Theoretical Background¶

This page explains the backmapping method implemented in the package.

References

Krajniak et al., "Generic Adaptive Resolution Method for Reverse Mapping of Polymers from Coarse-Grained to Atomistic Descriptions", J. Chem. Theory Comput. 2016, 12, 5549--5562. DOI: 10.1021/acs.jctc.6b00595

Krajniak, Zhang et al., "Reverse Mapping Method for Complex Polymer Systems", J. Comput. Chem. 2018. DOI: 10.1002/jcc.25129

Overview¶

Backmapping (reverse mapping) is the process of reintroducing atomistic detail into a coarse-grained simulation. The method implemented here uses a time-dependent, smooth transition controlled by a resolution parameter λ that is ramped from 0 (fully CG) to 1 (fully AT) during the simulation.

Unlike instantaneous mapping approaches that place atoms geometrically and then energy-minimize, this method gradually introduces atomistic interactions while fading out coarse-grained ones. The system relaxes dynamically, avoiding the need for aggressive minimization steps.

The Lambda Parameter¶

Each atom carries a per-atom resolution parameter λ:

λ = 0: the atom is at purely coarse-grained resolution
λ = 1: the atom is at fully atomistic resolution
0 < λ < 1: the atom is in the transition region

Lambda increases linearly at each timestep:

\[ \lambda(t + \Delta t) = \min\bigl(\lambda(t) + \alpha,\ 1\bigr) \]

where α is the ramp rate. A smaller α produces a slower, gentler transition. Typical values are 10^-4 to 10^-3.

Uniform vs Non-uniform Lambda¶

In uniform mode (default), all atoms start at the same initial λ value (λ₀) and increase together.

In non-uniform mode, each atom receives a random offset to its initial lambda, creating a staggered transition where different parts of the system reach full AT resolution at different times. This can help reduce artifacts from a sudden global transition.

Force Weighting¶

All interactions (pair, bond, angle) are weighted by lambda. The weighting scheme uses the product of the lambda values of the interacting atoms.

Non-bonded Pair Interactions¶

For a pair of atoms i and j:

AT interactions: weighted by \( w_\text{AT} = \lambda_i \times \lambda_j \)
CG interactions: weighted by \( w_\text{CG} = 1 - \lambda_i \times \lambda_j \)

This ensures:

At λ = 0: only CG interactions are active (\( w_\text{CG} = 1 \), \( w_\text{AT} = 0 \))
At λ = 1: only AT interactions are active (\( w_\text{CG} = 0 \), \( w_\text{AT} = 1 \))
During the transition: both sets of interactions contribute with complementary weights

Bonded Interactions (Bonds and Angles)¶

Cross-CG bonded interactions (bonds and angles that span CG bead boundaries) use the same weighting scheme:

AT cross bonds/angles: \( w = \lambda_i \times \lambda_j \)
CG cross bonds/angles: \( w = 1 - \lambda_i \times \lambda_j \)

For angles, the weight is computed from the first and last atoms of the angle triplet (i-j-k), using \( \lambda_i \) and \( \lambda_k \).

Intra-bead AT bonds and angles (within a single CG bead) use standard LAMMPS styles without lambda weighting, since they exist at both resolutions.

Cross Interactions¶

Cross interactions are bonded terms that span CG bead boundaries. They are essential for maintaining structural integrity during the transition.

CG Cross Bonds¶

These are the bonds that connect neighboring CG beads. They are typically derived from iterative Boltzmann inversion (IBI) or other coarse-graining methods, and are provided as tabulated potentials. They fade out as λ increases:

\[ F_\text{CG bond} = (1 - \lambda_i \lambda_j) \times F_\text{table}(r) \]

AT Cross Bonds¶

These are the atomistic bonds between atoms in different CG beads. They fade in as λ increases:

\[ F_\text{AT bond} = \lambda_i \lambda_j \times (-k)(r - r_0) \]

AT Cross Angles¶

Same weighting as AT cross bonds, applied to angle potentials between atoms in different CG beads.

CG Bead Management¶

During the backmapping simulation, each CG bead coexists with the AT atoms it represents. The fix backmap style manages the relationship:

Position tracking: After each timestep, the CG bead position is updated to the center-of-mass (COM) of its constituent AT atoms.
Force distribution: Forces on CG beads (from CG pair interactions and CG cross bonds) are redistributed to AT atoms proportional to their mass fraction:

\[ \mathbf{F}_i^\text{AT} \mathrel{+}= \frac{m_i}{M_\text{CG}} \mathbf{F}_\text{CG} \]

where \( m_i \) is the mass of AT atom i and \( M_\text{CG} \) is the total mass of all AT atoms in the bead.
Velocity zeroing: CG bead velocities are set to zero at each step to prevent them from drifting independently of their AT atoms.

Simulation Phases¶

A typical backmapping simulation has three phases:

Phase 1: CG Equilibration¶

Lambda ramp is inactive (fix_modify bm active no). The system runs at CG resolution to equilibrate the starting configuration. AT atoms move under their intra-bead potentials but inter-bead interactions are purely CG.

Phase 2: Backmapping¶

Lambda ramp is active (fix_modify bm active yes). Lambda increases by α each timestep. CG interactions fade out while AT interactions fade in. This is the core of the backmapping process.

Phase 3: AT Production¶

Once λ reaches 1.0 everywhere, the system is fully atomistic. CG interactions have zero weight. This phase runs standard AT dynamics for equilibration or production.