Notes on "Adaptive single replica multiple state transition interface sampling" (Du and Bolhuis, 2013)

• http://scitation.aip.org/content/aip/journal/jcp/139/4/10.1063/1.4813777

Abstract / Introduction

• Goal: sample rare transitions between a set of metastable states
• Strategy: two-ended methods, e.g. transition path sampling (TPS), nudged elastic band, action minimization, string method
• Tactic: TPS: collect an ensemble of dynamically unbiased paths between two specified endpoints
• Challenges in path sampling:
  • Dependence on initial path $\to$ solved by Replica Exchange Transition Interface Sampling (RETIS)
    • Drawback: requires a large number of simultaneous replicas
  • Metastable intermediates require inefficiently long paths $\to$ Multi State TPS/TIS (MSTPS/MSTIS)
    • Drawback: many interfaces must be treated simultaneously, and pathways have different lengths, rendering parallel implementation inefficient
• Proposed solution:
  • Use a single replica to sample all interfaces using MSTIS framework:
    • Initial path in an initial interface
    • Monte Carlo swapping move:
      • Proposal: switch to neighboring replica
      • Acceptance: If path fulfills criteria for new interface
    • Monte Carlo shooting move:
      • Proposal: create new paths
      • Proposal bias: Wang-Landau approach $\to$ use density of paths (DOP) at each interface, bias distribution from stable states to other basins of attraction
        • Notes: Wang-Landau converges slowly, use only to build up the DOP, then use fixed bias

Outline:

• Section II:
  • Review of MSTIS
  • introduce single-replica algorithms
    • One based Wang-Landau
    • Fixed bias
  • Adaptive sampling
• Section III:
  • applications to model systems:
    • 2D potential, LJ cluster isomerization, isomerization of alanine dipeptide in explicit solvent
• Section IV:
  • conclusion

II. Theoretical background

A. Multiple State Transition Interface Sampling (MSTIS)

• Discretize a dynamical trajectory $\newcommand{\x}{\mathbf{x}} \x^L \equiv \{ \x_0, \x_1, \dots, \x_L\}$ where consecutive phase points $\x = \{\mathbf{r}^N, \mathbf{r}^N \}$
  • Coordinates: $\mathbf{r}$
  • Momentum: $\mathbf{p}$
  • $N$ particles
  • Timestep: $\Delta T$
  • Total time duration $\mathcal{T} = L \Delta t$

• Probability of an unbiased trajectory $\x^L$: $$ \pi [ \x^L ] = \rho(\x_0) \prod_{i=0}^{L-1} \rho (\x_i \to \x_{i+1}) $$

  • $\rho(\x)$: steady-state distribution
  • $\rho(\x \to \mathbf{y})$: Markov transition probability in one timestep

  • Normalize using a "partition-function"-like factor: $$ Z \equiv \int \mathcal{D} \x^L \pi [ \x^ L ]$$ (integrated over all possible paths of all lengths)

    to yield normalized path probability $$ \mathcal{P}[\x^L] = \frac{\pi [\x^L]}{Z}$$

• Consider a set of $M$ states $\mathbf{S} = \{0,1,\dots,I,\dots,M \}$, where each state $I$ is defined as $\{ \x : \lambda_I (\x) < \lambda_{0 I} \} $ where:
  • $\lambda(\x)$ is a function of the phase point $\x$
    • frequently acts as a metric: defines the distance of a point $\x$ to a reference point characterizing the stable state
  • $\lambda_{0 I}$ defines the boundary of $I$
• Transition Interface Sampling (TIS)
  • For each state $I$, introduce $m_I + 1$ non-intersecting hypersurfaces defined by $\{ x : \lambda(\x) = \lambda_{i I} \}$ ($\lambda_{0I} \equiv$ the boundary of state $I$, $\lambda_{m I } \equiv$ "outermost" interface)
  • TIS path ensemble for interface $i$ of state $I$ contains paths that:
    • start in $I$,
    • end in any state $\in \mathbf{S}$, and
    • cross interface $\lambda_i$
  • TIS path probabilities
    • Define region of phase space beyond interface $i$ by $\Lambda_{i I}^+ \equiv \{ \x : \lambda(\x) > \lambda_{i I} \}$
    • TIS path probability for state $I$ is $$ \mathcal{P}_{I \Lambda_{i I}} [\x^L ] = \frac{\tilde{h}_i^I[\x^L] \pi[\x^L]}{Z_{I \Lambda_{i I}}}$$ where $\tilde{h}_i^I[\x^L] \pi[\x^L]$ is an indicator function for paths that begin in $I$, end in any stable state in $\mathbf{S}$ and cross the interface $\lambda_i$
  • Multiple state path ensemble: all trajectories from all states $I \in \mathbf{S}$ that cross the outermost interface of $I$ $$ \mathcal{P}_\text{MSTIS} [\x^L] = \sum_I^M \frac{\tilde{h}_i^I[\x^L] \pi[\x^L] \pi [\x^L]}{Z}$$
  • Shooting algorithm:
    • Proposal: c.f. "A novel path sampling method for the calculation of rate constants" ()

B. Single replica MSTIS

Appendix: Wang-Landau algorithm

[to-do]

Appendix: Weighted-Histogram Analysis Method (WHAM)

[to-do]