Entropy Production in Nonequilibrium Systems at Stationary States

We present a stochastic approach to nonequilibrium thermodynamics based on the expression of the entropy production rate advanced by Schnakenberg for systems described by a master equation. From the microscopic Schnakenberg expression we get the macroscopic bilinear form for the entropy production rate in terms of fluxes and forces. This is performed by placing the system in contact with two reservoirs with distinct sets of thermodynamic fields and by assuming an appropriate form for the transition rate. The approach is applied to an interacting lattice gas model in contact with two heat and particle reservoirs. On a square lattice, a continuous symmetry breaking phase transition takes place such that at the nonequilibrium ordered phase a heat flow sets in even when the temperatures of the reservoirs are the same. The entropy production rate is found to have a singularity at the critical point of the linear-logarithm type.

Modeling the gluon propagator in Landau gauge: Lattice estimates of pole masses and dimension-two condensates

We present an analytic description of numerical results for the Landau-gauge SU(2) gluon propagator D(p(2)), obtained from lattice simulations (in the scaling region) for the largest lattice sizes to date, in d = 2, 3 and 4 space-time dimensions. Fits to the gluon data in 3d and in 4d show very good agreement with the tree-level prediction of the refined Gribov-Zwanziger (RGZ) framework, supporting a massive behavior for D(p(2)) in the infrared limit. In particular, we investigate the propagator's pole structure and provide estimates of the dynamical mass scales that can be associated with dimension-two condensates in the theory. In the 2d case, fitting the data requires a noninteger power of the momentum p in the numerator of the expression for D(p(2)). In this case, an infinite-volume-limit extrapolation gives D(0) = 0. Our analysis suggests that this result is related to a particular symmetry in the complex-pole structure of the propagator and not to purely imaginary poles, as would be expected in the original Gribov-Zwanziger scenario.

STRUCTURE AND DYNAMICS: THE TRANSITION FROM NONEQUILIBRIUM TO EQUILIBRIUM IN INTEGRATE-AND-FIRE DYNAMICS

The transient and equilibrium properties of dynamics unfolding in complex systems can depend critically on specific topological features of the underlying interconnections. In this work, we investigate such a relationship with respect to the integrate-and-fire dynamics emanating from a source node and an extended network model that allows control of the small-world feature as well as the length of the long-range connections. A systematic approach to investigate the local and global correlations between structural and dynamical features of the networks was adopted that involved extensive simulations (one and a half million cases) so as to obtain two-dimensional correlation maps. Smooth, but diverse surfaces of correlation values were obtained in all cases. Regarding the global cases, it has been verified that the onset avalanche time (but not its intensity) can be accurately predicted from the structural features within specific regions of the map (i.e. networks with specific structural properties). The analysis at local level revealed that the dynamical features before the avalanches can also be accurately predicted from structural features. This is not possible for the dynamical features after the avalanches take place. This is so because the overall topology of the network predominates over the local topology around the source at the stationary state.

Mean-Field critical behavior and ergodicity break in a nonequilibrium one-dimensional RSOS growth model

We investigate the nonequilibrium roughening transition of a one-dimensional restricted solid-on-solid model by directly sampling the stationary probability density of a suitable order parameter as the surface adsorption rate varies. The shapes of the probability density histograms suggest a typical Ginzburg-Landau scenario for the phase transition of the model, and estimates of the "magnetic" exponent seem to confirm its mean-field critical behavior. We also found that the flipping times between the metastable phases of the model scale exponentially with the system size, signaling the breaking of ergodicity in the thermodynamic limit. Incidentally, we discovered that a closely related model not considered before also displays a phase transition with the same critical behavior as the original model. Our results support the usefulness of off-critical histogram techniques in the investigation of nonequilibrium phase transitions. We also briefly discuss in the appendix a good and simple pseudo-random number generator used in our simulations.

