4 the solutions are stable while for s<4 are stable but with much smaller amplitudes than the solutions for s>4. The main interest of this work was the statistical properties of the nonlinear systems. For this purpose we thermalized nonlinear lattices through Langevin equations of motion. We found that due to thermal fluctuations DBs are generated and annihilated. Once they are generated, they survive under Langevin forces with limited lifetime. The presence of discrete breathers is indicated by the slow decay of the energy correlation functions as a function of time. We studied further a one-dimentional model with a specific form of nonlinear on-site potential and found that this model exhibits an energy threshold; there is aspectral region where breathers are unstable and in order to excite stable breathers it is necessary to overcome the energy threshold. Although this property has been observed in three-dimentional models, this is the first evidence in one-dimentional problems. We determined numerically the spectral gap and then thermalized the lattice. We found that the gap is clearly manifested at small couplings through a large spectral contribution in the linerized phonon region that is not shifted strongly with temperature. We then investigated the equilibrium thermodynamics of a lattice with hard 0 on-site potential using the Transfer Integral technique that gives exact results in the thermodynamic limit. we computed the specific heat as a function of temperature that shows a smooth behavior and consequently no phase transition can occur. The issue of phase transition is nonlinear models that exhibit breather solutions is not very clear; a model that describe thermal denaturation of DNA involving nearest-neighbor nonlinear interactions and a Morse on-site potential clearly exhibits an entropy-driven transition. This is due however to the plateau that the Morse on-site potential contains and makes the particle dynamics unbounded. Other studies in the Discrete Nonlinear Schrodinger Equation demonstrate that DBs can induce a phase transition. In our work we computed the eigenvalues of Transfer operator for a nonlinear ladder model that shows different behavior for strong enough coupling between the chains in the low and higher temperature regime that may lead to a phase transition. finally we constructed a pseudospin Ising model motivated by the natural bimodality that DBs induce in nonlinear thermalized lattices. we observed that a thermalized lattice is split into regions of high energy accumulation where breathers dominate as well as regions of low energy accumulation where linear and quasilinear modes exist.We followed standard techniques from the theory of spin glasses and found that the nonlinear model has "glassy" character in high enough temperatures where breathers dominate in the system. The computations have been done in thermalized lattices that were not in thermal equilibrium. we showed that the high temperature phase corresponds to an average over the specific sectors of the phase space that is related to the presence of the nonlinear localized modes and not to the entire system phase space as we have done using the Transfer Integral technique. Even though we performed much computational analysis and introduced new ideas in the field we believe that several new questions ghave appeared that need to be addressed in the future. Specifically, the application of the TI technique in multiple ladders would give a clearer indication to the possibility to describe nonlinear lattice thermodynamics through replicas. Furthermore, a computational approach needs to be developed that will handle correctly the return to the single replica limit. In the contex of the pseudospin Ising model on the other hand a more detailed investigation is needed through a Potts model that we think might be more realistic. the latter is a generalization of the Ising model by