Pub Date : 2023-11-03DOI: 10.1007/s00332-023-09993-z
V. D. Camiola, V. Romano, G. Vitanza
Abstract Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the acoustic, optical and Z phonons are deduced. The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal’s calculus and its properties, the pseudo-differential operators are expanded up to the second order in $$hbar $$ ħ . An energy transport model is obtained by using the moment method with closure relations based on a quantum version of the Maximum Entropy Principle by employing a relaxation time approximation for the production terms of energy and energy flux. An explicit form of the thermal conductivity with quantum correction up to $$hbar ^2$$ ħ2 order is obtained under a long-time scaling for the most relevant phonon branches.
{"title":"Wigner Equations for Phonons Transport and Quantum Heat Flux","authors":"V. D. Camiola, V. Romano, G. Vitanza","doi":"10.1007/s00332-023-09993-z","DOIUrl":"https://doi.org/10.1007/s00332-023-09993-z","url":null,"abstract":"Abstract Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the acoustic, optical and Z phonons are deduced. The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal’s calculus and its properties, the pseudo-differential operators are expanded up to the second order in $$hbar $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ħ</mml:mi> </mml:math> . An energy transport model is obtained by using the moment method with closure relations based on a quantum version of the Maximum Entropy Principle by employing a relaxation time approximation for the production terms of energy and energy flux. An explicit form of the thermal conductivity with quantum correction up to $$hbar ^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>ħ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> order is obtained under a long-time scaling for the most relevant phonon branches.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"65 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135868671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-02DOI: 10.1007/s00332-023-09986-y
Jiawei Hu, Ari Stern
{"title":"Hamiltonian Mechanics and Lie Algebroid Connections","authors":"Jiawei Hu, Ari Stern","doi":"10.1007/s00332-023-09986-y","DOIUrl":"https://doi.org/10.1007/s00332-023-09986-y","url":null,"abstract":"","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"89 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135933196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-30DOI: 10.1007/s00332-023-09990-2
Jason J. Bramburger, Giovanni Fantuzzi
{"title":"Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization","authors":"Jason J. Bramburger, Giovanni Fantuzzi","doi":"10.1007/s00332-023-09990-2","DOIUrl":"https://doi.org/10.1007/s00332-023-09990-2","url":null,"abstract":"","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"320 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136102995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1007/s00332-023-09983-1
K. U. Kristiansen
Abstract In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity vanishes, hysteresis converges in an appropriate sense to Filippov’s sliding vector-field. Recently (2022), however, Bonet and Seara showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper—which was developed by Bonet et al in a paper from 2017 as an attempt to unify different regularizations of piecewise smooth systems—involves two singular perturbation parameters and includes a combination of slow–fast and nonsmooth effects. The description of this model is therefore—from the perspective of singular perturbation theory—challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov’s sliding vector-field to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).
{"title":"Blowup Analysis of a Hysteresis Model Based Upon Singular Perturbations","authors":"K. U. Kristiansen","doi":"10.1007/s00332-023-09983-1","DOIUrl":"https://doi.org/10.1007/s00332-023-09983-1","url":null,"abstract":"Abstract In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity vanishes, hysteresis converges in an appropriate sense to Filippov’s sliding vector-field. Recently (2022), however, Bonet and Seara showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper—which was developed by Bonet et al in a paper from 2017 as an attempt to unify different regularizations of piecewise smooth systems—involves two singular perturbation parameters and includes a combination of slow–fast and nonsmooth effects. The description of this model is therefore—from the perspective of singular perturbation theory—challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov’s sliding vector-field to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"26 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134908712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-21DOI: 10.1007/s00332-023-09970-6
Lucas Bouck, Ricardo H. Nochetto, Vladimir Yushutin
{"title":"A Hydrodynamical Model of Nematic Liquid Crystal Films with a General State of Orientational Order","authors":"Lucas Bouck, Ricardo H. Nochetto, Vladimir Yushutin","doi":"10.1007/s00332-023-09970-6","DOIUrl":"https://doi.org/10.1007/s00332-023-09970-6","url":null,"abstract":"","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"67 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s00332-023-09959-1
Richard M. Höfer, Amina Mecherbet, Richard Schubert
Abstract We consider a suspension of spherical inertialess particles in a Stokes flow on the torus $$mathbb {T}^3$$ T3 . The particles perturb a linear extensional flow due to their rigidity constraint. Due to the singular nature of this perturbation, no mean-field limit for the behavior of the particle orientation can be valid. This contrasts with widely used models in the literature such as the FENE and Doi models and similar models for active suspensions. The proof of this result is based on the study of the mobility problem of a single particle in a non-cubic torus, which we prove to exhibit a nontrivial coupling between the angular velocity and a prescribed strain.
{"title":"Non-existence of Mean-Field Models for Particle Orientations in Suspensions","authors":"Richard M. Höfer, Amina Mecherbet, Richard Schubert","doi":"10.1007/s00332-023-09959-1","DOIUrl":"https://doi.org/10.1007/s00332-023-09959-1","url":null,"abstract":"Abstract We consider a suspension of spherical inertialess particles in a Stokes flow on the torus $$mathbb {T}^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . The particles perturb a linear extensional flow due to their rigidity constraint. Due to the singular nature of this perturbation, no mean-field limit for the behavior of the particle orientation can be valid. This contrasts with widely used models in the literature such as the FENE and Doi models and similar models for active suspensions. The proof of this result is based on the study of the mobility problem of a single particle in a non-cubic torus, which we prove to exhibit a nontrivial coupling between the angular velocity and a prescribed strain.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"37 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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