Albert Einstein y la revolución de la Física al inicio del Siglo XX

Esta charla se enmarca dentro de la celebración de las XX Jornadas Blascabrerianas (de homenaje a y recuerdo de Blas Cabrera Felipe

Untersuchungen zum nichtstörungstheoretischen Renormierungsverhalten der Quanten-Einstein-Gravitation

Deutsche Version: Zunächst wird eine verallgemeinerte Renormierungsgruppengleichung für die effektiveMittelwertwirkung der EuklidischenQuanten-Einstein-Gravitation konstruiert und dann auf zwei unterschiedliche Trunkierungen, dieEinstein-Hilbert-Trunkierung und die$R^2$-Trunkierung, angewendet. Aus den resultierendenDifferentialgleichungen wird jeweils die Fixpunktstrukturbestimmt. Die Einstein-Hilbert-Trunkierung liefert nebeneinem Gaußschen auch einen nicht-Gaußschen Fixpunkt. Diesernicht-Gaußsche Fixpunkt und auch der Fluß in seinemEinzugsbereich werden mit hoher Genauigkeit durch die$R^2$-Trunkierung reproduziert. Weiterhin erweist sichdie Cutoffschema-Abhängigkeit der analysierten universellenGrößen als äußerst schwach. Diese Ergebnisse deuten daraufhin, daß dieser Fixpunkt wahrscheinlich auch in der exaktenTheorie existiert und die vierdimensionaleQuanten-Einstein-Gravitation somit nichtperturbativ renormierbar sein könnte. Anschließend wird gezeigt, daß der ultraviolette Bereich der$R^2$-Trunkierung und somit auch die Analyse des zugehörigenFixpunkts nicht von den Stabilitätsproblemen betroffen sind,die normalerweise durch den konformen Faktor der Metrikverursacht werden. Dadurch motiviert, wird daraufhin einskalares Spielzeugmodell, das den konformen Sektor einer``$-R+R^2$''-Theorie simuliert, hinsichtlich seinerStabilitätseigenschaften im infraroten (IR) Bereichstudiert. Dabei stellt sich heraus, daß sich die Theorieunter Ausbildung einer nichttrivialen Vakuumstruktur auf dynamische Weise stabilisiert. In der Gravitation könnteneventuell nichtlokale Invarianten des Typs $intd^dx,sqrt{g}R (D^2)^{-1} R$ dafür sorgen, daß der konformeSektor auf ähnliche Weise IR-stabil wird.

Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Einstein e Minkowski sulla relatività ristretta

La tesi è relativa al confronto tra Minkowski e Einstein in merito all'esposizione della teoria della relatività.

Cosmological predictions for a scalar tensor dark energy model by a dedicated Einstein-Boltzmann code

We have extended the Boltzmann code CLASS and studied a specific scalar tensor dark energy model: Induced Gravity

I raggi di Einstein come strumento per la cosmologia

La Tesi presenta uno studio sulla distribuzione dei raggi di Einsten compiuta su campioni di ammassi simulati. Il codice utilizzato, MOKA, consente la costruzione di vasti campioni di ammassi in differenti cosmologie e con differenti parametri strutturali. I risultati ottenuti sono stati confrontati con quelli ottenuti dalla simulazione cosmologica N-body ad alta risoluzione MUSIC. Sono stati quindi prodotti campioni di ammassi per sette valori diversi della normalizzazione dello spettro di potenza e 7 valori diversi del parametro di densità della materia mantenendo la geometria piatta

Max Abraham's and Tullio Levi-Civita's approach to Einstein Theory of Relativity

This work deals with the theory of Relativity and its diffusion in Italy in the first decades of the XX century. Not many scientists belonging to Italian universities were active in understanding Relativity, but two of them, Max Abraham and Tullio Levi-Civita left a deep mark. Max Abraham engaged a substantial debate against Einstein between 1912 and 1914 about electromagnetic and gravitation aspects of the theories. Levi-Civita played a fundamental role in giving Einstein the correct mathematical instruments for the General Relativity formulation since 1915. This work, which doesn't have the aim of a mere historical chronicle of the events, wants to highlight two particular perspectives: on one hand, the importance of Abraham-Einstein debate in order to clarify the basis of Special Relativity, to observe the rigorous logical structure resulting from a fragmentary reasoning sequence and to understand Einstein's thinking; on the other hand, the originality of Levi-Civita's approach, quite different from the Einstein's one, characterized by the introduction of a method typical of General Relativity even to Special Relativity and the attempt to hide the two Einstein Special Relativity postulates.