extending the number of directions of the spins, consequently we would have to descretize the energies not only by a two state system as we performed in pseudospin Ising representation but in a larger state system. Finally we have to mention that in this work we used mainly autocorrelation functions. In general the nonequilibrium processes described more efficiently by two-time correlation functions.In out of equilibrium systems the correlation functions depend on the waiting or "aging" time. Further investigation is required by using two-time correlation functions that will give better information for the long-lived nonlinear localized modes that induce slow relaxation phenomena.. Types: Τύπος Εργασίας--Διδακτορικές διατριβές, text" />
Στατιστικές ιδιότητες κλασσικών μη γραμμικών πλεγμάτων
δείτε την πρωτότυπη σελίδα τεκμηρίου στον ιστότοπο του αποθετηρίου του φορέα για περισσότερες πληροφορίες και για να δείτε όλα τα ψηφιακά αρχεία του τεκμηρίου*
Statistical properties of classical nonlinear lattices
Στατιστικές ιδιότητες κλασσικών μη γραμμικών πλεγμάτων
Eleftheriou, Maria
Ελευθερίου, Μαρία
Τσιρώνης, Γεώργιος
Σ'αυτήν την εργασία ασχολούμαστε με τη στατιστική μελέτη των διακριτών "πνοών" (Discrete Breathers). Πιο συγκεκριμένα επικεντρωνόμαστε σε μη γραμμικά Χαμιλτονιανά συστήματα καθώς και σε μη γραμμικά συστήματα με θερμοκρασία όπου οι "πνοές" (DBs)δημιουργούνται αυθόρμητα από τις θερμικές διακυμάνσεις. Στο δεύτερο κεφάλαιο αναπτύσσουμε μια καινούργια τεχνική στο πεδίο των μεθόδων κατασκευής των DBs η οποία μειώνει τον υπολογιστικό χρόνο. Εισάγουμε επίσης μια καινούργια τεχνική για να κινήσουμε αυτές τις λύσεις στο πλέγμα η οποία είναι αρκετά απλή. Μελετάμε επίσης μοντέλα με αλληλεπιδράσεις μεταξύ των σωματίων μη γραμμικού τύπου και βρίσκουμε λύσεις DBs εντοπισμένες στο χώρο με μη εκθετικό τρόπο. Μελετήσαμε εκτενέστερα το σχήμα τους και βρήκαμε οτι η ουρά των DBs διέπεται από έναν super-εκθετικό νόμο, ενώ ο πυρήνας χαρακτηρίζεται από νόμο συνιμητόνου. Επίσης μελετήσαμε μοντέλα με αλληλεπιδράσεις μεταξύ των σωματίων μη γραμμικού τύπου και μακράς εμβέλειας με εξάρτηση r^s όπου r η απόσταση. Εξάγαμε εντοπισμένες χωρικά λύσεις στο συνεχές όριο χρησιμοποιώντας σειρές Taylor στις διακριτές εξισώσεις. Στο τρίτο κεφάλαιο μελετάμε μη γραμμικά πλέγματα με θερμοκρασία και αρχικά εξετάζουμε ένα σύνολο ασύζευκτων σωματίων και έπειτα αλυσίδες συζευγμένων μη γραμμικών ταλαντωτών. Οι DBs γεννούνται αυθόρμητα και επιβιώνουν υπο την παρουσία των θερμικών διακυμάνσεων. Υπολογίζουμε τη χρονική εξάρτηση των συναρτήσεων συσχέτισης της ενέργειας καθώς αναλύουμε και επιβεβαιώνουμε ένα πολύ πρόσφατο πείραμα όπου ένας πιεζοηλεκτρικός οδηγός διεγείρει τον οπτικό τρόπο ταλάντωσης ενός διατομικού συστήματος μικροδοκών και δημιουργούντσι DBs υπο την παρουσία θερμικών διακυμάνσεων και τριβής. Στο τέταρτο κεφάλαιο μελετάμε ένα μονοδιάστατο υψηλής μη γραμμικότητας μοντέλο. Βρίσκουμε οτι χρειάζεται να εισαχθεί πεπερασμένη ενέργεια ώστε να διεγείρουμε DBs, υπάρχει δηλαδή ενεργειακό κατώφλι, ιδιότητα που συναντάται σε τρισδιάστατα μοντέλα. Προσδιορίζουμε το χάσμα όπου οι DBs είναι ασταθείς και αφού θερμάνουμε το πλέγμα υπολογίζουμε φάσματα ισχύος. Σ'αυτά τα φάσματα ισχύος το χάσμα εμφανίζεται για χαμηλές τιμές της σταθεράς σύζευξης. Στο πέμπτο κεφάλαιο μελετάμε στην θερμοδυναμική ισορροπία διάφορες θερμοδυναμικές ποσότητες όπως τη θερμοχωρητιθκότητα των μη γραμμικών συστημάτων με τη μέθοδο Transfer-Integral η οποία δίνει ακριβή αποτελέσματα στο θερμοδυναμικό όριο. Τέλος στο έκτο κεφάλαιο μετατρέπουμε το μη γραμμικό πλέγμα σε πλέγμα με spins τύπου Ising. Εισάγουμε το σύνολο ομοίων μακροσκοπικά συστημάτων και εξετάζουμε την επιδεκτικότητα του συστήματος μέσω της κατανομής επικάλυψης. Βρίσκουμε οτι οι χρονικές συναρτήσεις συσχέτισης συμφωνούν με την μέγιστη επικάλυψη για μεγάλες τιμές της θερμοκρασίας δείχνοντας οτι το μη γραμμικό σύστημα λόγω της παρουσίας DBs παρουσιάζει χαρακτηριστικά spin-glass.
