scholarly journals The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory

2019 ◽  
Vol 5 (5) ◽  
pp. eaav8490 ◽  
Author(s):  
Jesse Berezovsky
Keyword(s):  
Phase Transitions ◽  
Statistical Mechanics ◽  
Music Theory ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Mean Field Model ◽  
Musical Harmony ◽  
Emergent Order ◽  
The Mean

Music, while allowing nearly unlimited creative expression, almost always conforms to a set of rigid rules at a fundamental level. The description and study of these rules, and the ordered structures that arise from them, is the basis of the field of music theory. Here, I present a theoretical formalism that aims to explain why basic ordered patterns emerge in music, using the same statistical mechanics framework that describes emergent order across phase transitions in physical systems. I first apply the mean field approximation to demonstrate that phase transitions occur in this model from disordered sound to discrete sets of pitches, including the 12-fold octave division used in Western music. Beyond the mean field model, I use numerical simulation to uncover emergent structures of musical harmony. These results provide a new lens through which to view the fundamental structures of music and to discover new musical ideas to explore.

Partially Disordered State in the Frustrated Heisenberg Ferrimagnet

2014 ◽  
Vol 215 ◽  
pp. 55-60 ◽  
Author(s):  
Sergey N. Martynov
Keyword(s):  
Phase Diagram ◽  
Phase Transitions ◽  
Magnetic Phase ◽  
Mean Field ◽  
Exchange Interactions ◽  
Field Approximation ◽  
Magnetic Phase Transitions ◽  
Mean Field Approximation ◽  
The Mean ◽  
Competition Parameter

A model for the description of two-subsystem Heisenberg ferrimagnet with frustrated intersubsystem exchange and competition between exchange interactions in a subsystem is proposed. The conditions of the existence of noncollinear Yafet-Kittel state and partially ordered magnetic structure are investigated. The phase diagram of competition parameter vs temperature is obtained in the mean field approximation. The peculiarities of the succesive magnetic phase transitions are considered.

A MEAN FIELD STUDY OF THE PHASE TRANSITIONS IN A CLASS OF SYSTEMS WITH COMPLEX ORDER PARAMETERS

1994 ◽  
Vol 08 (28) ◽  
pp. 3963-3986
Author(s):  
EVGENIA J. BLAGOEVA
Keyword(s):  
Phase Diagram ◽  
Phase Transitions ◽  
Order Parameter ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Unconventional Superconductors ◽  
Complex Order ◽  
The Mean ◽  
Expansion Coefficients

A generalized Landau free energy for a complex order parameter expanded up to sixth-order is investigated using group theoretical arguments and the mean-field approximation. Results for the phase transitions that occur are presented. The phase diagram for all allowed values of the expansion coefficients is constructed with an emphasis placed on the influence of the anisotropy in the order parameter space. The results can be used in discussions of unconventional superconductors and modulated structural and magnetic orderings.

scholarly journals Mean field approximation of network of biophysical neurons driven by conductance-based ion exchange dynamics

2021 ◽  
Author(s):  
Abhirup Bandyopadhyay ◽  
Spase Petkoski ◽  
Viktor Jirsa
Keyword(s):  
Ion Exchange ◽  
Molecular Mechanisms ◽  
Field Model ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Mean Field Model ◽  
Ion Concentrations ◽  
The Mean ◽  
Locally Homogeneous

Changes in extracellular ion concentrations are known to modulate neuronal excitability and play a major role in controlling the neuronal firing rate, not just during the healthy homeostasis, but also in pathological conditions such as epilepsy. The microscopic molecular mechanisms of field effects are understood, but the precise correspondence between the microscopic mechanisms of ion exchange in the cellular space of neurons and the macroscopic behavior of neuronal populations remains to be established. We derive a mean field model of a population of Hodgkin Huxley type neurons. This model links the neuronal intra- and extra-cellular ion concentrations to the mean membrane potential and the mean synaptic input in terms of the synaptic conductance of the locally homogeneous mesoscopic network and can describe various brain activities including multi-stability at resting states, as well as more pathological spiking and bursting behaviors, and depolarizations. The results from the analytical solution of the mean field model agree with the mean behavior of numerical simulations of large-scale networks of neurons. The mean field model is analytically exact for non-autonomous ion concentration variables and provides a mean field approximation in the thermodynamic limit, for locally homogeneous mesoscopic networks of biophysical neurons driven by an ion exchange mechanism. These results may provide the missing link between high-level neural mass approaches which are used in the brain network modeling and physiological parameters that drive the neuronal dynamics.

