Interlinking motifs and entropy landscapes of statistically interacting particles

Abstract

The s = 1/2 Ising chain with uniform nearest-neighbor and next-nearest-neighbor coupling is used to construct a system of floating particles characterized by motifs of up to six consecutive local spins. The spin couplings cause the assembly of particles which, in turn, remain free of interaction energies even at high density. All microstates are configurations of particles from one of three different sets, excited from pseudo-vacua associated with ground states of periodicities one, two, and four. The motifs of particles and elements of pseudo-vacuum interlink in two shared site variables. The statistical interaction between particles is encoded in a generalized Pauli principle, describing how the placement of one particle modifies the options for placing further particles. In the statistical mechanical analysis arbitrary energies can be assigned to all particle species. The entropy is a function of the particle populations. The statistical interaction specifications are transparently built into that expression. The energies and structures of the particles alone govern the ordering at low temperature. Under special circumstances the particles can be replaced by more fundamental particles with shorter motifs that interlink in only one shared site variable. Structures emerge from interactions on two levels: particles with shapes from coupled spins and long-range ordering tendencies from statistically interacting particles with shapes.

Molecular chains under tension: Thermal and mechanical activation of statistically interacting extension and contraction particles

“…The statistical interaction is governed by combinatorial rules that are captured in Haldane's generalized exclusion principle, originally proposed in a quantum many-body context [25]. This concept of statistical interaction as a tool in statistical mechanical modeling has proven useful for a broad field of applications [9][10][11][12][13][14][26][27][28][29][30][31][32][33].…”

“…A significant broadening in scope of this methodology resulted from its extension to particle nesting [11,14,[30][31][32][33]. The generalized Pauli principle (1), in its original version, was meant to be an exclusion principle, implying that ∆d m ≤ 0 if ∆N m ′ > 0.…”

“…II E and II F belong to these types. Compact and nested particles at level 2 were introduced to describe ferromagnetic domains antiferromagnetic domain walls [30][31][32][33]. They were also employed to describe the coil-helix transition between conformations of a polypeptide [14].…”

“…The generic mathematical structure of our approach is based on an idea of Haldane [25] and has been developed by many contributors [26][27][28][29][30][31][32][33]. The inner workings of the methodology are outlined in Sec.…”

Jammed disks in a narrow channel: criticality and ordering tendencies

“…Therefore, all microstates with given particle content constitute a macrostate in the sense discussed earlier. The entropy of a macrostate as derived from the multiplicity expression (7) for N m ≫ 1 via S = k B ln W reads [18,22]:…”