Markovian RSA in Dimer Adsorption

• Markovian RSA is a process where dimer adsorption occurs irreversibly, with each trial based solely on the current blocked configuration.
• The method uses kinetic observables like Feder’s law and the surface blocking function to estimate a jamming coverage near 0.547.
• Finite-size and orientational effects influence local deposition structure, yet the overall long-range pair correlations remain close to isotropic behavior.

Markovian RSA denotes the irreversible random sequential adsorption framework in which particles are deposited one at a time onto a collector, each trial is independent of the past except for the existing blocked or occupied configuration, and accepted particles remain fixed forever. In that sense, the evolution is Markovian: the configuration at adsorption “time” $t$ fully determines the probabilities of future successful insertions. In the dimer case studied in "RSA Study of Dimers" (Ciesla et al., 2012), the adsorbing object is a non-spherical particle modeled in two dimensions as two touching disks of radius $r$, deposited onto a flat, homogeneous surface.

1. Markovian formulation in the dimer RSA model

The dimer model is defined on a homogeneous collector where every location is statistically equivalent before deposition begins. A single dimer covers area

$S_D = 2\pi r^2,$

with $r$ taken as the unit length in the simulations. The RSA procedure is the standard continuum algorithm: a virtual dimer is generated with a random position on the collector and a random in-plane orientation drawn from a uniform probability distribution; the trial configuration is checked for overlap with already adsorbed dimers; if there is no overlap, the dimer is adsorbed irreversibly and remains fixed; if there is overlap, the trial is rejected and discarded; and the process is repeated many times (Ciesla et al., 2012).

This formulation is history-dependent but memoryless at the trial level. The only information needed for the next acceptance or rejection event is the current adsorption state. A plausible implication is that “Markovian RSA” is best understood here not as a distinct algorithmic family, but as the ordinary irreversible RSA dynamics viewed through its state-based evolution law.

Two normalized observables organize the kinetics. The dimensionless RSA time is

$t_0 = N \frac{S_D}{S_C},$

where $N$ is the number of attempted insertions and $S_C$ is the collector area. The instantaneous coverage is

$\theta = n_d \frac{S_D}{S_C},$

where $n_d$ is the number of adsorbed dimers. These quantities make the adsorption process comparable across different collector sizes and simulation lengths.

2. Saturation kinetics and random jamming coverage

The principal asymptotic quantity is the random jamming coverage, or maximal random coverage, $\theta_{\max}$, defined as the limiting coverage as $r$0 in an infinite system. Because the simulations are finite in both system size and time, $r$1 is estimated by extrapolating the late-time approach to saturation. Two forms are compared: Feder’s law for irreversible deposition of random objects on a $r$2-dimensional collector,

$r$3

and, for ordered deposition, a logarithmically corrected form. For the two-dimensional dimer problem, both fits describe the data reasonably well, but the $r$4 form is slightly better (Ciesla et al., 2012).

From these extrapolations, the estimated jamming coverage is approximately $r$5 or $r$6, depending on the fitting form, and the inferred value is

$r$7

This value is very close to the known jamming coverage for circular particles in two dimensions. The paper interprets that proximity as evidence that, for the purpose of random packing, dimers and circles behave similarly in this dimension.

Finite-size effects were checked by varying collector size and boundary conditions. A quadratic fit of the number adsorbed versus collector size showed that the largest system used has less than about $r$8 bias. This supports treating the quoted $r$9 as effectively close to the infinite-system limit.

3. Surface blocking function and fluctuation-based characterization

A central kinetic observable in RSA is the surface blocking function, also called the available surface function (ASF). It measures the fraction of attempted insertions that succeed at a given coverage and is computed directly as

$S_D = 2\pi r^2,$0

At low coverage, the dimer data are fitted by the standard quadratic form

$S_D = 2\pi r^2,$1

with

$S_D = 2\pi r^2,$2

For hard circles, the quoted values are

$S_D = 2\pi r^2,$3

The dimer coefficients are therefore noticeably larger. The stated interpretation is that dimers block the surface more efficiently than circles do, because successful deposition requires not only the right position but also the right orientation (Ciesla et al., 2012).

The ASF is also fitted by a form consistent with asymptotic jamming,

$S_D = 2\pi r^2,$4

again implying a jamming coverage near $S_D = 2\pi r^2,$5. Combined with the kinetic relation

$S_D = 2\pi r^2,$6

this links the vanishing of available surface near saturation to the Feder-type law $S_D = 2\pi r^2,$7. In this sense, the ASF is not only descriptive but directly tied to the long-time adsorption kinetics.