Phase Transitions and the Ising Model

2019 ◽  
pp. 423-446
Author(s):  
Robert H. Swendsen
Keyword(s):  
Phase Transitions ◽  
Ising Model ◽  
Landau Theory ◽  
Magnetic Moments ◽  
Mean Field ◽  
Higher Dimensions ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Ising Chain ◽  
The Mean

Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution for the Ising chain (one-dimensional model), including the introduction of the transfer matrix method. Higher dimensions are treated in the Mean Field Approximation (MFA), which is also extended to Landau theory. The Ising model is deceptively simple. It can be defined in a few words, but it displays astonishingly rich behavior. It originated as a model of ferromagnetism in which the magnetic moments were localized on lattice sites and had only two allowed values.

ANALYTICAL EXPRESSIONS OF THE ORDER PARAMETERS NEAR THE TRANSITION TEMPERATURES IN THE SPIN-3/2 ISING SYSTEM WITH BILINEAR AND BIQUADRATIC INTERACTIONS

2006 ◽  
Vol 20 (04) ◽  
pp. 455-467 ◽  
Author(s):  
OSMAN CANKO ◽  
MUSTAFA KESKIN
Keyword(s):  
Phase Transitions ◽  
Mean Field ◽  
Field Approximation ◽  
Irreversible Thermodynamics ◽  
Order Parameters ◽  
Mean Field Approximation ◽  
Transition Temperatures ◽  
Ising System ◽  
The Mean ◽  
Analytical Expressions

Analytical expressions of the order parameters near the transition temperatures in the spin-3/2 Ising system with bilinear (J) and biquadratic (K) interactions are presented for various values of J/K. First, we obtain the free energy expression and the equations to determine order parameters by using the mean-field approximation. Then, the order parameters are expressed in the vicinity of the transition temperatures in which these expressions are very important to study the dynamics of the system by means of Onsager's theory of irreversible thermodynamics. Hence, we investigate the phase transitions occurring in the system and also obtain two tricritical points analytically. Finally, the specific heat and magnetic susceptibility are calculated and an argument about the critical exponents at the second-order phase transitions and tricritical points is given.

scholarly journals Short-ranged interaction effects on Z2 topological phase transitions: The perturbative mean-field method

2015 ◽  
Vol 29 (06) ◽  
pp. 1530005 ◽  
Author(s):  
Hsin-Hua Lai ◽  
Hsiang-Hsuan Hung
Keyword(s):  
Phase Transitions ◽  
Analytical Approach ◽  
Mean Field ◽  
Field Approximation ◽  
Quantum Spin ◽  
Mean Field Approximation ◽  
Topological Phase ◽  
Self Consistent ◽  
The Mean ◽  
Topological Phase Transitions

Time-reversal symmetric topological insulator (TI) is a novel state of matter that a bulk-insulating state carries dissipationless spin transport along the surfaces, embedded by the Z2 topological invariant. In the noninteracting limit, this exotic state has been intensively studied and explored with realistic systems, such as HgTe/(Hg, Cd)Te quantum wells. On the other hand, electronic correlation plays a significant role in many solid-state systems, which further influences topological properties and triggers topological phase transitions. Yet an interacting TI is still an elusive subject and most related analyses rely on the mean-field approximation and numerical simulations. Among the approaches, the mean-field approximation fails to predict the topological phase transition, in particular at intermediate interaction strength without spontaneously breaking symmetry. In this paper, we develop an analytical approach based on a combined perturbative and self-consistent mean-field treatment of interactions that is capable of capturing topological phase transitions beyond either method when used independently. As an illustration of the method, we study the effects of short-ranged interactions on the Z2 TI phase, also known as the quantum spin Hall (QSH) phase, in three generalized versions of the Kane–Mele (KM) model at half-filling on the honeycomb lattice. The results are in excellent agreement with quantum Monte Carlo (QMC) calculations on the same model and cannot be reproduced by either a perturbative treatment or a self-consistent mean-field treatment of the interactions. Our analytical approach helps to clarify how the symmetries of the one-body terms of the Hamiltonian determine whether interactions tend to stabilize or destabilize a topological phase. Moreover, our method should be applicable to a wide class of models where topological transitions due to interactions are in principle possible, but are not correctly predicted by either perturbative or self-consistent treatments.

scholarly journals SUSCEPTIBILITIES AND THE PHASE STRUCTURE OF AN EFFECTIVE CHIRAL MODEL WITH POLYAKOV LOOPS

2007 ◽  
Vol 16 (07n08) ◽  
pp. 2319-2324
Author(s):  
CHIHIRO SASAKI ◽  
BENGT FRIMAN ◽  
KRZYSZTOF REDLICH
Keyword(s):  
Phase Transitions ◽  
Phase Structure ◽  
Mean Field ◽  
Field Approximation ◽  
Order Parameters ◽  
Mean Field Approximation ◽  
Chiral Model ◽  
Chiral Phase ◽  
The Mean ◽  
Polyakov Loops

In the Nambu–Jona-Lasinio model with Polyakov loops, we explore the relation between the deconfinement and chiral phase transitions within the mean-field approximation. We focus on the phase structure of the model and study the susceptibilities associated with corresponding order parameters.

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