(EL)
In the first part of this thesis we studied Hamiltonian systems where we developed new numerical techniques, such as the adaptive method that reduce significantly the computational time in the context of construction of DBs as well as the derivative gradient vector technique, that induces mobility. Until now tha standard method in the domain of mobility of DBs was the use of an antisymmetric pinning eigenvector of the Floquet matrix. On the other hand the application of the derivative gradient vector technique not only is much simpler compared to the calculations that determine the pinning mode through the Floquet matrix, but also renders mobile breather solutions that possibly do not exhibit pinning modes.Additionally, we studied systems with nonlinear interactions and found breather solutions with nonexponential spatial shape. These breather solutions have a more compact spatial shape that is described by a superexponential law in the tails. We found also the continuoum versions of the discrete breather modes in a lattice with long range nonlinear interactions that are localized and their existence and stability depend on the value of long range interaction s, their amplitude and their width. In general for s>4 the solutions are stable while for s<4 are stable but with much smaller amplitudes than the solutions for s>4. The main interest of this work was the statistical properties of the nonlinear systems. For this purpose we thermalized nonlinear lattices through Langevin equations of motion. We found that due to thermal fluctuations DBs are generated and annihilated. Once they are generated, they survive under Langevin forces with limited lifetime. The presence of discrete breathers is indicated by the slow decay of the energy correlation functions as a function of time. We studied further a one-dimentional model with a specific form of nonlinear on-site potential and found that this model exhibits an energy threshold; there is aspectral region where breathers are unstable and in order to excite stable breathers it is necessary to overcome the energy threshold. Although this property has been observed in three-dimentional models, this is the first evidence in one-dimentional problems. We determined numerically the spectral gap and then thermalized the lattice. We found that the gap is clearly manifested at small couplings through a large spectral contribution in the linerized phonon region that is not shifted strongly with temperature. We then investigated the equilibrium thermodynamics of a lattice with hard 0 on-site potential using the Transfer Integral technique that gives exact results in the thermodynamic limit. we computed the specific heat as a function of temperature that shows a smooth behavior and consequently no phase transition can occur. The issue of phase transition is nonlinear models that exhibit breather solutions is not very clear; a model that describe thermal denaturation of DNA involving nearest-neighbor nonlinear interactions and a Morse on-site potential clearly exhibits an entropy-driven transition. This is due however to the plateau that the Morse on-site potential contains and makes the particle dynamics unbounded. Other studies in the Discrete Nonlinear Schrodinger Equation demonstrate that DBs can induce a phase transition. In our work we computed the eigenvalues of Transfer operator for a nonlinear ladder model that shows different behavior for strong enough coupling between the chains in the low and higher temperature regime that may lead to a phase transition. finally we constructed a pseudospin Ising model motivated by the natural bimodality that DBs induce in nonlinear thermalized lattices. we observed that a thermalized lattice is split into regions of high energy accumulation where breathers dominate as well as regions of low energy accumulation where linear and quasilinear modes exist.We followed standard techniques from the theory of spin glasses and found that the nonlinear model has "glassy" character in high enough temperatures where breathers dominate in the system. The computations have been done in thermalized lattices that were not in thermal equilibrium. we showed that the high temperature phase corresponds to an average over the specific sectors of the phase space that is related to the presence of the nonlinear localized modes and not to the entire system phase space as we have done using the Transfer Integral technique. Even though we performed much computational analysis and introduced new ideas in the field we believe that several new questions ghave appeared that need to be addressed in the future. Specifically, the application of the TI technique in multiple ladders would give a clearer indication to the possibility to describe nonlinear lattice thermodynamics through replicas. Furthermore, a computational approach needs to be developed that will handle correctly the return to the single replica limit. In the contex of the pseudospin Ising model on the other hand a more detailed investigation is needed through a Potts model that we think might be more realistic. the latter is a generalization of the Ising model by extending the number of directions of the spins, consequently we would have to descretize the energies not only by a two state system as we performed in pseudospin Ising representation but in a larger state system. Finally we have to mention that in this work we used mainly autocorrelation functions. In general the nonequilibrium processes described more efficiently by two-time correlation functions.In out of equilibrium systems the correlation functions depend on the waiting or "aging" time. Further investigation is required by using two-time correlation functions that will give better information for the long-lived nonlinear localized modes that induce slow relaxation phenomena.
(EN)
Σχολή/Τμήμα--Σχολή Θετικών και Τεχνολογικών Επιστημών--Τμήμα Φυσικής--Διδακτορικές διατριβές
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