The paper further connects ASF to experimentally accessible number fluctuations. The reduced variance of the number of adsorbed particles in a test area is defined by

$S_D = 2\pi r^2,$8

In the low-coverage limit,

$S_D = 2\pi r^2,$9

The simulations confirm this agreement at small $r$0, whereas the two quantities begin to differ significantly for $r$1. This provides a limited but concrete fluctuation-based route to estimating the blocking function when direct measurement is difficult.

4. Positional correlations and jammed-layer structure

The jammed layer is characterized through a radial autocorrelation function $r$2, that is, the pair-distance distribution or positional correlation between adsorbed particles. For hard circles in RSA, the function has a known short-distance singularity and damped oscillatory structure due to excluded volume. The small-distance behavior is described by the universal RSA form

$r$3

For dimers, the near-contact structure differs in detail because of shape anisotropy. The first maximum is shifted to larger separations, around $r$4, and the near-contact region is broader. Even so, when the dimers are coarse-grained as a set of circles, the correlation function becomes very similar to that of circle adsorption (Ciesla et al., 2012).

The resulting structural picture is that the monolayer is random and short-ranged. Correlations decay rapidly and even superexponentially at larger distances. This suggests that non-convexity and orientational degrees of freedom modify local structure more strongly than they modify long-range positional organization.

5. Orientational statistics and finite-size effects

Because dimers are anisotropic, orientational ordering is a distinct question. The paper defines an orientation order function

$r$5

where $r$6 is the unit orientation vector of the $r$7-th dimer. For a completely random orientation distribution, $r$8 for all $r$9. For perfect alignment, $t_0 = N \frac{S_D}{S_C},$0 oscillates between $t_0 = N \frac{S_D}{S_C},$1 and $t_0 = N \frac{S_D}{S_C},$2.

The observed global orientational order is weak and strongest in small collectors. The paper attributes this to two effects: geometrically, when space becomes tight, parallel placement is more likely than perpendicular placement; and near fixed boundaries, the insertion procedure creates a slight preference for certain orientations (Ciesla et al., 2012). Local orientational correlations nevertheless decay very rapidly, essentially disappearing beyond distances of about $t_0 = N \frac{S_D}{S_C},$3.

The stated conclusion is that the apparent ordering is mostly a finite-size effect rather than a genuine long-range ordered phase. For macroscopic collectors, the monolayer is effectively isotropic. This is an important correction to a possible misconception that anisotropic adsorbates necessarily generate substantial orientational order under RSA dynamics.

6. Experimental comparison, higher-dimensional extension, and terminological disambiguation

The dimer study compares its simulation results with both established theory and experiment. The approach to jamming is consistent with Feder’s law in two dimensions, and the short-distance correlation behavior follows the same universal logarithmic form known from circle RSA. The measured jamming coverage is also compared with adsorption experiments on insulin, treated approximately as a dimer-like object: using the insulin molecular mass and effective size, $t_0 = N \frac{S_D}{S_C},$4 is translated to about $t_0 = N \frac{S_D}{S_C},$5, close to reported experimental values in the range $t_0 = N \frac{S_D}{S_C},$6–$t_0 = N \frac{S_D}{S_C},$7 (Ciesla et al., 2012). The remaining differences are attributed to the simpler geometry of the model and to the fact that real insulin can occur as monomers or hexamers, which would allow somewhat higher coverages.

The study also extends the analysis to three dimensions, modeling dimers as two touching spheres. There again, the jamming coverage is found to be close to that for spheres and the autocorrelation behavior similar to the two-dimensional case. This suggests that, for $t_0 = N \frac{S_D}{S_C},$8, random packing of dimers and spheres may be governed by the same broad RSA principles.

A separate terminological issue arises from the acronym “RSA.” In adsorption physics, RSA denotes random sequential adsorption, and “Markovian RSA” refers to the irreversible, state-determined deposition dynamics discussed above. By contrast, "A Continued Fraction-Hyperbola based Attack on RSA cryptosystem" (Bansimba et al., 2023) concerns RSA cryptanalysis and explicitly does not discuss Markovian RSA in the standard sense of Markov chains or a known RSA variant with that name. Similarly, "An analogue of the ElGamal scheme based on the Markovski algorithm" (Malyutina et al., 2021) concerns a quasigroup-based analogue of ElGamal encryption based on the Markovski algorithm, not adsorption. This suggests that literature searches for “Markovian RSA” require careful disambiguation between random sequential adsorption and cryptographic RSA.

Taken together, the dimer results establish a specific meaning for Markovian RSA in the adsorption literature: a Markovian, irreversible continuum deposition process whose principal effects of anisotropy appear in the blocking function and local orientational statistics, while the overall jamming coverage and long-distance pair correlations remain close to those of spheres or circles.