(1240) Gaussian integrals

(1) Write a Gaussian integral expression for the normalization constant \( Z \) of the distribution
\( P(x_1, x_2, \cdots, x_N) = \frac{1}{Z} \exp[- \beta A(x_1, x_2, \cdots, x_N)] \)
where \( A \) is obtained from time discretization of
\( A[x] = \frac{1}{2} \int_0^t ( \dot{x}^2 + \omega^2 x^2 ) dt \)
with boundary conditions \( x(0)=x(t)=0 \), and \( N \) equally-spaced midpoints.

(2) Using an appropriate procedure obtain a result for \( N \gg 1 \). Clarify how the result depends on \( N \) in the \( N \rightarrow \infty \) limit.

(3) Consider a 1D chain of \( N \) beads. Each bead is connected to the ground with a spring that has a spring-constant \( \alpha_{\parallel} \). Additionally the beads are interconnected by springs that have spring constant \( \alpha_{\perp} \). Write the potential energy \( V(x_1, x_2, \cdots, x_N) \) of the system, where \( x_n \) is the displacement of the \( n\text{-th} \) bead from its equilibrium position. Based on what you have obtained above, write an expression for the partition function \( Z \).

(4) What is the condition for the validity of the approximation that you have used for the partition function of the chain. What is the "small parameter" that controls the accuracy.

Note: Optional calculation method is via diagonalization. In "spring" language this means to find the normal modes of vibrations.