Publications
Growing collection of my publications, and selected conference papers, as well as my M.Sc. theses at the bottom of the page. You can browse abstracts or access and read the full papers via interactive buttons under each entry. Feel free to download any material, if you find it valuable, but please remember to cite each work (i.e. with BIB button). If you have any questions or wish to discuss further, I’d be happy to hear from you!
2024
- P.R.ETesting for Markovian Character of Transfer of Fluctuations in Solar Wind Turbulence on Kinetic ScalesDariusz Wójcik, and Wiesław M. MacekPhysical Review E., Aug 2024
We apply statistical analysis to search for processes responsible for turbulence in physical systems. In our previous studies, we have shown that solar wind turbulence in the inertial range of large magnetohydrodynamic scales exhibits Markov properties. We have recently extended this approach on much smaller kinetic scales. Here we are testing for the Markovian character of stochastic processes in a kinetic regime based on magnetic field and velocity fluctuations in the solar wind, measured onboard the Magnetospheric Multiscale (MMS) mission: behind the bow shock, inside the magnetosheath, and near the magnetopause. We have verified that the Chapman-Kolmogorov necessary conditions for Markov processes is satisfied for local transfer of energy between the magnetic and velocity fields also on kinetic scales. We have confirmed that for magnetic fluctuations, the first Kramers-Moyal coefficient is linear, while the second term is quadratic, corresponding to drift and diffusion processes in the resulting Fokker-Planck equation. It means that magnetic self-similar turbulence is described by generalized Ornstein-Uhlenbeck processes. We show that for the magnetic case, the Fokker-Planck equation leads to the probability density functions of the kappa distributions, which exhibit global universal scale invariance with a linear scaling and lack of intermittency. On the contrary, for velocity fluctuations, higher order Kramers-Moyal coefficients should be taken into account and hence scale invariance is not observed. However, the nonextensity parameter in Tsallis entropy provides a robust measure of the departure of the system from equilibrium. The obtained results are important for a better understanding of the physical mechanism governing turbulent systems in space and laboratory.
@article{PRE24, title = {Testing for Markovian Character of Transfer of Fluctuations in Solar Wind Turbulence on Kinetic Scales}, author = {W\'ojcik, Dariusz and Macek, Wies\l{}aw M.}, journal = {Physical Review E.}, volume = {110}, issue = {2}, pages = {025203}, dimensions = {true}, numpages = {16}, year = {2024}, month = aug, publisher = {American Physical Society}, doi = {10.1103/PhysRevE.110.025203}, url = {https://link.aps.org/doi/10.1103/PhysRevE.110.025203}, } - EGUProbing Small Scale Solar Wind Turbulence: Markovian Analysis and Scale Interactions from Inertial to Kinetic RegimesDariusz Wójcik, and Wiesław M. MacekEGU General Assembly, Apr 2024
Based on the data collected by the Magnetospheric Multiscale (MMS) mission’s satellites, we delve into the subject of turbulence on inertial, sub-ion, and kinetic scales. Building upon prior Markovian analysis of turbulence of the transfer of magnetic-to-magnetic field fluctuations in the near-Earth space environment [10.1093/mnras/stad2584, 10.3847/1538-4357/aca0a0], we also extend our investigation to ion velocity-to-velocity and magnetic-to-velocity cases. However, we direct our focus towards the purer statistical facet of the analysis, joint with the elements of dynamical approach. We analyze whether the transfer of increments exhibits ‘local‘ or ‘non-local‘ character, which in this context, they describe the scales involved in interactions that lead to the turbulent cascade. Additionally, we observe a global scale-invariance in relation to the Fokker-Planck equation, for a magnetic field case. Finally, we briefly discuss a potential non-parametric approach, namely a stochastic dynamical jump-diffusion model, or alternatively a multi-fractal approach, which can be useful to describe the underlying process accurately. We believe that such a comparative approach spanning diverse conditions is meaningful, as it aims to unveil any underlying universality within the statistical properties of the near-Earth solar wind space plasma at the intricate kinetic and sub-ion scales.
- S.R.Fractal Scaling Laws in the Universe Distribution of Galaxies (In Review)Wiesław M. Macek, and Dariusz WójcikScientific Reports, Nature, Oct 2024
We have recently argued that a simple nonlinear law could possibly be important for the origin of the Universe resulting in fractal or multifractal features. Various fractal scaling models of the large-scale mass distribution have already been proposed. The expected universal multifractal function for galaxies is similar to that identi ed by NASA’s Voyager mission in the Solar System. Hence we now apply the similar method for determination of the reliable multifractal spectrum of distribution of galaxies on cosmological scales, based on selected observations from a million of galaxies in the Redshift Catalog updated in June 2008. We show that the observed spectrum is consistent with the weighted one- or two-scale Cantor set models characteristic for turbulence in laboratory and inside the Sun’s heliosphere immersed in the very local interstellar medium. However, the total degree of multifractality Delta=0.2 is smaller than that inside the heliosphere. This would be characteristic for a simple linear fractal scaling of galaxy distribution, but somewhat varying for nearby (Delta=0.1) and the most remote galaxies (Delta=0.2) receding from our Solar System. The parameters p=0.45 and lambda<=1/2 for one-scale model are apparently related to some voids in the large-scale distribution of matter. A possible asymmetry (A 3/4) of the total spectrum for the two-scale weighted Cantor set (A != 1) could admittedly be attributed to some deviations from the Hubble’s law for the ideal uniform expansion of the Universe.