Konstruktion und Analyse einer funktionalen Renormierungsgruppengleichung für Gravitation im Einstein-Cartan-Zugang

Während das Standardmodell der Elementarteilchenphysik eine konsistente, renormierbare Quantenfeldtheorie dreier der vier bekannten Wechselwirkungen darstellt, bleibt die Quantisierung der Gravitation ein bislang ungelöstes Problem. In den letzten Jahren haben sich jedoch Hinweise ergeben, nach denen metrische Gravitation asymptotisch sicher ist. Das bedeutet, daß sich auch für diese Wechselwirkung eine Quantenfeldtheorie konstruieren läßt. Diese ist dann in einem verallgemeinerten Sinne renormierbar, der nicht mehr explizit Bezug auf die Störungstheorie nimmt. Zudem sagt dieser Zugang, der auf der Wilsonschen Renormierungsgruppe beruht, die korrekte mikroskopische Wirkung der Theorie voraus. Klassisch ist metrische Gravitation auf dem Niveau der Vakuumfeldgleichungen äquivalent zur Einstein-Cartan-Theorie, die das Vielbein und den Spinzusammenhang als fundamentale Variablen verwendet. Diese Theorie besitzt allerdings mehr Freiheitsgrade, eine größere Eichgruppe, und die zugrundeliegende Wirkung ist von erster Ordnung. Alle diese Eigenschaften erschweren eine zur metrischen Gravitation analoge Behandlung.rnrnIm Rahmen dieser Arbeit wird eine dreidimensionale Trunkierung von der Art einer verallgemeinerten Hilbert-Palatini-Wirkung untersucht, die neben dem Laufen der Newton-Konstante und der kosmologischen Konstante auch die Renormierung des Immirzi-Parameters erfaßt. Trotz der angedeuteten Schwierigkeiten war es möglich, das Spektrum des freien Hilbert-Palatini-Propagators analytisch zu berechnen. Auf dessen Grundlage wird eine Flußgleichung vom Propertime-Typ konstruiert. Zudem werden geeignete Eichbedingungen gewählt und detailliert analysiert. Dabei macht die Struktur der Eichgruppe eine Kovariantisierung der Eichtransformationen erforderlich. Der resultierende Fluß wird für verschiedene Regularisierungsschemata und Eichparameter untersucht. Dies liefert auch im Einstein-Cartan-Zugang berzeugende Hinweise auf asymptotische Sicherheit und damit auf die mögliche Existenz einer mathematisch konsistenten und prädiktiven fundamentalen Quantentheorie der Gravitation. Insbesondere findet man ein Paar nicht-Gaußscher Fixpunkte, das Anti-Screening aufweist. An diesen sind die Newton-Konstante und die kosmologische Konstante jeweils relevante Kopplungen, wohingegen der Immirzi-Parameter an einem Fixpunkt irrelevant und an dem anderen relevant ist. Zudem ist die Beta-Funktion des Immirzi-Parameters von bemerkenswert einfacher Form. Die Resultate sind robust gegenüber Variationen des Regularisierungsschemas. Allerdings sollten zukünftige Untersuchungen die bestehenden Eichabhängigkeiten reduzieren.

Interacting Bose-Fermi mixtures in optical lattices

In this thesis, we investigate mixtures of quantum degenerate Bose and Fermi gases of neutral atoms in threedimensional optical lattices. Feshbach resonances allow to control interspecies interactions in these systems precisely, by preparing suitable combinations of internal atomic states and applying external magnetic fields. This way, the system behaviour can be tuned continuously from mutual transparency to strongly interacting correlated phases, up to the stability boundary.rnThe starting point for these investigations is the spin-polarized fermionic band insulator. The properties of this non-interacting system are fully determined by the Pauli exclusion principle for the occupation of states in the lattice. A striking demonstration of the latter can be found in the antibunching of the density-density correlation of atoms released from the lattice. If bosonic atoms are added to this system, isolated heteronuclear molecules can be formed on the lattice sites via radio-frequency stimulation. The efficiency of this process hints at a modification of the atom number distribution over the lattice caused by interspecies interaction.rnIn the following, we investigate systems with tunable interspecies interaction. To this end, a method is developed which allows to assess the various contributions to the system Hamiltonian both qualitatively and quantitatively by following the quantum phase diffusion of the bosonic matter wave.rnBesides a modification of occupation number statistics, these measurements show a significant renormalization of the bosonic Hubbard parameters. The final part of the thesis considers the implications of this renormalization effect on the many particle physics in the mixture. Here, we demonstrate how the quantum phase transition from a bosonic superfluid to a Mott insulator state is shifted towards considerably shallower lattices due to renormalization.

