Microscopic work distribution of small systems in quantum isothermal processes and the minimal work principle

For a two-level quantum mechanical system, we derive microscopically the exact expression for the fluctuation of microscopic work in. a multistep nonequilibrium process, and we rigorously prove that in an isothermal process, the fluctuation is vanishingly small and the most probabilistic work is just equal to the difference of the free energy. Our study demonstrates that the convergence of the microscopic work in the isothermal process is due to the nature of the isothermal process rather than the usual thermodynamic limit condition. Our investigation justifies the validity of a minimum work principle formulation of the second law even for a small system far from the thermodynamic limit.