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6

Doi makes this slightly more confusing because just after these equations, he writes: Here for simplicity we have dropped the subscript q. So all of the pieces of your Eqs \eqref{7} and \eqref{8} are actually the Fourier transforms of the pieces of your \eqref{1} and \eqref{2}, so they should all have q subscripts. Its also common to write Fourier ...

5

I'll try to summarize the argument mentioned in the comments from Daniel Schroeder's Introduction to Thermal Physics. Your derivation is correct within a certain approximation commonly made when dealing with atomic sized systems. Consider an isolated system of an hydrogen atom in thermal equilibrium with a reservoir (which could in principle be the rest of ...

2

Since you are interested in quantum simulation, I would actually suggest reading Feynman and Hibbs Quantum Mechanics and Path Integrals. It is very cheap which is a big plus and also remarkably readable as everything Feynman wrote is. Even though the book would be considered advanced because path integrals are considered advanced, much of what the book does ...

8

It looks Doi makes some extra simplifications (beyond expanding the exponential) that are valid when the external field is weak. Let's start with what you wrote and make one simplification,  \begin{eqnarray} \overline{\delta \phi _a} &= \frac{\langle\delta \phi_a \rangle -\langle \delta \phi _a \beta U_{ext}\rangle}{\langle 1-\beta U_{ext}\rangle} \tag{...

5

For context to future readers, Doi starts with a model of an $N$ unit polymer formed by a random walk along a uniform grid of lattice length $b$. It seems to be implicitly assumed in the described derivation that the lattice used is $b\mathbb{Z}^3$, that is the uniform 3D grid. Note there are other types of lattices for which Eq.$\,(\ref{2})$ is not true. ...

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