# Event-chain Monte Carlo for long-range interaction

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## Non-lifting irreversible Monte Carlo

While developing ECMC, we always wonder what role the lifting variable that indicates active particle plays for the efficiency of ECMC. Such a lifting variable makes the active particle motion persistent, and by imposing very short chains, the performance of ECMC returns to the case of traditional Monte Carlo. However, advancing active particles in the same direction is not equivalent to traveling in phase space without turning back. On the other hand, keeping explicit directions in lifting variable does not always help, as suggested by the example of skew detailed balance (see section 1.4.2.2).
By presenting three examples of non-lifting Monte Carlo followed by two with lifting, we review people’s effort in achieving irreversibility. Actually, two of three non-lifting irreversible Monte Carlo keep additional implicit information: The next variable to relax in a sequential scheme, and the momentum in hybrid Monte Carlo. Checking how different degrees of irreversibility improve mixing provides us instructive (though feeble) clues over the role of lifting in ECMC’s excellent performance.

### Hybrid Monte Carlo

Hybrid Monte Carlo, also known as Hamiltonian Monte Carlo, was introduced by Duane et al. [61]. It applies only to continuous configuration space because it uses a molecular dynamics integrator to generate proposal moves. The Metropolis-Hastings filter is used to accept the move (see [62], chapter 5). Hybrid Monte Carlo process can be described as:
1. Extend the state space s = (q1, …,qn) to a phase space with velocity so that s = (q,v), q = (q1, …,qn), and v = (p1, …,pn). Define the Hamiltonian H(q,p) = U(q)+Pni =1 p2i /2.
2. (a) For the current (q,p), perform Hamiltonian mechanics of Eq. 1.2.1 for time t, which reduces to q˙i = pi and p˙i = −@U/@qi. Then let p0 = −p(t), q0 = q(t). (b) Accept (q0,p0) with the Metropolis-Hastings filter, and the energy is replaced with H.
3. Flip p0 −p0.
4. Disturb p by p00 = p0 p1−+N, where N is a normal distribution with same dimension of p centering around zero and has variance corresponding to the temperature, is a tuning parameter.
5. Take (q0,p00) as a new state and go to step 2 until halted.

#### Lifting irreversible Monte Carlo

Reversible Monte Carlo methods employ symmetric proposal probabilities fp(i ! j) = fp(j ! i) so that tuning the acceptance rate allows us to realize detailed balance. Many systems achieve fast exploration with explicit use of direction. For example, particle momentum counteracts diffusive dynamics to make components in a fluid mix faster. Angular momentum drives a molecule under constant rotation.
Persistence in flipping up spins or flipping down will also lead to rapid decorrelation between spin configurations.
Lifting variables allow people to add information in Monte Carlo moves so that the program has the memory of previous moves. A state si is extended with a lifting variable si = (xi,ai), and the proposal depends heavily upon the lifting variable ai. Here we present several lifting irreversible Monte Carlo ideas, including onedimensional random walk, skew detailed balance, and our ECMC.

One-dimensional random walk

A chain of N sites with uniform desired probability is one of the few models with lifting variables that rigorous mathematics can apply. Diaconis et al. [64] established the first rigorous scaling of convergence for lifting irreversible Monte Carlo, proving that its mixing time converges within O(N) steps compared to O(N2) with an undirected random walk.
Previously, an unbiased random walk may jump from site i to i−1 or i+1 each with probability 1/2. The end of the chain needs special treatment if periodic boundary conditions are not applied. Diaconis et al. added the direction to the state represented by (i,d) with d = ±1, and the transition rule is T(i,d)!(i+d,d) = 1− 1 N .

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Fast implementation

If ECMC stays in the time-driven approach, it will hardly be capable of competing with other well-established methods. It consumes a great deal of time to integrate over small timesteps and introduces imprecision that is hard to analyze. Molecular dynamics has accumulated progress to overcome this inefficiency over the years, by parallel computing forces throughout the entire system, or altering potentials into piecewise-constant ones (see section 1.2.3) to which event-driven approaches can apply. ECMC is a stochastic process with a natural event-driven formulation with whatever potential the system has. In section 2.3.1 we describe how to convert small step integration into inverting potential functions. Then in section 2.3.2, we introduce the bounding potential that simplifies the event calculation so that only the derivative of the original potential is called.

