content_image/336392.jpg content_image/336391.jpg content_image/336390.jpg content_image/336388.jpg content_image/336389.jpg content_image/336387.jpg content_image/336386.jpg
In other words, in terms of 

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the Hamiltonians of individual subsystems can be expressed as H$_{X}$$_{k}$ = h ( ν$_{k}$ ). The bath Hamiltonian H$_{B}$ is the sum of the subsystem Hamiltonians and therefore the energy eigen- value of the bath state | n 〉$_{B}$ = | i$_{1}$, i$_{2}$, i$_{3}$, · · · 〉$_{B}$ is 

<img src='content_image/336391.jpg'>

Note that if { i$_{k}$ } had infinitely many nonzero entries, then the corresponding energy would be infinite. This justifies the definition of the Hilbert space H$_{B}$ in the way described above. 

Let σ ( ν ) = exp( − βh ( ν )) /ζ ( ν ) be the parameter-dependent density matrix and ζ ( ν ) be the corresponding partition function. The shorthand notation σ$_{k}$ = σ ( ν$_{k}$ ) will be used for the thermal equilibrium states of X$_{k}$. The partition function of the bath is then given by 

<img src='content_image/336390.jpg'>

Note that the convergence of this product depends only on the large k behavior of the sequence { ν$_{k}$ } . By adjusting the way ν$_{k}$ converges to 1, it is possible obtain a finite Z$_{B}$ . 

Let us now compute the HTO. In the following expressions, the subscripts are used for indicating the subsystem a given density matrix applies to. Suppose that the initial state of the device A is ρ . The initial and final states of AB are given by the following self-explanatory expressions. 

The change in the average energy of the bath is given by 

where J = J ( βQ ) and 

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