Analysis of the random packing of identical particles. VI. Disordering of a network of bonds

Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.

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Random Packing Performance in Continuous Foam Fractionation

Chemical Engineering & Technology ◽

10.1002/ceat.202100097 ◽

2021 ◽

Author(s):

Qianfeng Gu ◽

Xiaochen Xue ◽

Osama M. Darwesh ◽

Pascal Habimana ◽

Wei Liu ◽

...

Keyword(s):

Random Packing ◽

Foam Fractionation

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Optimal Random Packing of Spheres and Extremal Effective Conductivity

Symmetry ◽

10.3390/sym13061063 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1063

Author(s):

Vladimir Mityushev ◽

Zhanat Zhunussova

Keyword(s):

Effective Conductivity ◽

Dimensional Space ◽

Elementary Function ◽

Voronoi Tessellation ◽

Random Packing ◽

Periodicity Cell ◽

Algebraic Equations ◽

Optimal Packing ◽

Linear Algebraic Equations ◽

Initial Location

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

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Decay amplitudes to three hadrons from finite-volume matrix elements

Journal of High Energy Physics ◽

10.1007/jhep04(2021)113 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Maxwell T. Hansen ◽

Fernando Romero-López ◽

Stephen R. Sharpe

Keyword(s):

Finite Volume ◽

Decay Amplitude ◽

Weak Decay ◽

Matrix Elements ◽

Isospin Breaking ◽

Final State ◽

Identical Particles ◽

Hadron Decay ◽

Finite State ◽

Particle Decays

Abstract We derive relations between finite-volume matrix elements and infinite-volume decay amplitudes, for processes with three spinless, degenerate and either identical or non-identical particles in the final state. This generalizes the Lellouch-Lüscher relation for two-particle decays and provides a strategy for extracting three-hadron decay amplitudes using lattice QCD. Unlike for two particles, even in the simplest approximation, one must solve integral equations to obtain the physical decay amplitude, a consequence of the nontrivial finite-state interactions. We first derive the result in a simplified theory with three identical particles, and then present the generalizations needed to study phenomenologically relevant three-pion decays. The specific processes we discuss are the CP-violating K → 3π weak decay, the isospin-breaking η → 3π QCD transition, and the electromagnetic γ* → 3π amplitudes that enter the calculation of the hadronic vacuum polarization contribution to muonic g − 2.

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