Pair Persistence & Residence Times¶

This page is a self-contained, textbook-style introduction to pair persistence analysis — how MolPy measures how long two particles stay "bonded" (within a distance cutoff) and turns that into a residence-time correlation function. The canonical applications are hydrogen-bond dynamics in water and ion-pair lifetimes in electrolytes. No background beyond the idea of a time-correlation function is assumed; if you have read the Diffusion & Ionic Transport guide you already have everything you need.

As always, the per-pair, per-frame bookkeeping runs in Rust (molrs); the MolPy layer extracts per-species coordinates and box dimensions and returns a typed result.

Conventions used throughout

A pair is one reference particle \(i\) and one partner particle \(j\).
A pair is bonded at a frame if their minimum-image distance is within a cutoff. Two cutoffs may be used: an inner \(r_0\) (formation) and an outer \(r_1\ge r_0\) (breaking).
\(\langle\cdots\rangle_t\) is an average over time origins; \(\tau\) is the lag.
Units: length Å, time ps. The correlation \(C(\tau)\) is dimensionless.

1. Why lifetimes need their own tool¶

Diffusion and conductivity (transport guide) tell you how far things move, but not how long a specific association survives. Yet many properties — proton transfer, ion-pair stability, solvation-shell exchange — are governed by the lifetime of a contact, not by bulk mobility.

The difficulty is that a contact is rarely clean: two particles sitting near the cutoff will rattle in and out many times per picosecond without the pair meaningfully breaking. A good lifetime measure has to take a stand on what counts as "the same bond surviving". That choice is the whole subject of this page.

2. The survival correlation function¶

Define a survival indicator for the pair \((i,j)\) that is born at time \(t\) and observed at time \(t+\tau\):

\[ S_{ij}(t, t+\tau) = \begin{cases} 1 & \text{the pair is considered still alive at } t+\tau,\\ 0 & \text{otherwise.} \end{cases} \]

The persistence (residence-time) correlation function averages this over all reference particles and time origins, normalized per reference particle:

\[ \boxed{\;C(\tau) = \Big\langle\,\frac{1}{N_i}\sum_i\sum_j S_{ij}(t,\,t+\tau)\,\Big\rangle_t\;} \]

Two limits make this concrete:

\(C(0)\) is the mean coordination number — the average number of partners within the formation cutoff. (At \(\tau=0\) every freshly born pair is alive.)
\(C(\tau)\) decays as bonds break; its decay time is the residence time. Fitting \(C(\tau)\approx C(0)\,e^{-\tau/\tau_\text{res}}\) (or integrating it) gives a single lifetime.

The only thing left to pin down is the rule inside \(S_{ij}\).

3. Three definitions of survival¶

The literature offers several survival rules; MolPy implements the three that are well defined and widely used. All share the same birth condition — a pair is born at \(t\) only if it is within the inner cutoff \(r_0\) — and differ in how they decide a born pair is still alive at \(t+\tau\).

3.1 Continuous survival¶

The strictest rule (Rapaport, 1983): the pair must stay within the survival cutoff \(r_1\) at every frame from \(t\) to \(t+\tau\). The first time it leaves, it is dead forever for that origin:

\[ S^\text{cont}_{ij}(t,t+\tau) = \prod_{s=0}^{\tau}\Theta\!\big(r_1 - r_{ij}(t+s)\big), \]

with \(\Theta\) the step function. This yields the continuous residence time — short, because every transient excursion ends the bond. Use a single cutoff (\(r_1=r_0\)) for the classic definition.

3.2 Intermittent survival¶

The most permissive rule (Luzar & Chandler, 1996): the pair is alive at \(t+\tau\) if it is within \(r_1\) at that frame, regardless of whether it left in between. Re-formation is allowed:

\[ S^\text{int}_{ij}(t,t+\tau) = \Theta\!\big(r_1 - r_{ij}(t+\tau)\big). \]

This yields the intermittent correlation, whose decay reflects the structural lifetime including re-crossings. The continuous and intermittent correlations bracket the truth, and their ratio characterises how much a bond rattles.