Interacting bosons and fermions in three-dimensional optical lattice potentials

This thesis reports on the realization, characterization and analysis of ultracold bosonic and fermionic atoms in three-dimensional optical lattice potentials. Ultracold quantum gases in optical lattices can be regarded as ideal model systems to investigate quantum many-body physics. In this work interacting ensembles of bosonic 87Rb and fermionic 40K atoms are employed to study equilibrium phases and nonequilibrium dynamics. The investigations are enabled by a versatile experimental setup, whose core feature is a blue-detuned optical lattice that is combined with Feshbach resonances and a red-detuned dipole trap to allow for independent control of tunneling, interactions and external confinement. The Fermi-Hubbard model, which plays a central role in the theoretical description of strongly correlated electrons, is experimentally realized by loading interacting fermionic spin mixtures into the optical lattice. Using phase-contrast imaging the in-situ size of the atomic density distribution is measured, which allows to extract the global compressibility of the many-body state as a function of interaction and external confinement. Thereby, metallic and insulating phases are clearly identified. At strongly repulsive interaction, a vanishing compressibility and suppression of doubly occupied lattice sites signal the emergence of a fermionic Mott insulator. In a second series of experiments interaction effects in bosonic lattice quantum gases are analyzed. Typically, interactions between microscopic particles are described as two-body interactions. As such they are also contained in the single-band Bose-Hubbard model. However, our measurements demonstrate the presence of multi-body interactions that effectively emerge via virtual transitions of atoms to higher lattice bands. These findings are enabled by the development of a novel atom optical measurement technique: In quantum phase revival spectroscopy periodic collapse and revival dynamics of the bosonic matter wave field are induced. The frequencies of the dynamics are directly related to the on-site interaction energies of atomic Fock states and can be read out with high precision. The third part of this work deals with mixtures of bosons and fermions in optical lattices, in which the interspecies interactions are accurately controlled by means of a Feshbach resonance. Studies of the equilibrium phases show that the bosonic superfluid to Mott insulator transition is shifted towards lower lattice depths when bosons and fermions interact attractively. This observation is further analyzed by applying quantum phase revival spectroscopy to few-body systems consisting of a single fermion and a coherent bosonic field on individual lattice sites. In addition to the direct measurement of Bose-Fermi interaction energies, Bose-Bose interactions are proven to be modified by the presence of a fermion. This renormalization of bosonic interaction energies can explain the shift of the Mott insulator transition. The experiments of this thesis lay important foundations for future studies of quantum magnetism with fermionic spin mixtures as well as for the realization of complex quantum phases with Bose-Fermi mixtures. They furthermore point towards physics that reaches beyond the single-band Hubbard model.

Quantum Einstein gravity - advancements of heat kernel-based renormalization group studies

The asymptotic safety scenario allows to define a consistent theory of quantized gravity within the framework of quantum field theory. The central conjecture of this scenario is the existence of a non-Gaussian fixed point of the theory's renormalization group flow, that allows to formulate renormalization conditions that render the theory fully predictive. Investigations of this possibility use an exact functional renormalization group equation as a primary non-perturbative tool. This equation implements Wilsonian renormalization group transformations, and is demonstrated to represent a reformulation of the functional integral approach to quantum field theory.rnAs its main result, this thesis develops an algebraic algorithm which allows to systematically construct the renormalization group flow of gauge theories as well as gravity in arbitrary expansion schemes. In particular, it uses off-diagonal heat kernel techniques to efficiently handle the non-minimal differential operators which appear due to gauge symmetries. The central virtue of the algorithm is that no additional simplifications need to be employed, opening the possibility for more systematic investigations of the emergence of non-perturbative phenomena. As a by-product several novel results on the heat kernel expansion of the Laplace operator acting on general gauge bundles are obtained.rnThe constructed algorithm is used to re-derive the renormalization group flow of gravity in the Einstein-Hilbert truncation, showing the manifest background independence of the results. The well-studied Einstein-Hilbert case is further advanced by taking the effect of a running ghost field renormalization on the gravitational coupling constants into account. A detailed numerical analysis reveals a further stabilization of the found non-Gaussian fixed point.rnFinally, the proposed algorithm is applied to the case of higher derivative gravity including all curvature squared interactions. This establishes an improvement of existing computations, taking the independent running of the Euler topological term into account. Known perturbative results are reproduced in this case from the renormalization group equation, identifying however a unique non-Gaussian fixed point.rn