A three-level vibrational system has energies

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Non-equilibrium statistical mechanics and stochastic thermodynamics

Error, Chance and the Machine, Ch Random walks and Markov chains

Problem

A three-level vibrational system has energies E1=0E_1=0, E2=kBTE_2=k_BT and E3=3kBTE_3=3k_BT, measured in units of the thermal energy kBTk_BT. Once every observation interval the system may hop to an adjacent level only (there is no direct 1↔31\leftrightarrow3 transition). The upward hop probabilities are fixed by the experiment at P12=P23=α=0.10P_{12}=P_{23}=\alpha=0.10; detailed balance with the Boltzmann distribution at temperature TT then fixes the downward hop probabilities, Pi+1,i=α e(Ei+1−Ei)/kBT.P_{i+1,i} = \alpha\,e^{(E_{i+1}-E_i)/k_BT}. Any interval in which no hop is attempted leaves the system in its current level (a self-loop). ((a)) Write out the full 3×33\times3 transition matrix P\mathsf P. ((b)) Find the stationary distribution π\boldsymbol\pi from the Boltzmann weights πi∝e−Ei/kBT\pi_i\propto e^{-E_i/k_BT}, and confirm that it satisfies detailed balance, πiPij=πjPji\pi_iP_{ij}=\pi_jP_{ji}, for both pairs of neighbouring levels. ((c)) For a reversible chain the symmetrised matrix S=D1/2PD−1/2\mathsf S=\mathsf D^{1/2}\mathsf P\mathsf D^{-1/2}, with D=diag(π)\mathsf D=\mathrm{diag}(\boldsymbol\pi), is symmetric. Show this for P\mathsf P above and explain why it forces every eigenvalue of P\mathsf P to be real. ((d)) State whether the chain is irreducible and aperiodic, and hence whether it settles to π\boldsymbol\pi from any starting distribution.

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