Two identical unit masses

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

Two identical unit masses, each tied to a wall by a spring of stiffness k=2 N m−1k=2\,\mathrm{N\,m^{-1}} and to each other by a coupling spring of stiffness kc=1 N m−1k_c=1\,\mathrm{N\,m^{-1}}, have potential energy U(x1,x2)=12xTKxU(x_1,x_2) = \tfrac12\mathbf{x}^{\mathrm T}\mathsf{K}\mathbf{x} with K=(k+kc−kc−kck+kc)=(3−1−13) N m−1.\mathsf{K} = \begin{pmatrix}k+k_c & -k_c\\ -k_c & k+k_c\end{pmatrix} = \begin{pmatrix}3&-1\\-1&3\end{pmatrix}\,\mathrm{N\,m^{-1}}. At temperature T=250 KT=250\,\mathrm{K}, the Boltzmann probability density for the displacements is proportional to e−U/kBT=e−xTAxe^{-U/k_BT} = e^{-\mathbf{x}^{\mathrm T}\mathsf{A}\mathbf{x}}, with A=K/2kBT\mathsf{A}=\mathsf{K}/2k_BT positive definite. Using ⟨xixj⟩=kBT(K−1)ij\langle x_ix_j\rangle = k_BT(\mathsf{K}^{-1})_{ij}, find the thermal correlation ⟨x1x2⟩\langle x_1x_2\rangle in m2\mathrm{m^2}, taking kB=1.380649×10−23 J K−1k_B = 1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}.

Browse the problem library