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**Sample text**

By a generalization of the Liouville–von Neumann equation, the statistical thermodynamic theory of the ensemble of the damped oscillators in contact with a thermal bath is given. By introducing the quantum entropy of the ensemble, it is shown that the entropy of the ensemble grows in a dissipative process and in thermal equilibrium for the probability distribution of the quantum states, such that Gibbs’ canonical distribution is valid. Finally, a wave equation of the linearly damped oscillator is given.

By the second law of nonequilibrium thermodynamics, the entropy S (U , q ) satisfies the balanced equation: dS dSr dSi = + , dt dt dt dSi ³0 dt (3) d Sr where dt is the reversible rate of change of the entropy or entropy flux on the rigid diathermal dSi wall and dt is the so-called entropy production. 5772/61010 Also, entropy production and the so-called rate of energy dissipation R =T dSi ¶S dq =T ³0 ¶q dt dt (10) decrease the free energy of the oscillator. In what follows, we shall give the actual form of the rate of energy dissipation.

M. Prausnitz. Local compositions in thermodynamic excess func‐ tions for liquid mixtures. , vol. 14, pp. 135-144, January 1968. [7] D. S. Abrams and J. M. Prausnitz. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. , vol. 21, pp. 116-128, January 1975. [8] A. Fredenslund, R. L. Jones, J. M. Prausnitz. Group contribution estimation of activi‐ ty coefficients in nonideal liquid mixtures. , 1975, 21 (6), 1086-1099.