Dear Xiaowey,
(1) I'd need to know the parameters with which you generated the GLE matrices. In general it's a good idea to paste into the
input.xyz the comment section that is generated by GLE4MD
(2) I don't know how you're restarting your simulations - if the simulation ends gracefully (wall-clock time limit elapsed, touch EXIT or SIGTRM) you can just use the RESTART file as input (i-pi RESTART &> log). that way the time should increment smoothly, unlike in your output files
(3) The two simulations cannot be compared - the GLE thermostat you use is a classical thermo so the simulation is classical. PIGLET is quantized so there may be very large differences in behavior
(4) Seems the quantum simulations are melting, hence the steady increase in enthalpy. Good news, you stumbled upon a system for which quantum effects matter!
So bottom line: no problem with i-PI, only minor aestetic problems with your setup. What you're seeing is the physics of this system as described by DFT without (GLE) and with (PIGLET) quantum effects.
Michele