ICRANet Newsletter

ICRANet Newsletter
June/July 2021

English - French - Italian - Russian

SUMMARY
1. The Sixteenth Marcel Grossmann virtual Meeting (MG16), July 5 - 10, 2021
2. The 17^th Italian-Korean Symposium (IK17), August 2-6, 2021
3. Annular solar eclipse, June 10, 2021
4. Renewal of the cooperation agreement ICRANet-UDEA (Colombia), June 9, 2021
5. New cooperation Protocol ICRANet - Damghan University (Iran), June 23, 2021
6. Scientific visits to ICRANet
7. Recent publications

1. The Sixteenth Marcel Grossmann virtual Meeting (MG16), July 5 - 10, 2021

The Sixteenth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG16) took place online from July 5 to 10, 2021. The meeting exceeded every expectation and confirmed once again its world leading role in the field of Relativistic Astrophysics, developed in the years since 1985 by ICRA at Sapienza University, and, in the most recent years, thanks to the collaboration with ICRANet center in Pescara.
More than 1200 participants from 51 different countries in the world, joined the conference and presented the most relevant recent results on the understanding of the Universe, achieved thanks to Albert Einstein's equations of general relativity. Thanks to the virtual format, a lot of scientists from developing countries had also the possibility to attend the conference. The rich program of the conference was articulated in 46 plenary presentations, 3 public lectures, 5 round tables and 81 parallel session, each one with about 9 speakers.
The meeting started on Monday morning, July 5 with the opening remarks by Prof. Remo Ruffini (Director of ICRANet), immediately followed by the MG16 Awards ceremony, presented by Prof. Roy P. Kerr. This year, the MG16 Individual Awards went to:

• Prof. Demetrios Christodoulou (ETH Zurich) “For his many lasting contributions to the foundation of mathematical physics including the dynamics of relativistic gravitational fields. Notably for: contributing in 1971, at the age of 19, to derive with Remo Ruffini the mass-energy formula of black holes as a function of their angular momentum, charge and irreducible mass. Christodoulou turned then to the study of partial differential equations and mathematical physics, to which he remained dedicated for the rest of his career. Highlights in this area include the theoretical discovery of the nonlinear memory effect of gravitational waves (Phys. Rev. Letters 1991), the monograph (1993) in collaboration with Sergiu Klainerman on the global nonlinear stability of the Minkowski spacetime, the monograph (2009) on the formation of black holes in pure general relativity by imploding gravitational waves, and the monographs (2007 and 2019) on the formation and further development of shocks in fluids";

• Prof. Gerard ‘t Hooft (Utrecht University) “for his persistent devotion to the study of the quantum field theory boundary conditions at the black hole horizon";

• Prof. Tsvi Piran (Hebrew University of Jerusalem) “for extending Relativistic Astrophysics across international frontiers, a true companion in the search for the deeper meaning of Einstein's great theory";

• Prof. Steven Weinberg (University of Texas at Austin) “for unwavering support for the MG meetings since their inception, a true companion in the search for the deeper meaning of Einstein's great theory".


The MG16 Institution Awards to the SRG mission “for the creation of the world's best X-ray map of the entire sky, for the discovery of millions of previously unknown accreting supermassive black holes at cosmological redshifts, for the detection of X-rays from tens of thousands of galaxy clusters, filled mainly with dark matter, and for permitting the detailed investigation of the growth of the large-scale structure of the universe during the era of dark energy dominance" went to:

• Prof. Alexander Shirshakov (on behalf of S.A. Lavochkin Association);

• Prof. Peter Predehl (on behalf of Max Planck Institute for Extraterrestrial Physics - MPE);

• Prof. Rashid Sunyaev (on behalf of Space Research Institute IKI of the Russian Academy of Sciences).


The MG16 Awards booklet is available here: http://www.icra.it/mg/mg16/mg16_awards.pdf
During this six day conference, a variety of topics were discussed in the plenary sessions, beginning with events in relativistic astrophysics on Monday, Black holes and the Quantum on Tuesday, Lambda CDM tensions on Wednesday, Black holes in GRBs and Precision tests on Thursday, Massive stars and Physics behind stellar collapse on Friday and Current and future missions on Saturday.
Up to 30 parallel sessions per days took place in every parallel sessions block, covering the following topics: Accretion, Active Galactic Nuclei, Alternative Theories, Black Holes, Binaries, Boson Stars, Cosmic Microwave background, Cosmic Strings, Dark Energy and Large Scale Structure, Dark Matter, Education, Exact solutions, Early Universe, Fundamental interactions and Stellar evolution, Fast transient, Gravitational Waves, High Energy, History of Relativity, Neutron Stars, Precision Tests, Quantum Gravity, Strong Field and White Dwarfs. All the abstracts submitted for the parallel sessions have been collected in the book of abstracts, available on the Indico platform for MG16 at the following link: https://indico.icranet.org/event/1/book-of-abstracts.pdf


Three public lectures have been presented by Razmik Mirzoyan, Asghar Qadir and Mohammad Bagheri (Ulugh Beg lectures) and Francis Halzen. Five roundtables have been organized by Andrea Merloni on the new results from SRG/eRosita, by Wick Haxton and Gianpaolo Bellini on Solar neutrinos and Borexino, by Marc Kamionkowski, Piero Rosati and Licia Verde on Precision cosmology, by Eleonora Troja, Jorge Rueda, Liang Li and Rahim Moradi on GRB 170817A and by Reinhard Genzel (A 40-Year Journey), Carlos Arguelles (A dark matter nature of SgrA*?), Eduar Becerra, Andreas Krut and Jorge Rueda on what is in our Galactic center.
The electronic proceedings of the 16th Marcel Grossmann meeting held on July 5-10, 2021 in virtual format, will be published by World Scientific. As done previously, the contributions from plenary speakers will be published in IJMPD. The proceedings submission is now started. Please consult Instructions for authors: http://www.icra.it/mg/mg16/doc/instructions_for_authors.pdf. The page limit is 20 pages and the deadline for submission has been scheduled on September 30, 2021.
The recordings of the different sessions (plenary and parallel session, official award ceremony, public lectures and round tables) are now available on the ICRANet YouTube channel at the following link: http://www.icranet.org/video_mg16
3 satellite meetings to MG16, celebrating the 50th anniversary of “Introducing the Black Hole" are going to be organized by ICRANet:
• the 17th Italian-Korean Symposium on Relativistic Astrophysics (IK17), August 2 - 6, 2021, Kunsan National University (Korea) and online (https://www.apctp.org/plan.php/kis2021);
• the ICRANet - Isfahan Astronomy Meeting “From the ancient Persian Astronomy to recent developments in theoretical and experimental general relativity and Astrophysics", November 3 - 5, 2021, Isfahan University of Technology (IUT - Iran) and online;
• the 3rd Julio Garavito Armero Meeting, December 1 - 3, 2021, Colombia.
For more information on MG16 meeting, please consult MG16 official website (http://www.icra.it/mg/mg16/) and MG16 meeting on Indico (https://indico.icranet.org/event/1/).
The meeting has been also advertised:
• on ICRANet website: http://www.icranet.org/communication/
• on Hyperspace: https://hyperspace.uni-frankfurt.de/?s=MG16&submit=Search
• on INSPIRE: https://inspirehep.net/conferences/1861415
• on 3DNews Daily Digital Digest (in Russian): https://3dnews.ru/1043464/observatoriya-spektrrg-udostoena-prestignoy-premii-v-oblasti-astrofiziki
• on the Canadian Astronomy Data center: https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/meetings/getMeetings.html?number=6564

2. The 17^th Italian- Korean Symposium (IK17), August 2 - 6, 2021

It gives us great pleasure to announce the 17th Italian-Korean Symposium on Relativistic Astrophysics, that will be held from August 2 to 6, 2021 in person at Kunsan National University (Gunsan, Korea) and online. The meeting has been co organized by Kunsan National University, CQUeST and Sogang University (on the Korean side) and, on the italian side, by ICRANet. Members of IK17 International Organizing Committee (IOC) are Remo Ruffini (ICRA-Sapienza University Rome/ICRANet, Co-Chair), Rong-Gen Cai (ITP, China), Pisin Chen(LeCosPA, National Taiwan University), Misao Sasaki(IPMU, Japan), Jun Luo (Sun Yat-Sen University), Sang Pyo Kim (Kunsan National University), Bum-Hoon Lee (CQUeST, Sogang University, Chair), Changhwan Lee (Pusan National University) and Hyungwon Lee (Inje University). Members of IK17 Local Organizing Committee (LOC) are Stefano Scopel (CQUeST, Sogang University, Chair), Wontae Kim (CQUeST, Sogang University), Jeong-Hyuck Park (CQUeST, Sogang University), Wonwoo Lee (CQUeST, Sogang University), Jin Young Kim (Kunsan National University), Jiwan Kim (Kunsan National University), Bogeun Gwak (Dongguk University) and Imtak Jeon (APCTP). The IK17 Program Committee is composed by Remo Ruffini (ICRA-Sapienza University Rome/ICRANet), Gregory Vereshchagin (ICRANet), Jorge A Rueda (ICRANet), Rahim Moradi (ICRANet), She-Sheng Xue (ICRANet), Simonetta Filippi (ICRANet), Yefei Yuan (ICRA/USTC, China), Marco Feroci (INFN IAPS), Narek Sahakyan (ICRANet Armeria), Sang Pyo Kim (Kunsan National University), Hyungwon Lee (Inje University), Stefano Scopel (CQUeST, Sogang University), Bum-Hoon Lee (CQUeST, Sogang University), Wontae Kim (CQUeST, Sogang University), Wonwoo Lee (CQUeST, Sogang University), Changhwan Lee (Pusan National University) and Bogeun Gwak (Dongguk University).
The Italian-Korean Symposia on Relativistic Astrophysics is a series of biannual meetings, alternatively organized in Italy and in Korea since 1987. The symposia discussions cover topics in astrophysics and cosmology, such as gamma-ray bursts and compact stars, high energy cosmic rays, dark energy and dark matter, general relativity, black holes, and new physics related to cosmology. The scientific program is under preparation, and more details about the event will be published on its webpage (http://45.120.69.181/plan.php/kis2021) as soon as available.

3. Annular solar eclipse, June 10, 2021

A special event on the occasion of the annular solar eclipse on June 10, 2021 took place virtually on that day, coordinated, as the previous ones, by and in the ICRANet center in Pescara at international level. During the eclipse, from h 12:00 to 13:00 (CEST), Prof. Costantino Sigismondi, ICRANet collaborator, in teleconference between Pescara and the astrophysical Observatory of Asiago, presented the opening lecture of the Summer School ASYAGO, speaking about the astrometric length of the solar diameter as well as about the importance of those data for the comprehension of the solar physics. The diameter will be measured starting from the 4 times of contact between the lunar and the solar disk: this opportunity occurs only on the occasion of those astronomical phenomena.
The eclipse will be an annular one in Greenland and partial, almost very low in northern Italy. A similar situation occurred on June 21, 2020 even in the center of Italy. The images with an accurate timing per second (data from the astrophysical Observatory of Asiago) will be used to extrapolate the contact times.
An exhaustive video, from the starting to the end, as the one taken on June 21, 2020 in Rome, has allowed to obtain a final resolution of 0.1", so 100 km on the solar diameter, equivalent to almost 1 million and a half km. This video is on YouTube and the relevant publication is in the online review Gerbertus. It is essential that a standard clock could be filmed during the video and that we could study how it evolves to then extrapolate the timing UTC (for example, on the website http://rime.inrim.it/labtf/tempo-legale-italiano/). In an eclipse, even if partial like this one is in northern Italy, with a duration of 1 hour, an accurate timing of few tenths of a second offers a relative precision of 1 part on 36000, equivalent to almost 40 km on the whole solar diameter.
During this eclipse, we will try to caught the phenomenon in all its technical and historical aspects, with particular attention to the scientific objective of the accurate measurement of the sola diameter. In particular, is has been clarified how, through the angular measurement of the length of the underlying rope between the intersection points in-between the two circles of the sun and of the moon (radical rope), it describes a parabolic function of the time, and the 2 zeros correspond to the first and to the last contact.
In presence of a sun slightly larger of its standard value equal to 1,392 millions of km, the eclipse started a bit before and ended a bit later compared to the ephemerides, calculated with regard to a standard sun. Vice versa if the sun is minor.
Implications in solar astrophysics
The high-resolution measurement of the solar diameter is very difficult already when we reach a scale of 1000 km, which is the smallest one discernable from the Earth, due to the turbulence of the atmosphere. The solar granulation is visible through the best telescopes only when the atmosphere is calm and has this dimension: those are the dimensions to the surface of the convective cells which bring the sun core energy towards the photosphere. The convective region concerns the internal part of the sun above the 70% of the sun ray, therefore the last 400.000 km. The standard solar model works, but we can't still predict exactly the trend of solar cycles, connected to the magnetism inside the sun, even if their recurrence each 11 years is well known. Even today, nobody can predict exactly when and how the maximum solar activity will be with regard to the cycle started in 2020. These kind of problems are common in the field of the stellar variability, where, for example, we have the same uncertainty for the Mira type variables.
The global oscillations of the sun or of its oblateness are extremely difficult to measure both on the ground and from the satellites: on the ground the difficulty is due to the continuous agitation of the atmosphere, while from the satellites it is difficult for the optic, limited in the size and by the systematic errors of the visual field.
The eclipse gives us the opportunity to use another celestial body, the Moon, its profile and its movement, as a term of comparison. Possible variations of the solar diameter on the scales less than 10 km would imply anyway some mechanism of redistribution of the energy between the magnetic one, the gravitational one and the thermal one, which could help to better define our knowledge of the nearest star to us (the only one of which we know quite well the surface details and whose activity directly affects the life and the climate on earth).
During the event of June 10, Prof. Sigismondi has clarified, together with the experts in this field, those technical aspects of the solar models and of the measurement of the solar diameter thanks to the eclipse. Prof. Sigismondi is an expert in the study of the high resolution measurement of the solar diameter, its astrophysics and relativistic implications; he has guided the observational campaign of 2 annular eclipses: in Valoria la Buena (Spain) on October 3, 200 and in Kourou (French Guyana) on September 22, 2006. The first observed annular eclipse was the one on May 30, 1984 in its initial phases at the sunset time in Rome. The partial and total eclipses observed or studied, for astrometrical purposes, have been dozens. One of the most important one is the annular-total eclipses (hybrid) of May 9, 1567 observed in Rome by Padre Cristoforo Clavio. This branch of the solar physics started in 1978 from that analysis.
The program, recording and podcast material on that event could be found on the webpage of the meeting: http://www.icranet.org/index.php?option=com_content&task=view&id=1375
On that occasion, Prof. Sigismondi has also prepared a press release (in Italian) available at this link: http://www.icranet.org/index.php?option=com_content&task=view&id=1032&Itemid=920

4. Renewal of the cooperation agreement ICRANet-UDEA (Colombia), June 9, 2021

On June 9, 2021, the Cooperation Agreement between ICRANet and the Universidad de Antioquia (UDEA) has been renewed. The renewal was signed by Prof. Adriana Echavarria Isaza (Dean of the Faculty of exact and natural sciences of UDEA) and by Prof. Remo Ruffini (Director of ICRANet). This agreement will be valid for further 5 years and the main joint activities to be developed under its framework include: the promotion of theoretical and observational activities within the field of Relativistic Astrophysics; the institutional exchange of faculty members, researchers, post-doctorate fellows and students; the promotion of technological developments; the development of Data Centers for Astrophysical data in all wavebands; the organization of training and teaching courses, seminars, conferences, workshops or short courses, the development of inter-institutional research areas associated to local graduate programs and joint publications.
For the text of the agreement see: http://www.icranet.org/index.php?option=com_content&task=view&id=1038

5. New cooperation Protocol ICRANet - Damghan University (Iran), June 23, 2021

On June 23, 2021, ICRANet has signed a new Cooperation protocol with the Damghan University in Iran. The Cooperation Protocol has been signed (both in English and in Persian) by Dr. Abdolali Basiri (President of the Damghan University), by Dr. Shahab Shahidi (School of Physics, Damghan University), by Prof. Remo Ruffini (Director of ICRANet) and by Prof. Narek Sahakyan (Director of ICRANet Seat in Yerevan).
The cooperation protocol will be valid for 5 years and the main joint activities to be developed under their framework include: the promotion of theoretical and observational activities within the field of Relativistic Astrophysics; the institutional exchange of faculty members, researchers, post-doctorate fellows and students; the promotion of technological developments; the development of Data Centers for Astrophysical data in all wavebands; the organization of training and teaching courses, seminars, conferences, workshops or short courses, the development of inter-institutional research areas associated to local graduate programs and joint publications. For the text of the Cooperation Protocol: http://www.icranet.org/index.php?option=com_content&task=view&id=1380

6. Scientific visits to ICRANet

• Carlos Raul Arguelles (Universidad Nacional de La Plata), from July 2, 2021 - ongoing
• Krzysztof Belczynski (Polish Academy of Sciences), July 1-16, 2021
• Yunlong Zheng (University of Sciences and Technology of China - USTC), from July 9, 2021 - ongoing

During their visit, those scientists have the opportunity to discuss their scientific research and to have fruitful exchange of ideas with other researchers from ICRANet and from different parts of the world. Prior and/or during their visit, those scientists had also the opportunity to participate to the 116th Marcel Grossmann meeting (online) from ICRANet Hq.

7. Recent publications

Yen Chen Chen, Classifying Seyfert galaxies with deep learning, accepted for publication in ApJS.
Traditional classification for subclass of the Seyfert galaxies is visual inspection or using a quantity defined as a flux ratio between the Balmer line and forbidden line. One algorithm of deep learning is Convolution Neural Network (CNN) and has shown successful classification results. We building a 1-dimension CNN model to distinguish Seyfert 1.9 spectra from Seyfert 2 galaxies. We find our model can recognize Seyfert 1.9 and Seyfert 2 spectra with an accuracy over 80% and pick out an additional Seyfert 1.9 sample which was missed by visual inspection. We use the new Seyfert 1.9 sample to improve performance of our model and obtain a 91% precision of Seyfert 1.9. These results indicate our model can pick out Seyfert 1.9 spectra among Seyfert 2 spectra. We decompose H{\alpha} emission line of our Seyfert 1.9 galaxies by fitting 2 Gaussian components and derive line width and flux. We find velocity distribution of broad H{\alpha} component of the new Seyfert 1.9 sample has an extending tail toward the higher end and luminosity of the new Seyfert 1.9 sample is slightly weaker than the original Seyfert 1.9 sample. This result indicates that our model can pick out the sources that have relatively weak broad H{\alpha} component. Besides, we check distributions of the host galaxy morphology of our Seyfert 1.9 samples and find the distribution of the host galaxy morphology is dominant by large bulge galaxy. In the end, we present an online catalog of 1297 Seyfert 1.9 galaxies with measurement of H{\alpha} emission line.
ArXiv: https://arxiv.org/abs/2107.06653


Li-Yang Gao, Ze-Wei Zhao, She-Sheng Xue, Xin Zhang, Relieving the H0 tension with a new interacting dark energy model, JCAP 07 (2021) 005.
We investigate an extended cosmological model motivated by the asymptotic safety of gravitational field theory, in which the matter and radiation densities and the cosmological constant receive a correction parametrized by the parameters δG and δΛ, leading to that both the evolutions of the matter and radiation densities and the cosmological constant slightly deviate from the standard forms. Here we explain this model as a scenario of vacuum energy interacting with matter and radiation. We consider two cases of the model: (i) Λ(t)CDM with one additional free parameter δG, with δG and δΛ related by a low-redshift limit relation and (ii) e Λ(t)CDM with two additional free parameters δG and δΛ that are independent of each other. We use two data combinations, CMB+BAO+SN (CBS) and CMB+BAO+SN+H0 (CBSH), to constrain the models. We find that, in the case of using the CBS data, neither Λ(t)CDM nor e Λ(t)CDM can effectively alleviate the H0 tension. However, it is found that using the CBSH data the H0 tension can be greatly relieved by the models. In particular, in the case of e Λ(t)CDM, the H0 tension can be resolved to 0.71σ. We conclude that as an interacting dark energy model, Λ(t)CDM is much better than Λ(t)CDM in the sense of both relieving the H0 tension and fitting to the current observational data.
DOI: https://doi.org/10.1088/1475-7516/2021/07/005


MAGIC collaboration, Multiwavelength variability and correlation studies of Mrk 421 during historically low X-ray and γ-ray activity in 2015-2016, MNRAS, Volume 504, Issue 1, June 2021.
We report a characterization of the multiband flux variability and correlations of the nearby (z = 0.031) blazar Markarian 421 (Mrk 421) using data from Metsähovi, Swift, Fermi-LAT, MAGIC, FACT, and other collaborations and instruments from 2014 November till 2016 June. Mrk 421 did not show any prominent flaring activity, but exhibited periods of historically low activity above 1 TeV (F_{>1 TeV} < 1.7 × 10⁻¹² ph cm⁻² s⁻¹) and in the 2-10 keV (X-ray) band (F_2−10keV<3.6×10⁻¹¹ erg cm⁻² s⁻¹), during which the Swift-BAT data suggest an additional spectral component beyond the regular synchrotron emission. The highest flux variability occurs in X-rays and very high-energy (E > 0.1 TeV) γ-rays, which, despite the low activity, show a significant positive correlation with no time lag. The HR_keV and HR_TeV show the harder-when-brighter trend observed in many blazars, but the trend flattens at the highest fluxes, which suggests a change in the processes dominating the blazar variability. Enlarging our data set with data from years 2007 to 2014, we measured a positive correlation between the optical and the GeV emission over a range of about 60 d centred at time lag zero, and a positive correlation between the optical/GeV and the radio emission over a range of about 60 d centred at a time lag of 43⁺⁹⁻⁶ d. This observation is consistent with the radio-bright zone being located about 0.2 parsec downstream from the optical/GeV emission regions of the jet. The flux distributions are better described with a lognormal function in most of the energy bands probed, indicating that the variability in Mrk 421 is likely produced by a multiplicative process.
DOI: https://doi.org/10.1093/mnras/staa3727


MAGIC collaboration, First detection of VHE gamma-ray emission from TXS 1515-273, study of its X-ray variability and spectral energy distribution, MNRAS, Stab1994, July 21, 2021.
We report here on the first multi-wavelength (MWL) campaign on the blazar TXS 1515-273, undertaken in 2019 and extending from radio to very-high-energy gamma rays (VHE). Up until now, this blazar had not been the subject of any detailed MWL observations. It has a rather hard photon index at GeV energies and was considered a candidate extreme high-synchrotron-peaked source. MAGIC observations resulted in the first-time detection of the source in VHE with a statistical significance of 7.6σ. The average integral VHE flux of the source is 6±1 per cent of the Crab nebula flux above 400 GeV. X-ray coverage was provided by Swift-XRT, XMM-Newton, and NuSTAR. The long continuous X-ray observations were separated by ∼9 h, both showing clear hour scale flares. In the XMM-Newton data, both the rise and decay timescales are longer in the soft X-ray than in the hard X-ray band, indicating the presence of a particle cooling regime. The X-ray variability timescales were used to constrain the size of the emission region and the strength of the magnetic field. The data allowed us to determine the synchrotron peak frequency and classify the source as a flaring high, but not extreme, synchrotron peaked object. Considering the constraints and variability patterns from the X-ray data, we model the broad-band spectral energy distribution. We applied a simple one-zone model, which could not reproduce the radio emission and the shape of the optical emission, and a two-component leptonic model with two interacting components, enabling us to reproduce the emission from radio to VHE band.
DOI: https://doi.org/10.1093/mnras/stab1994


Kh. Jafarzade, J. Sadeghi, B. Eslam Panah, S. H. Hendi, Geometrical thermodynamics and P-V criticality of charged accelerating AdS black holes, accepted for publication in Annals of Physics.
The unusual asymptotic structure of the accelerating black holes led to ambiguity in their geometric characteristics and thermodynamic behavior. Motivated by the interesting properties of such black holes and the significant role of electric charge and string tension on their structure, we study the thermodynamic behavior of these black holes by two methods and examine the changes of free parameters on the thermal behavior of the black holes. First, we investigate phase transition and thermal stability of the system through the use of heat capacity in the non-extended phase space. We examine the effects of electric charge, string tension and the cosmological constant on the phase transition and stability of the system. We also find that to have a phase transition, we have to apply some constraints on the free parameters. Then, we employ the geometrical thermodynamic (GT) method to study phase transition and compare the obtained results with those of the heat capacity. Next, we work in the extended phase space by considering the cosmological constant as a dynamical pressure and evaluate the existence of van der Waals like phase transition. We obtain critical quantities and study the effective role of electric charge and string tension on these quantities. Finally, we make use of the GT method in the extended phase space and find that the results of the GT method, heat capacity and P-V diagram lead to a consistent conclusion.
ArXiv: https://arxiv.org/abs/1711.04522