We study the quantization of mesoscopic inductance coupling circuit and discuss its time evolution. By means of the thermal field dynamics theory we study the quantum fluctuation of the system at finite temperature.
By extending the thermal field dynamics to the time dependent system, quantum fluctuations of the time dependent mesoscopic LC electric circuit at finite temperature were studied. While the up going temperature will increase the quantum fluctuations, time changes of the physical quantities such as the inductance and capacitance can suppress the quantum fluctuations. The electric source will not affect the quantum fluctuations, but will contribute to the averages of the charge and magnetic flux. It was also found that if there is a source, the averages will depend on the temperature; if there is no source, the averages will be zero and so temperature independent.