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The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
Germany experienced a 6-month second lockdown (November 2020–April 2021) during the COVID-19 pandemic, which included the closure of all physical activity (PA) facilities. The use of online exercise classes (OECs) was promoted by public health and exercise organizations. Using the present cross-sectional online survey, we assess the use of and opinion towards OECs in Germany during the second lockdown. We used contingency tables and the Chi2 test to calculate the frequency of awareness and use of OECs according to PA status, well-being and demographic data, and conducted a binary logistic regression with OEC awareness or use and dichotomized independent predictors. The associations between opinion and activity status, frequency of use, educational attainment, age and body mass index were calculated using Spearman correlations. A total of 993 datasets were analyzed in detail. Of the 785 (79.1%) participants reporting awareness of OECs, 536 tried them, and 262, 188 and 85 used them <1 per week, 1–2 per week and ≥3 per week, respectively. The users were typically active, female participants with poorer mental well-being. The opinions towards OECs varied according to participant characteristics, such as activity status, BMI and age. Overall, regular OEC use was quite limited, and, as such, cannot replace in-person exercise opportunities. Keeping physical activity facilities open and safe must be prioritized in the ongoing pandemic.
The aim of this thesis is finding a geometric configuration that allows electron insertion into a Gabor plasma lens in order to increase the density of the confined electrons and provide ignition conditions at parameters where ignition is not possible. First, simulations using CST and bender were conducted to investigate several geometric configurations in terms of their performance of inserting electrons manually. One particular design has been chosen as a basis for an experiment. In order to prepare the experiment, further simulations using the code bender have been conducted to investigate the density distribution that is formed inside the Gabor lens when inserting electrons transversally in compliance with the chosen design. Additionally, bender was used to investigate the impact of the initial electron energy on the distribution inside the lens. Simulations with and without space charge effects have shown a significant impact of the space charge effects on the resulting density dstribution. Therefore, space charge effects have proven to be the major electron redistribution process. A given electron source was characterised in order to find the performance under the conditions inside a Gabor lens. In particular, a transversal magnetic field that will be present in the experiment has to be compensated by shielding the inner regions of the source by a μ-metal layer. Using a μ-metal shield, transversal magnetic fields are sufficiently tolerable to perform measurements in a Gabor lens. Additionally, operating close to 100 eV electron energy yields a maximum in the emitted current. Adding a Wehnelt cylinder to the electron source furthermore improves the extracted current to roughly 1 mA. A test stand consisting of a newly designed anode for the Gabor lens, as well as a terminal for the electron source, was constructed. The electron source was thoroughly characterised in the environment of the Gabor lens and the ignition properties of the new system were evaluated. In further experiments, electron beam assisted ignition by increasing the residual gas pressure was observed and the impact of the position of the electron source on the ignition properties was investigated. In addition, ignition of a sub-critical state, that is a state consisting of potential, magnetic field and pressure that did not yet perform ignition by itself, was performed by increasing the extracted current from the electron source. Finally, the electron source was used to influence a pre-ignited plasma. The density was measured, which was increased by the use of the electron source in most cases. This project is part of the EDEN collaboration (Electron DENsity boosting) of the NNP Group at IAP Frankfurt with INFN institutes in Bologna and Catania.
Lockdown measures during the COVID-19 pandemic have led to reductions in physical activity (PA) worldwide. Leading public health organizations have recommended the use of online exercise classes (OEC) to compensate the loss of regular exercise classes. As of now, no data are available on the uptake of OEC and on users’ attitudes. The aim of the current online survey was to assess the use of and attitudes towards OEC in Germany. Respondents indicated awareness and use of OEC, and levels of agreement with statements on OEC. Frequency of awareness and use of OEC according to PA status were calculated with contingency tables and the Χ2 test. Differences between users and non-users were tested with the Student’s t-test and the Mann–Whitney U test. Data on attitudes are presented as percentages, and Spearman correlations were calculated between attitudes and activity status, frequency of use, educational attainment, age and body mass index. A total of 979 datasets were analyzed. Of the respondents, 681 were aware of and 180, 118 and 84 used them <1 per week, 1–2 per week and ≥3 per week, respectively. Significantly more active respondents were aware of and used OEC compared to less active respondents. All in all, regular OEC use was quite limited. OEC was differentially attractive to people according to PA status, frequency of use, BMI and age. Tailoring OEC to current non-users and adding motivational support might enhance the regular use of OEC.