@article{SR24, author = {Macek, Wiesław M. and Wójcik, Dariusz}, year = {2024}, month = oct, journal = {Scientific Reports, Nature}, title = {Fractal Scaling Laws in the Universe Distribution of Galaxies (In Review)}, pages = {26}, }
2023
- AP.J.Magnetospheric Multiscale Observations of Markov Turbulence on Kinetic ScalesWiesław M. Macek, Dariusz Wójcik, and James L. BurchThe Astrophysical Journal, Feb 2023
In our previous studies we have examined solar wind and magnetospheric plasma turbulence, including Markovian character on large inertial magnetohydrodynamic scales. Here we present the results of the statistical analysis of magnetic field fluctuations in the Earth’s magnetosheath, based on the Magnetospheric Multiscale mission at much smaller kinetic scales. Following our results on spectral analysis with very large slopes of about −16/3, we apply a Markov-process approach to turbulence in this kinetic regime. It is shown that the Chapman–Kolmogorov equation is satisfied and that the lowest-order Kramers–Moyal coefficients describing drift and diffusion with a power-law dependence are consistent with a generalized Ornstein–Uhlenbeck process. The solutions of the Fokker–Planck equation agree with experimental probability density functions, which exhibit a universal global scale invariance through the kinetic domain. In particular, for moderate scales we have the kappa distribution described by various peaked shapes with heavy tails, which, with large values of the kappa parameter, are reduced to the Gaussian distribution for large inertial scales. This shows that the turbulence cascade can be described by the Markov processes also on very small scales. The obtained results on kinetic scales may be useful for a better understanding of the physical mechanisms governing turbulence.
@article{APJ23, doi = {10.3847/1538-4357/aca0a0}, url = {https://dx.doi.org/10.3847/1538-4357/aca0a0}, year = {2023}, month = feb, publisher = {The American Astronomical Society}, volume = {943}, number = {2}, dimensions = {true}, pages = {152}, author = {Macek, Wiesław M. and Wójcik, Dariusz and Burch, James L.}, title = {Magnetospheric Multiscale Observations of Markov Turbulence on Kinetic Scales}, journal = {The Astrophysical Journal} } - MNRASStatistical Analysis of Stochastic Magnetic Fluctuations in Space Plasma Based on the MMS MissionWiesław M. Macek, and Dariusz WójcikMonthly Notices of the Royal Astronomical Society, Sep 2023
Based on the Magnetospheric Multiscale (MMS) mission we look at magnetic field fluctuations in the Earth’s magnetosheath. We apply the statistical analysis using a Fokker–Planck equation to investigate processes responsible for stochastic fluctuations in space plasmas. As already known, turbulence in the inertial range of hydromagnetic scales exhibits Markovian features. We have extended the statistical approach to much smaller scales in space, where kinetic theory should be applied. Here we study in detail and compare the characteristics of magnetic fluctuations behind the bow shock, inside the magnetosheath, and near the magnetopause. It appears that the first Kramers–Moyal coefficient is linear and the second term is a quadratic function of magnetic increments, which describe drift and diffusion, correspondingly, in the entire magnetosheath. This should correspond to a generalization of Ornstein–Uhlenbeck process. We demonstrate that the second-order approximation of the Fokker–Planck equation leads to non-Gaussian kappa distributions of the probability density functions. In all cases in the magnetosheath, the approximate power-law distributions are recovered. For some moderate scales, we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this shape is reduced to the normal Gaussian distribution. It is worth noting that for smaller kinetic scales the rescaled distributions exhibit a universal global scale invariance, consistently with the stationary solution of the Fokker–Planck equation. These results, especially on kinetic scales, could be important for a better understanding of the physical mechanism governing turbulent systems in space and astrophysical plasmas.
