A discrete-time Markov chain (Chapter 8) models a molecule hopping between three…

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 processes in continuous time

Problem

A discrete-time Markov chain (Chapter 8) models a molecule hopping between three conformations

1

2

3

once every

\Delta t = 1\,\mu{\rm s}

, with stationary distribution

\boldsymbol\pi = (0.5,\ 0.3,\ 0.2)

satisfying detailed balance

\pi_i P_{ij} = \pi_j P_{ji}

. The forward one-step probabilities are

P_{12} = 0.02

P_{13} = 0.01

and

P_{23} = 0.015

. ((a)) Use detailed balance to find

P_{21}

P_{31}

and

P_{32}

. ((b)) Taking the continuous-time master equation's rates as

W_{mn} = P_{mn}/\Delta t

for

m\ne n

, write the generator's three diagonal entries

G_{11}

G_{22}

G_{33}

(in

{\rm s^{-1}}

), using the convention that every row of the generator sums to zero. ((c)) Confirm that

\boldsymbol\pi\mathsf{G} = 0

, the equilibrium condition of the master equation. ((d)) Treat

\pi_i \propto e^{-E_i/k_{\rm B}T}

as an effective Boltzmann weight at

T = 300\,{\rm K}

, the discrete analogue of the stationary solution of the Fokker-Planck equation for a particle in a potential. Find the energy differences

E_2 - E_1

and

E_3 - E_1

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