Lockdown measures including the closure of physical activity facilities were installed against the spread of the novel coronavirus in March 2020. The aim of the current online survey was to assess the lockdown effects on physical activity in German adults. We assessed physical activity using the European Health Interview Survey (EHIS) questionnaire. Pre-lockdown vs. lockdown differences were tested with the X2 test and the Student’s t-test for paired data. Predictor variables to explain compliance with physical activity recommendations were identified using a fixed effects binary logistic regression analysis. Data of 979 respondents were analyzed. Transport related and leisure time physical activity decreased (p < 0.001, d = 0.16; p < 0.001, d = 0.22, respectively). Compliance with physical activity recommendations decreased from 38.1% to 30.4% (chi2 [1, 1958] = 12.754, p < 0.001, V = 0.08). In the regression analysis, BMI (OR 0.944, 95% CI 0.909–0.981; p = 0.003), education (OR 1.111, 95% CI 1.021–1.208; p = 0.015), transport related (OR 1.000, 95% CI 1.000–1.000; p = 0.008) and leisure time physical activity (OR 1.004, 95% CI 1.003–1.004; p < 0.001), muscle strengthening (OR 5.206, 95% CI 4.433–6.114; p < 0.001), as well as the ‘lockdown vs. normal’ categorical variable (OR 0.583, 95% CI 0.424–0.802; p = 0.001) showed a contribution, while sex (p = 0.152), age (p = 0.266), work related physical activity (p = 0.133), and remote working (p = 0.684) did not. Physical activity declined in German adults, and should also be promoted in light of the emerging evidence on its protective effects of against COVID-19. Special attention should be given to muscle strengthening activities and groups with lower educational attainment.
Physical activity and well-being during the second COVID19-related lockdown in Germany in 2021
(2021)
In the second wave of the COVID-19 pandemic in Germany, lockdown measures were reinstalled and were in place between November 2020 and April 2021, including the closure of physical activity facilities. The aim of the current online survey was to assess the lockdown effects on physical activity and well-being in the general population. Pre-lockdown vs. lockdown differences were tested with the Χ2 test and the Student’s t-test for paired data. Predictor variables to explain compliance with physical activity recommendations were identified using a fixed-effects binary logistic regression analysis. Data of 993 respondents were analyzed. Transport-related and leisure-time physical activity decreased (p < 0.001, d = 0.25, and p < 0.001, d = 0.33, respectively). Compliance with physical activity recommendations decreased from 42.2% to 29.4% (chi2 (1, 1986) = 35.335, p < 0.001, V = 0.13). Well-being decreased significantly (t (990) = 23.405, p < 0.001) by 16.3 points (d = 0.74). Physical activity and well-being declined in German adults during the second COVID-19-related lockdown. Physical activity should be promoted also in light of the emerging evidence on its protective effects against COVID-19.
The spread of the COVID-19 virus was met by a strict lockdown in many countries around the world, with the closure of all physical activity (PA) facilities and limitations on moving around freely. The aim of the present online survey was to assess the effect of lockdown on physical activity in Italy. Physical activity was assessed using the European Health Interview Survey questionnaire. A total of 1500 datasets were analyzed. Differences between conditions were tested with a chi2-based (χ2) test for categorical variables, and with the Student’s t-test for paired data. A fixed effects binary logistic regression analysis was conducted to identify relevant predictor variables to explain the compliance with World Health Organisation (WHO) recommendations. We found a substantial decline in all physical activity measures. Mean differences in walking and cycling metabolic equivalent of task minutes per week (METmin/week), respectively, were 344.4 (95% confidence interval (95% CI): 306.6–382.2; p < 0.001) and 148.5 (95% CI: 123.6–173.5; p < 0.001). Time spent in leisure time decreased from 160.8 to 112.6 min/week (mean difference 48.2; 95% CI: 40.4–56.0; p < 0.001). Compliance with WHO recommendations decreased from 34.9% to 24.6% (chi2 (1, 3000) = 38.306, p < 0.001, V = 0.11). Logistic regression showed a reduced chance (OR 0.640, 95% CI: 0.484–0.845; p = 0.001) to comply with WHO PA recommendations under lockdown conditions. Measures to promote physical activity should be intensified to limit detrimental health effects.
A selection of recent data referring to Pb+Pb collisions at the SPS CERN energy of 158 GeV per nucleon is presented which might describe the state of highly excited strongly interacting matter both above and below the deconfinement to hadronization (phase) transition predicted by lattice QCD. A tentative picture emerges in which a partonic state is indeed formed in central Pb+Pb collisions which hadronizes at about T = 185 MeV, and expands its volume more than tenfold, cooling to about 120 MeV before hadronic collisions cease. We suggest further that all SPS collisions, from central S+S onward, reach that partonic phase, the maximum energy density increasing with more massive collision systems.