General Introduction
1 Simulation method review
1.1 The need for large-scale simulation
1.1.1 Examples
1.1.2 State of the art
1.2 Molecular dynamics
1.2.1 Integrator
1.2.2 Long-range algorithms
1.2.3 Event-driven molecular dynamics
1.3.1 Properties of transition matrix
1.3.2 Tricks by modifying target distribution
1.3.3 Several reversible algorithms
1.4 Irreversible Monte Carlo
1.4.1 Non-lifting irreversible Monte Carlo
1.4.2 Lifting irreversible Monte Carlo
1.5 Models and measurements
1.5.1 Molecular models
1.5.2 Measurements
2 Event-chain Monte Carlo
2.1 Introduction
2.2 Basics
2.2.1 Lifted configurations
2.2.2 Factor event rate
2.2.3 Factorization
2.3 Fast implementation
2.3.1 Event-driven approach
2.3.2 Bounding potential
2.3.3 Sampling
2.3.4 Direction switching and restarts
3 Event-chain Monte Carlo for long-range interaction
3.1 Long-range system
3.2 Calculation of pairwise Coulomb interaction
3.2.1 Ewald summation
3.2.2 Line-charge method
3.2.3 Higher-order methods
3.2.4 Interpolation
3.2.5 Separate-image method
3.3 ECMC cell-veto algorithm
3.3.1 Description
3.3.2 Cell-veto event rate
3.3.3 Walker’s method
3.4 Coulomb bounding potential
3.5 Dipole factors
3.5.1 Two-dipole case
3.5.2 Homogeneous systems
3.5.3 Dipole lifting schemes
3.6 Numerical tests
3.6.1 Dipole system
3.6.2 SPC/Fw water model
4 JeLLyFysh architecture
4.1 Issues of application development
4.2 Features of JELLYFYSH-V1.0
4.3 Basic concepts
4.3.1 Event flow
4.3.2 Units
4.3.3 Global state and internal state
4.4 Essential modules
4.4.1 Mediator
4.4.2 Event handler
4.4.3 Activator
4.4.4 State handler
4.4.5 Scheduler
4.4.6 Input-output handler
4.5 Important tools
4.5.1 Globally used modules
4.5.2 Cells and cell occupancy
4.5.3 Initializer
4.5.4 Potentials, estimator
4.5.5 Lifting schemes
5 JELLYFYSH simulation
5.1 Event handlers of JELLYFYSH-V1.0
5.1.1 Event handler for two-body invertible potentials
5.1.2 Event handlers for non-invertible potentials
5.1.3 Cell-veto event handler
5.1.4 Event handlers for composite object motion
5.1.5 Sampling event handler and end-of-chain event handler .
5.1.6 Other helping event handlers
5.2 Factory
5.2.1 Running
5.2.2 Customization
5.3 Verifications
5.3.1 Coulomb atoms
5.3.2 Dipoles
5.3.3 SPC/Fw water
6 Realistic water system
6.1 Speeding up JELLYFYSH
6.1.1 Fast track for unconfirmed events
6.1.2 Other improvements
6.2 Coulomb event profiling
6.2.1 Charge-water estimator
6.2.2 Optimizing cell-veto events
6.2.3 Dipole factor vs. atomic factor
6.3 Water system benchmark
6.3.1 Chain state
6.3.2 Molecular translation
6.3.3 Discussions
7 Sequential Monte Carlo
7.1 Single-dipole system
7.1.1 Configuration
7.1.2 Sequential Monte Carlo method
7.1.3 Validation
7.2 Rotational dynamics
7.2.1 Zigzag and excursion
7.2.2 Large-time limit
7.3 Mixing of orientation
General Conclusion
Bibliography

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