3.3 Stable-states picture (SSP)¶

A middle ground (Laage & Hynes, 2008) that suppresses rattling without discarding genuine survival: use two cutoffs. A pair is born within the inner \(r_0\) ("stable reactant") and is counted alive as long as it stays within the outer \(r_1\) ("stable product"); it must cross the outer cutoff to be declared broken:

\[ S^\text{SSP}_{ij}(t,t+\tau) = \Theta\!\big(r_0 - r_{ij}(t)\big)\prod_{s=0}^{\tau}\Theta\!\big(r_1 - r_{ij}(t+s)\big), \qquad r_1\ge r_0. \]

The gap between \(r_0\) and \(r_1\) is a buffer: a particle wandering just past the contact distance is not immediately counted as having left. SSP is the recommended default for ion pairs.

Relation to tame's definitions

This mirrors the tame persist recipe. tame names two definitions: IMM (Impey–Madden–McDonald, 1983), a single cutoff with a tolerance time that ignores escapes shorter than \(t^*\), and SSP (Laage–Hynes). MolPy's continuous/intermittent/ssp cover the same physics with explicit, time-tolerance-free criteria; the IMM tolerance-time variant is intentionally not reproduced (its published implementation threaded an out-of-scope timestep).

4. Using `Persist`¶

Persist takes tags of the form "t1,t2:method:r0[,r1]":

t1,t2 — reference and partner atom types (e.g. cation and anion).
method — continuous, intermittent, or ssp.
r0[,r1] — inner (and optional outer) cutoff in Å. A single value sets \(r_1=r_0\).

from molpy.compute import Persist

p = Persist(
    tags=[
        "3,4:ssp:3.0,4.0",        # cation-anion, stable-states picture
        "1,1:continuous:3.5",      # like-species, single-cutoff continuous
    ],
    max_dt=30.0,    # longest lag, ps
    dt=0.01,        # timestep, ps
)
result = p(trajectory)

C = result.correlations["3,4:ssp:3.0,4.0"]
C[0]            # mean coordination number (partners per reference ion)
result.time     # lag times tau, ps

When t1 == t2 the like-species self-pair (\(i=j\)) is dropped automatically.

5. From persistence to pairing diffusion¶

Persistence is not only descriptive — it lets you decompose a transport coefficient by bonding state. Combining the distinct-diffusion correlation of §2 of the transport guide with a survival function \(f(r_{ij};s)\) yields the pairing contribution to the diffusion coefficient:¹

\[ D^\text{d,pairing}_{\alpha\beta} = \lim_{\tau\to\infty}\frac{1}{6\tau N} \sum_i\sum_{j\ne i}\big\langle\Delta\mathbf{r}_{i,\alpha}(\tau)\cdot\Delta\mathbf{r}_{j,\beta}(\tau)\,f(r_{ij};s)\big\rangle, \]

i.e. only the displacement of partners that remain associated (the "in" channel) is counted. This separates how paired ions diffuse from how free ions diffuse — directly relevant to whether ion pairs carry net charge. Interpret the result only where both the persistence count has converged and the MDC is linear in time.

6. Pitfalls checklist¶

Single cutoff with rattling → the continuous lifetime collapses to the frame spacing. Use ssp with a buffer (\(r_1 > r_0\)), or intermittent.
Cutoff chosen off the RDF → pick \(r_0\) at the first coordination-shell minimum of \(g(r)\); an arbitrary cutoff makes the lifetime meaningless.
max_dt shorter than the lifetime → \(C(\tau)\) never decays within the window and no residence time can be read off.
Sparse sampling → both survival products miss fast re-crossings; sample densely enough to resolve the contact dynamics.
Comparing across definitions → continuous, intermittent, and SSP give different numbers by construction; always report which one.

7. References¶

D. C. Rapaport, Mol. Phys. 50, 1151 (1983) — continuous vs intermittent hydrogen-bond correlation functions.
A. Luzar, D. Chandler, Nature 379, 55 (1996); Phys. Rev. Lett. 76, 928 (1996) — intermittent correlation and hydrogen-bond kinetics.
A. Luzar, J. Chem. Phys. 113, 10663 (2000) — persistence/residence-time definitions.
R. W. Impey, P. A. Madden, I. R. McDonald, J. Phys. Chem. 87, 5071 (1983) — residence time with a tolerance time (IMM).
D. Laage, J. T. Hynes, J. Phys. Chem. B 112, 14230 (2008) — stable-states picture (SSP).