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Large-scale Three-dimensional Imaging of Cellular Organization in the Mouse Neocortex

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Published on: September 5, 2018

Octupolar order in two dimensions.

Epifanio G Virga¹

¹Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, I-27100, Pavia, Italy. eg.virga@unipv.it.

The European Physical Journal. E, Soft Matter

|July 1, 2015

Summary

This summary is machine-generated.

Octupolar order in 2D is described using maxima of probability density for a specific tensor. This representation is equivalent to diagonalizing the tensor, but only in two dimensions.

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Area of Science:

Condensed matter physics
Statistical mechanics
Tensor analysis

Background:

Octupolar order is a complex phenomenon in condensed matter systems.
Describing such order in two dimensions presents unique challenges.
Third-rank tensors are often used to characterize complex ordering phenomena.

Purpose of the Study:

To develop a novel description of octupolar order in two spatial dimensions.
To establish an equivalence between probability density maxima and tensor diagonalization for octupolar order.
To investigate the limitations of this equivalence in different dimensionalities.

Main Methods:

Utilizing the maxima (and conjugated minima) of probability density.
Associating these maxima with a third-rank, fully symmetric, and traceless tensor.
Comparing this representation with the diagonalization of the relevant tensor.

Main Results:

A new method for describing two-dimensional octupolar order is presented.
The probability density maxima representation is shown to be equivalent to tensor diagonalization.
This equivalence is demonstrated to be specific to the two-dimensional case.

Conclusions:

The maxima of probability density offer a viable way to describe octupolar order in 2D.
Tensor diagonalization provides an equivalent mathematical framework for this description.
The dimensionality of the system is a critical factor for this equivalence.