The Metropolis method is simply the power method for computing the right eigenvector of M with the largest magnitude eigenvalue. By construction, the correct probability distribution is a right eigenvector with eigenvalue 1 . Therefore, for the Metropolis method to converge to this result, one has to show that M has only one eigenvalue with this magnitude, and all other eigenvalues are smaller. 12.6 Langevin and Fokker-Planck Equations We end this chapter with a discussion and derivation of the Fokker-Planck and Langevin equations. These equations will in turn be used in our discussion on advanced Monte Carlo methods for quantum mechanical systems, see chapter for example chapter 16. 12.6.1 Fokker-Planck Equation For many physical systems initial distributions of a stochastic variable y tend to an equilibrium distribution w equilibrium ( y ) , that is w ( y , t ) → w equilibrium ( y ) as t → ∞ . In equilibrium, detailed balance constrains the transition rates W ( y → y ′ ) w ( y )= W ( y ′ → y ) w equilibrium ( y ) , where W ( y ′ → y ) is the probability per unit time that the system changes from a state | y ) , characterized by the value y for the stochastic variable Y , to a state | y ′ ) . Note that for a system in equilibrium the transition rate W ( y ′ → y ) and the reverse W ( y → y ′ ) may be very different. Let us now assume that we have three probability distribution functions for times t 0 < t ′ < t , that is w ( x 0 , t 0 ) , w ( x ′ , t ′ ) and w ( x , t ) . We have then
396 12 Random walks and the Metropolis algorithm Initialize: Establish an initial state, for example a position x ( i ) Suggest a move y t Compute ac- ceptance ratio A ( x ( i ) → y t ) Generate a uniformly distributed variable r Is A ( x ( i ) → y t ) ≥ r ? Reject move: x ( i + 1 ) = x ( i ) Accept move: x ( i ) = y t = x ( i + 1 ) Last move? Get local expecta- tion values Last MC step? Collect samples End yes no yes yes no Fig. 12.7 Chart flow for the Metropolis algorithm.
12.6 Langevin and Fokker-Planck Equations 397 w ( x , t )= integraldisplay ∞ − ∞ W ( x . t | x ′ . t ′ ) w ( x ′ , t ′ ) d x ′ , and w ( x , t )= integraldisplay ∞ − ∞ W ( x . t | x 0 . t 0 ) w ( x 0 , t 0 ) d x 0 , and w ( x ′ , t ′ )= integraldisplay ∞ − ∞ W ( x ′ . t ′ | x 0 , t 0 ) w ( x 0 , t 0 ) d x 0 . We can combine these equations and arrive at the famous Einstein-Smoluchenski-Kolmogorov- Chapman (ESKC) relation W ( x t | x 0 t 0 )= integraldisplay ∞ − ∞ W ( x , t | x ′ , t ′ ) W ( x ′ , t ′ | x 0 , t 0 ) d x ′ . We can replace the spatial dependence with a dependence upon say the velocity (or momen- tum), that is we have W ( v , t | v 0 , t 0 )= integraldisplay ∞ − ∞ W ( v , t | v ′ , t ′ ) W ( v ′ , t ′ | v 0 , t 0 ) d x ′ . We will now derive the Fokker-Planck equation. We start from the ESKC equation W ( x , t | x 0 , t 0 )= integraldisplay ∞ − ∞ W ( x , t | x ′ , t ′ ) W ( x ′ , t ′ | x 0 , t 0 ) d x ′ . We define s = t ′ − t 0 , τ = t − t ′ and t − t 0 = s + τ . We have then W ( x , s + τ | x 0 )= integraldisplay ∞ − ∞ W ( x , τ | x ′ ) W ( x ′ , s | x 0 ) d x ′ .
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