@article{MNRAS23, author = {Macek, Wiesław M. and Wójcik, Dariusz}, title = {{Statistical Analysis of Stochastic Magnetic Fluctuations in Space Plasma Based on the MMS Mission}}, journal = {Monthly Notices of the Royal Astronomical Society}, volume = {526}, number = {4}, pages = {5779-5790}, dimensions = {true}, year = {2023}, month = sep, issn = {0035-8711}, doi = {10.1093/mnras/stad2584}, url = {https://doi.org/10.1093/mnras/stad2584}, eprint = {https://academic.oup.com/mnras/article-pdf/526/4/5779/52600435/stad2584.pdf} } - EGUComparative MMS Analysis of Markov Turbulence in the Magnetosheath on Kinetic ScalesWiesław M. Macek, and Dariusz WójcikEGU General Assembly, Apr 2023
We apply Fokker-Planck equation to investigate processes responsible for turbulence in space plasma. In our previous studies, we have shown that turbulence in the inertial range of hydromagnetic scales exhibits Markov properties. We have also extended this statistical approach on much smaller scales, where kinetic theory should be applied. Namely, we have already obtained the results of the statistical analysis of magnetic field fluctuations in the Earth’s magnetosheath based on the Magnetospheric Multiscale (MMS) mission. Here we compare the characteristics of turbulence behind the bow shock, inside the magnetosheath, and near the magnetopause. We check whether the second order approximation of the Fokker-Planck equation leads to kappa distribution of the probability density function provided that the first Kramers-Moyal coefficient is linear and the second term is quadratic, describing drift and diffusion correspondingly, which is a generalization of Ornstein-Uhlenbeck process. In some cases the power-law distributions are recovered. For moderate scales we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this is reduced to the normal Maxellian distribution. The obtained results on kinetic scales could be important for a better understanding of the physical mechanism governing turbulent systems in laboratory and space.
- AGUMarkov Analysis of Magnetic Turbulence on Kinetic Scales Based on MMS DataWiesław M. Macek, and Dariusz WójcikAGU Chapman Conference, May 2023
Based on the THEMIS mission in the Earth’s magnetosheath, we have verified that turbulence at shocks is well described by inward- and outward-propagating Alfvén waves. Recently, we have also presented results of statistical analysis of magnetic field fluctuations in the magnetosheath using the data from the Magnetospheric Multiscale (MMS) mission on extremely small kinetic scales. We have shown that magnetic turbulence exhibits features characteristic for Markov processes. It is interesting to note that on kinetic scales the turbulence cascade is consistent with a generalized Ornstein-Uhlenbeck process. The solutions of the Fokker-Planck equation agree with experimental probability density functions, from the kappa distributions to the normal Gaussian distribution for large inertial scales, which exhibit a universal global scale invariance of the collapsing PDFs through the kinetic domain. Here we compare the characteristics of turbulence at various regions in Earth’s space environment: behind the bow shock, inside the magnetosheath, and near the magnetopause. The obtained results especially on kinetic scales may be important for a better understanding of the physical mechanism governing turbulent systems in laboratory and space plasmas.
2021
- M.Sc.Cybersecurity Insurance Modeling (Master’s thesis, in Polish)Dariusz WójcikWarsaw University of Technology, Faculty of Mathematics and Information Science, Dec 2021
Cyber security refers to procedures and techniques design to protect computer systems (e.g. computer networks, devices and programs) and primarily confidential personal data from threats of potential attacks (e.g., actions that compromise the confidentiality, integrity, availability, and authenticity of processed data). Large enterprises and corporations, as well as government institutions, are increasingly exposed to cyber threats. Hence, we can observe not only an increasing number of legal regulations concerning cyber security, among others in the context of data protection, but also the rapid development of the cybersecurity insurance sector. However, due to the rather unusual risk characteristics in this case, the modeling and analysis in this domain is not only an interesting but still developing research topic. The purpose of my thesis was to review and systematize the "state of the art" in cybersecurity insurance business, with particular emphasis on modeling techniques proposed in the literature.
- M.Sc.Stability Analysis in the Generalized Lorenz System (Master’s thesis, in Polish)Dariusz WójcikCardinal Stefan Wyszyński University in Warsaw, Dec 2021
I start with the derivation of the well-known Edward Norton Lorenz system, which is a system of three nonlinear ordinary differential equations, showing sensitivity to initial conditions, which makes it impossible to predict the behavior of this system in the long term. Then I derived the characteristic polynomials in order to be able to study its qualitative behavior in order to investigate the stability of a given system. Rayleigh - Bénard convection can be observed under certain specific conditions. Further, I introduced the convection equations to proceed to the derivation of the generalized Lorenz model, which was proposed by Prof. Wiesław Macek and Dr. Marek Strumik in 2010. The authors obtained a system of 4 nonlinear ordinary differential equations based on the Rayleigh - Bénard scenario using the general magnetohydrodynamic approach. The above mentioned system depends on 5 parameters. Variable X is proportional to the intensity of convection movements as in the standard Lorenz model, while Y and Z describe the temperature profile. Additionally, in the generalized Lorenz model, the authors introduced a new time - dependent variable W, which describes the profile of the magnetic field caused by the convection of a magnetized fluid. Then, I analyzed the model by examining the stability of the critical points and examining the occurrence of Hopf bifurcation depending on the control parameters. Various types of Hopf bifurcation can be expected. In addition, I briefly described the long - term behavior of the circuit. This analysis suggests that the linear stability range of the generalized Lorenz system must be much wider than for the standard Lorenz model.