On a derivation of the Boltzmann equation in Quantum Field Theory

The Boltzmann equation (BE) is a commonly used tool for the study of non-equilibrium many particle systems. It has been introduced in 1872 by Ludwig Boltzmann and has been widely generalized throughout the years. Today it is commonly used in physical applications, from the study of ordinary fluids to problems in particle Cosmology where Quantum Field Theoretical techniques are essential. Despite its numerous experimental successes, the conceptual basis of the BE is not entirely clear. For instance, it is well known that it is not a fundamental equation of physics like, say, the Heisenberg equation (HE). A natural question then arises whether it is possible to derive the BE from physical first principles, i.e. the Heisenberg equation in Quantum Field Theory. In this work we attempted to answer this question and succeeded in deriving the BE from the HE, thus further clarifying its conceptual status. In particular, the results we have obtained are as follows. Firstly, we establish the non-perturbative validity of what we call the "pre-Boltzmann equation". The crucial point here is that this latter equation is equivalent to the Heisenberg equation. Secondly, we proceed to consider various limits of the pre-Boltzmann equation, namly the "low density" and the "weak coupling" limits, to obtain two equations that can be considered as generalizations of the BE. These limits are always taken together with the "long time" limit, which allows us to interpret the BE as an appropriate long time limit of the HE. The generalization we obtain consists in additional "correction" terms to the usual Boltzmann collision factor, and can be associated to multiple particle scattering. Unlike the pre-Boltzmann equation, these latter results are only valid pertubatively. Finally, we briefly consider the possibility to extend these results beyond said limits and outline some important aspects in this case

On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity

The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion.Comment: 7 pages, 3 figure

Sentinel node biopsy for conjunctival melanoma

Purpose: To evaluate the role of sentinel lymph node (SLN) biopsy in patients with conjuctival melanoma (CjM). Study design: Retrospective observational cohort study and literature review. Subjects: Slovenian patients with CjM are included in the study. Methods: Prospectively collected data of CjM patients treated from June 2005 to December 2016 were retrospectively analyzed. Main outcome measures: The numbers of SLN biopsy procedures, positive and false positive SLN, and local and regional relapses have been described together with overall survival. Results: From June 2005 until December 2016, 24 patients with CjM were treated. The median follow-up time was 65.3 months. The mean Breslow thickness was 1.5 mm (sd = 1.8 mm), and ulceration was present in 29% of cases. Altogether, 14/24 (58%) SLN biopsy procedures were performed. SLN was positive in 2/14 (14%) cases. The estimated 5-year overall survival (OS) of the group was 72.5%, with a median survival of 151 months (95% CI 77–224). From January 2013 to January 2020, five (5/140, 3%) authors reported results comparable to our study. Conclusion: Our results confirm that CjM is a rare disease with approximately 14% of positive SLN. At the moment, there are no firm conclusions regarding who would benefit most from SLN biopsy or whether or not CLND should be offered. Data from literature emphasize the need for consistent and uniform staging and future multicentric studies