetry can also
be reflected from the effects under the nonequilibrium environments. The
spatial asymmetric two-qubit system coupled with nonequilibrium bosonic
environments shows the thermal rectification effect, which can be witnessed by
the Bell nonlocality. Different spatial asymmetric factors can be linearly
cancelled with each other in the thermal rectification effect, which is also
reflected on the changes of the entanglement and the Bell nonlocality. Our
study demonstrates that the nonequilibrium environments are both valuable for
the entanglement and Bell nonlocality resources, based on different optimal
nonequilibrium conditions though.
