Doubt about a time convolution master equation

[Ark236]

TL;DR 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?

 

[azntoon]

No, the two equations are not equivalent. The equation from "The Theory of Open Quantum Systems, Breuer" is more general and can be used for a wider range of systems than the equation from "Quantum Effects in Biology, Mohseni". The difference between the two equations is that the second equation includes an additional term, which is the tensor product of the system state at time $\tau$ and the bath state at time 0. This additional term is necessary to account for the full dynamics of the system-bath interaction.