Thermodynamics of inhomogenous imperfect quantum gases in harmonic traps

We discuss thermodynamic properties of harmonically trapped imperfect quantum gases. The spatial inhomogeneity of these systems imposes a redefinition of the mean-field interparticle potential energy as compared to the homogeneous case. In our approach, it takes the form $\frac{a}{2} N^2 \, \omega^d$, where $N$ is the number of particles, $\omega$ - the harmonic trap frequency, $d$ - system's dimensionality, and $a$ is a parameter characterizing the interparticle interaction. We provide arguments that this model corresponds to the limiting case of a long-ranged interparticle potential of vanishingly small amplitude. This conclusion is drawn from a computation similar to the well-known Kac scaling procedure, which is presented here in a form adapted to the case of an isotropic harmonic trap. We show that within our model, the imperfect gas of trapped repulsive bosons undergoes the Bose-Einstein condensation provided $d>1$. The main result of our analysis is that in $d=1$ the gas of attractive imperfect fermions with $a=-a_{F}<0$ is thermodynamically equivalent to the gas of repulsive bosons with $a=a_{B}>0$ provided the parameters $a_{F}$ and $a_{B}$ fulfill the relation $a_{B}+a_{F}=\hbar$. This result supplements similar recent conclusion about thermodynamic equivalence of two-dimensional uniform imperfect repulsive Bose and attractive Fermi gases.Comment: Revised version, comments adde

IST Austria Thesis

The polaron model is a basic model of quantum field theory describing a single particle interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned with models applicable in the context of an impurity atom in a Bose-Einstein condensate as well as the problem of electrons moving in polar crystals. The model has a simple structure in which the interaction of the particle with the field is given by a term linear in the field’s creation and annihilation operators. In this work, we investigate the properties of this model by providing rigorous estimates on various energies relevant to the problem. The estimates are obtained, for the most part, by suitable operator techniques which constitute the principal mathematical substance of the thesis. The first application of these techniques is to derive the polaron model rigorously from first principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak interactions as a low-energy effective theory for this problem. In the second part, we investigate rigorously the ground state of the model at fixed momentum and for large values of the coupling constant. Qualitatively, the system is expected to display a transition from the quasi-particle behavior at small momenta, where the dispersion relation is parabolic and the particle moves through the medium dragging along a cloud of phonons, to the radiative behavior at larger momenta where the polaron decelerates and emits free phonons. At the same time, in the strong coupling regime, the bosonic field is expected to behave purely classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to be asymptotically equal to the one obtained from the semiclassical counterpart of the problem, first studied by Landau and Pekar in the 1940s. For polaron models with regularized form factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove that for a large window of momenta below the radiation threshold, the energy-momentum relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the Landau–Pekar effective mass, as expected. For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is of the optical type and the form factor is formally UV–singular due to the nature of the point charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to the regular case, this requires the inclusion of the quantum fluctuations of the phonon field, which makes the problem considerably more difficult. The results are supplemented by studies on the absolute ground-state energy at strong coupling, a proof of the divergence of the effective mass with the coupling constant for a wide class of polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model and the application of the techniques used for its removal for the energy estimates

Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit

We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model

The three-dimensional structure of caenopore-5 from Caenorhabditis elegans determined by means of NMR spectroscopy

The caenopore-5 protein encoded by the spp-5 gene is one out of 23 caenopores identified in C.elegans. It belongs to the SAPLIP family whose members perform a variety of functions based on their ability to interact with lipids. It has been shown that caenopore-5 is responsible for the survival and growth of C.elegans on E.coli lawns. The main aim of this project was the determination of the three-dimensional structure of caenopore-5 by means of NMR spectroscopy. The sequential assignment revealed two conformers of caenopore-5. Separate structure calculations for the two sets of distance constraints, which correspond to the two different conformers of caenopore-5 revealed that they only differ in the isomerisation of Pro 99.The structure of caenopore-5 displays the saposin-like fold characteristic of the SAPLIP family. It consists of five amphiphatic alpha helices connected by short loops or a kink. They are all arranged in the tertiary structure which resembles a folded leaf stabilised by three disulphide bridges made of six conserved cysteines. Interestingly, the comparison of the caenopore-5 structure with structures of other members of the SAPLIP family revealed that two non-antimicrobial peptides saposin A and C are the most similar. Furthermore, caenopore-5 shows a remarkable preference for the interaction with negatively charged phosphatidylglycerol liposomes, which served as a model for bacterial membranes. This interaction induced a conformational change in caenopore-5. The knowledge of the tertiary structure now provides the basis for further studies of the Caenopore-5 function

Protist-Type Lysozymes of the Nematode Caenorhabditis elegans Contribute to Resistance against Pathogenic Bacillus thuringiensis

Pathogens represent a universal threat to other living organisms. Most organisms express antimicrobial proteins and peptides, such as lysozymes, as a protection against these challenges. The nematode Caenorhabditis elegans harbours 15 phylogenetically diverse lysozyme genes, belonging to two distinct types, the protist- or Entamoeba-type (lys genes) and the invertebrate-type (ilys genes) lysozymes. In the present study we characterized the role of several protist-type lysozyme genes in defence against a nematocidal strain of the Gram-positive bacterium Bacillus thuringiensis. Based on microarray and subsequent qRT-PCR gene expression analysis, we identified protist-type lysozyme genes as one of the differentially transcribed gene classes after infection. A functional genetic analysis was performed for three of these genes, each belonging to a distinct evolutionary lineage within the protist-type lysozymes (lys-2, lys-5, and lys-7). Their knock-out led to decreased pathogen resistance in all three cases, while an increase in resistance was observed when two out of three tested genes were overexpressed in transgenic lines (lys-5, lys-7, but not lys-2). We conclude that the lysozyme genes lys-5, lys-7, and possibly lys-2 contribute to resistance against B. thuringiensis, thus highlighting the particular role of lysozymes in the nematode's defence against pathogens

Human matrix metalloproteinases: An ubiquitarian class of enzymes involved in several pathological processes

Human matrix metalloproteinases (MMPs) belong to the M10 family of the MA clan of endopeptidases. They are ubiquitarian enzymes, structurally characterized by an active site where a Zn(2+) atom, coordinated by three histidines, plays the catalytic role, assisted by a glutamic acid as a general base. Various MMPs display different domain composition, which is very important for macromolecular substrates recognition. Substrate specificity is very different among MMPs, being often associated to their cellular compartmentalization and/or cellular type where they are expressed. An extensive review of the different MMPs structural and functional features is integrated with their pathological role in several types of diseases, spanning from cancer to cardiovascular diseases and to neurodegeneration. It emerges a very complex and crucial role played by these enzymes in many physiological and pathological processes

Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron

We consider the large polaron described by the Fröhlich Hamiltonian and study its energy-momentum relation defined as the lowest possible energy as a function of the total momentum. Using a suitable family of trial states, we derive an optimal parabolic upper bound for the energy-momentum relation in the limit of strong coupling. The upper bound consists of a momentum independent term that agrees with the predicted two-term expansion for the ground state energy of the strongly coupled polaron at rest and a term that is quadratic in the momentum with coefficient given by the inverse of twice the classical effective mass introduced by Landau and Pekar

On the global minimum of the energy–momentum relation for the polaron

For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling