Publications

2025

1. J.G.R

   Searching for Universality of Turbulence in the Earth’s Magnetosphere

   Dariusz Wójcik and Wiesław M. Macek

   Journal of Geophysical Research: Space Physics, 2025

   e2025JA034020 2025JA034020

   Abs DOI Bib HTML PDF

   Abstract Turbulence in space plasmas remains a fundamental challenge, and Earth’s magnetosphere (MSP) offers a natural laboratory for its study. Using high-resolution magnetic field data from the Magnetospheric Multiscale (MMS) mission, we extend a stochastic Markovian framework to analyze turbulence across 10 diverse magnetospheric regions, including key reconnection sites. We show that magnetic field fluctuations are well described as Markov processes in scale across the kinetic domain. Multi-scale conditional Probability Density Functions (PDFs) reveal Markovian properties beyond the Einstein-Markov scale. The Kramers-Moyal coefficients display linear and quadratic dependencies, and the Fokker-Planck equation accurately reproduces empirical conditional PDFs, with stationary solutions matching observed Kappa distributions. Scale invariance, characterized by power-law behavior and self-similar PDFs, holds up to {∼ 0.4 s in most regions but breaks in day-side reconnection jets. These findings support the universality of the Markovian cascade description across the magnetosphere, while identifying region-specific deviations that reflect differing energy dissipation mechanisms.

   @article{JGR25,
     author = {Wójcik, Dariusz and Macek, Wiesław M.},
     title = {Searching for Universality of Turbulence in the Earth's Magnetosphere},
     journal = {Journal of Geophysical Research: Space Physics},
     volume = {130},
     number = {10},
     pages = {e2025JA034020},
     keywords = {turbulence, magnetic fields, solar wind plasma, statistical methods, solar wind/magnetosphere interactions},
     doi = {10.1029/2025JA034020},
     url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2025JA034020},
     eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2025JA034020},
     dimensions = {true},
     note = {e2025JA034020 2025JA034020},
     year = {2025}
   }

2. SR.G

   Fractal Nature of Galaxy Clustering in the Updated CfA Redshift Catalog (In Review)

   Wiesław M. Macek and Dariusz Wójcik

   Scientific Reports: Galaxies, 2025

   DOI PDF
3. Ph.D.

   Analysis of Markov Processes in Space Environment

   D. Wójcik

   Space Research Centre, Polish Academy of Sciences, Sep 2025

   PDF
4. EGU25

   Testing For Universality of Markov Solar Wind Turbulence at the Earth’s Magnetosphere on Kinetic Scales Based on the MMS Mission

   Dariusz Wójcik and Wiesław M. Macek

   EGU General Assembly, Apr 2025

   Abs DOI HTML

   Understanding turbulence in space and astrophysical plasmas is critical for advancing our comprehension of complex systems governed by nonlinear dynamics. This study extends the application of the Markovian framework in small-scale turbulence in the Earth’s magnetosphere, with a particular focus on the solar wind – magnetosphere interaction observed by NASA Magnetospheric Multiscale (MMS) mission. We benefit from the exceptional resolution of the Fluxgate Magnetometer instrument as well as the Fast Plasma Investigation instrument onboard the MMS. The high temporal resolution, coupled with recent machine learning methods, allows one to identify the turbulent regions and magnetic reconnection events with great accuracy. Hence, the data are analyzed across diverse magnetospheric regions, enabling insights into turbulence-driven energy transfers. With the obtained measurements we could analyze the magnetic field gradients, turbulence intensity, and the plasma parameters. By employing the multi-scale probabilistic approach, we explore the turbulent cascade using conditional Probability Density Functions (cPDFs) and the Markovian properties of fluctuations, revealing new insights into the dynamics of energy transfer at sub-ion scales. Our results confirm the Markovian necessary and sufficient properties of the turbulent cascade across kinetic scales, emphasizing the significance of the Einstein-Markov (EM) scale and the intermittent nature of energy transfer to smaller scales. The derived Fokker-Planck equation in scale governs the evolution of cPDFs through drift and diffusion coefficients, which have been directly calculated from the empirical data. This employed framework captures key features of turbulence, including its hierarchical structure, deviations from self-similarity, and the phenomenon of intermittency, evidenced by non-Gaussian statistics and broadened PDF tails. These findings provide a robust description of the cascade process, from large-scale energy input to dissipation at smaller scales. By investigating turbulence in two electron diffusion regions, where magnetic reconnection may occur, the highlighted Markovian framework and Fokker-Planck methodology are interestingly still viable to describe the complexity of turbulence processes. This gives promising insight into understanding the stochastic nature of reconnection-driven turbulence. Despite some limitations, including the simplifying assumptions inherent to the Markovian framework and second-order Fokker-Planck equations, our results demonstrate its effectiveness in capturing the essence of kinetic-scale turbulence. The connection between scale-dependent statistics and underlying physical processes, such as intermittency and energy cascades, highlights the framework’s utility for both theoretical and observational studies. This work bridges statistical physics and plasma turbulence for analyzing scale-dependent phenomena in magnetospheric plasmas. We hope that by elucidating the interplay of order and randomness in these systems, our findings support the idea of extending stochastic modeling to higher-dimensional problems.

2024

1. P.R.E

   Testing for Markovian Character of Transfer of Fluctuations in Solar Wind Turbulence on Kinetic Scales

   Dariusz Wójcik and Wiesław M. Macek

   Physical Review E., Aug 2024

   Abs DOI Bib HTML PDF

   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 = {Wójcik, Dariusz and Macek, Wiesław 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},
   }

2. WhiteP

   Next Generation Solar Wind Facility for Discovery Plasma Science

   Seth Dorfman, Emily Lichko, Joseph Olson, James Juno, Evdokiya Kostadinova, Mel Abler, Saikat Chakraborty Thakur, Peter Heuer, and Dariusz) Wójcik

   Feb 2024

   White paper submitted to the Fusion Energy Sciences Advisory Committee (FESAC)
3. EGU24

   Probing Small Scale Solar Wind Turbulence: Markovian Analysis and Scale Interactions from Inertial to Kinetic Regimes

   Dariusz Wójcik and Wiesław M. Macek

   EGU General Assembly, Apr 2024

   Abs DOI HTML

   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.

2023

1. AP.J.

   Magnetospheric Multiscale Observations of Markov Turbulence on Kinetic Scales

   Wiesław M. Macek, Dariusz Wójcik, and James L. Burch

   The Astrophysical Journal, Feb 2023

   Abs DOI Bib HTML PDF

   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}
   }

2. MNRAS

   Statistical Analysis of Stochastic Magnetic Fluctuations in Space Plasma Based on the MMS Mission

   Wiesław M. Macek and Dariusz Wójcik

   Monthly Notices of the Royal Astronomical Society, Sep 2023

   Abs DOI Bib HTML PDF

   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}
   }

3. EGU23

   Comparative MMS Analysis of Markov Turbulence in the Magnetosheath on Kinetic Scales

   Wiesław M. Macek and Dariusz Wójcik

   EGU General Assembly, Apr 2023

   Abs DOI HTML

   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.

4. AGU23

   Markov Analysis of Magnetic Turbulence on Kinetic Scales Based on MMS Data

   Wiesław M. Macek and Dariusz Wójcik

   AGU Chapman Conference, May 2023

   Abs HTML

   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

1. M.Sc.

   Cybersecurity Insurance Modeling (Master’s thesis, in Polish)

   Dariusz Wójcik

   Warsaw University of Technology, Faculty of Mathematics and Information Science, Dec 2021

   Abs HTML PDF

   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.

2. M.Sc.

   Stability Analysis in the Generalized Lorenz System (Master’s thesis, in Polish)

   Dariusz Wójcik

   Cardinal Stefan Wyszyński University in Warsaw, Dec 2021

   Abs HTML PDF

   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.