Abstract
We explore the consequences of metrically decomposing a finite phase space, modeled as a d-dimensional lattice, into disjoint subspaces (lattices). Ergodic flows of a test particle undergoing an unbiased random walk are characterized by implementing the theory of finite Markov processes. Insights drawn from number theory are used to design the sublattices, the roles of lattice symmetry and system dimensionality are separately considered, and new lattice invariance relations are derived to corroborate the numerical accuracy of the calculated results. We find that the reaction efficiency in a finite system is strongly dependent not only on whether the system is compartmentalized, but also on whether the overall reaction space of the microreactor is further partitioned into separable reactors. We find that the reaction efficiency in a finite system is strongly dependent not only on whether the system is compartmentalized, but also on whether the overall reaction space of the microreactor is further partitioned into separable reactors. The sensitivity of kinetic processes in nanoassemblies to the dimensionality of compartmentalized reaction spaces is quantified.
Highlights
To provide a physical motivation for the present study, understanding the factors influencing self-assembly in nanophase materials is a major experimental and theoretical challenge [1, 2]
We present results for two new lattice invariance relations that can be used to check the accuracy of the site-specific n(i) values that underlie the data given in Tables 1 and 2
In this contribution we have studied ergodic flows of a random walker undergoing unbiased displacements in a positional phase space represented by a host lattice, and characterized quantitatively the consequences of different metric decompositions of a given, finite parent lattice
Summary
To provide a physical motivation for the present study, understanding the factors influencing self-assembly in nanophase materials is a major experimental and theoretical challenge [1, 2]. How do the results change if one or more of the disjoint spaces are n × m lattices, that is, not approximately “square?” Is the difference in reaction efficiency magnified or suppressed? Fundamental results in number theory can be used to motivate the choices of lattices for which the metric decomposability of phase space can be studied. Given the computational demands in implementing the theory of finite Markov processes, these lattices are just too large; our results for dimension d = 3 presented in Section 3 are (much) more limited than for d = 2. We comment on the relevance of our results to self-assembly in nanosystems
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