# A Doubt about a time convolution master equation

#### Ark236

Summary
The same equation in 2 different books is different
I study of interaction between a system with a reservoir considering a weak coupling between them. I consider a bosonic bath, the initial state are separable and the operator of interaction between the system and bath is linear in the displacements of the oscillators.
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In the book "Quantum Effect in Biology, Mohseni" , show that the time convolution master equation is:

$$\dfrac{d}{dt}\rho_{s,I}(t)=-\int_{0}^{t}d\tau Tr_{B}[\cal{L}_{SB,I}(t)\cal{L}_{sB,I}(\tau)\rho_{B}(0)]\rho_{S,I}(\tau)$$
where $$\cal{L}_{SB,I}(t) \hat{A}=\dfrac{1}{\hbar}[H_{I}(t),\hat{A}]$$ and $$H_I$$ is the hamiltonian system in the interaction picture.

but in "The Theory of Open Quantum Systems, Breuer" show that the time convolution master equation is:

$$\dfrac{d}{dt}\rho_{s,I}(t)=-\int_{0}^{t}d\tau Tr_{B}[\cal{L}_{SB,I}(t)\cal{L}_{sB,I}(\tau)\rho_{S,I}(\tau)\otimes \rho_{B}(0)]$$

I think that bot equation are not equivalent?

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