The double-well chain of Section 11.5

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems

Error, Chance and the Machine, Ch Markov-chain Monte Carlo

Problem

The double-well chain of Section 11.5, with

U(x) = (x^2-1)^2

, barrier height

1

and target

p(x) \propto e^{-\beta U(x)}

, is run with the small proposal width

0.2

. The eigenvalue calculation gives its relaxation time as

t_{\rm rel} = 5.5\times10^5

steps at

\beta = 12

and

2.1\times10^7

steps at

\beta = 16

. ((a)) Treat

t_{\rm rel}

A\,e^{\beta\Delta U_{\rm eff}}

with a constant

A

. Estimate the effective barrier

\Delta U_{\rm eff}

from the two values and compare it with the true barrier. ((b)) Use the same form to predict

t_{\rm rel}

\beta = 20

, and compare the prediction with the eigenvalue result,

7.8\times10^8

steps. ((c)) With the wider proposal width

0.5

, runs of

10^6

steps changed sign about

24\,000

times at

\beta = 3

and

1200

times at

\beta = 8

. Read as

N(\beta) \approx A'\,e^{-\beta\Delta U_{\rm eff}}

, what effective barrier do these counts give? Explain why they do not measure the barrier.

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