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WWC review of this study
Effects of an elementary school program to enhance prosocial behavior on children's cognitive-social problem-solving skills and strategies.
Battistich, V., Solomon, D., Watson, M., Solomon, J., & Schaps, E. (1989). Journal of Applied Developmental Psychology, 10 (2), 147–169.
examining
Caring School Community (CSC) Intervention Report - Character Education
Reviewed: April 2007
At least one statistically significant positive finding
Meets WWC standards with reservations
Character Education Review Protocol 1.0
Review Standards 1.0
This review may not reflect the full body of research evidence for this intervention.
Evidence Tier rating based solely on this study. This intervention may achieve a higher tier when combined with the full body of evidence.
Please see the WWC summary of evidence for Caring School Community (CSC).
Academic achievement outcomes—Statistically significant positive effects found
Significant?
Holistic measure of reading comprehension
Caring School Community (CSC) vs. Unknown
Posttest
Grade 6;
More Outcomes
California Achievement Test (CAT)
Behavior outcomes—Statistically significant positive effects found
Spontaneous prosocial behavior
Grades K–4;
Supportive, friendly, and helpful behavior
Negative behavior
Harmoniousness
Social competence
Knowledge, attitudes, & values outcomes—Statistically significant positive effects found
Democratic values
Conflict resolution interview
Grades K, 2, and 4: Cohort 1;
Social problem solving interview
Grades 1 and 3: Cohort 1;
Concern for equality
Grade 4: Cohort 1;
Perceptual benevolence
Kindergarten: Cohort 2;
Social understanding
Motive to help others learn
Concern for others
Total self-esteem
Characteristics of study sample as reported by study author.
The study took place in six elementary schools located in a middle- to upper middle-class suburban community in northern California.
Study sample
The study included two cohorts of elementary school students from six elementary schools. Each cohort began with the project in kindergarten. The first cohort started kindergarten in 1982–83 and the second in 1985–86. The first cohort included 13 kindergarten classrooms that were followed through the elementary school years;2 the second cohort included 14 classrooms that were followed through the first grade. There were about 350 students a year in the first cohort (divided evenly between the intervention and comparison groups). Of those 350 students, about 165 students remained with the cohort all five years.
Intervention Group
The intervention schools implemented the Child Development Project (CDP) program. (For details about the connection between the CDP and the CSC, see the CSC intervention report).3 Students in the intervention group received the CDP program every year starting in kindergarten. Class meetings in the intervention condition included activities designed to promote core values. In the classroom, students learned group interaction skills and relevant values and worked in small groups toward mutual academic and nonacademic goals. Teachers identified and discussed exemplary behavior using examples from the classroom, television, literature, and movies. Developmental discipline, a classroom management approach, was applied to teach prosocial norms and values. In addition, children were encouraged to help others by doing classroom chores, tutoring younger students as part of the "buddies" programs, performing charitable community activities, and helping with activities in the school at large. An implementation check done by two independent observers indicated a high level of implementation and significantly different classroom experiences (with respect to classroom activities and teacher behavior) in the intervention classrooms compared with the comparison classrooms.
Comparison Group
The comparison group included three elementary schools in the same school district as the intervention schools and matched with the intervention schools on socioeconomic status and interest in the intervention. Comparison group students did not participate in the Caring School Community program. No information was provided on character education related practices in the comparison schools.
Outcome descriptions
Student outcomes in three domains were examined: behavior; knowledge, attitudes, and values; and academic achievement. Students' behavior was assessed using direct observations of students' behavior in the classroom. Students' knowledge, attitudes, and values were assessed using several self-report questionnaires. Academic achievement was assessed using standardized achievement tests. (See Appendices A2.1–A2.3 for more detailed descriptions of the outcome measures.)
Support for implementation
Teacher training consisted of a one-week summer institute, monthly workshops, frequent meetings with project staff who also observed the classrooms periodically, and supporting curriculum materials.
In the case of multiple manuscripts that report on one study, the WWC selects one manuscript as the primary citation and lists other manuscripts that describe the study as additional sources.
Battistich, V., Schaps, E., Watson, M., & Solomon, D. (1996). Prevention effects of the Child Development Project: Early findings from an ongoing multi-site demonstration trial. Journal of Adolescent Research, 11(1), 12–35.
Solomon, D., Watson, M., Delucchi, K., Schaps, E., & Battistich, V. (1988). Enhancing children's prosocial behavior in the classroom. American Educational Research Journal, 25 (4), 527–554.
Solomon, D., Battistich, V., Watson, M., Schaps, E., & Lewis, C. (2000). A six-district study of educational change: Direct and mediated effects of the child development project. Social Psychology of Education, 4 (1), 3–51.
Battistich, V. (2003). Effects of a school-based program to enhance prosocial development on children's peer relations and social adjustment. Journal of Research in Character Education, 1(1), 1-16.
Lewis, C., Watson, M., & Schaps, E. (2003). Building community in school. In M. J. Elias, H. Arnold, & C. S. Hussey (Eds.), EQ+IQ = Best leadership practices for caring and successful schools (pp. 100–108). Thousand Oaks, CA: Corwin Press.
Solomon, D., Watson, M., Battistich, V., Schaps, E., & Delucchi, K. (1996). Creating classrooms that students experience as communities. American Journal of Community Psychology, 24(6), 719-747.
Benninga, J. S., Tracz, S. M., Sparks, R. K., Jr., Solomon, D., Battistich, V., Delucchi, K. L., Sandoval, R., & Stanley, B. (1991). Effects of two contrasting school task and incentive structures on children's social development. Elementary School Journal, 92 (2), 149–167.
Battistich, V., Schaps, E., & Wilson, N. (2004). Effects of an elementary school intervention on students' connectedness to school and social adjustment during middle school. Journal of Primary Prevention, 24 (3), 243–262.
Solomon, D., Battistich, V., Delucchi, K., & Solomon, J. (1989, March). Evaluation of the Child Development Project: Research design, procedures, and findings. Paper presented at the annual meeting of the American Education Research Association. San Francisco, CA.
Solomon, D., Battistich, V., & Watson, M. (1993, March). A longitudinal investigation of the effects of a school intervention program on children's social development. Paper presented at the biennial meeting of the Society for Research in Child Development. New Orleans, LA.
Schaps, E., Battistich, V., & Solomon, D. (1997). School as a caring community: A key to character education. In A. Molnar (Ed.), The Construction of Children's Character, Part II, 96th Yearbook of the National Society for the Study of Education. Chicago: University of Chicago Press.
Battistich, V., Solomon, D., Watson, M., & Schaps, E. (1996, March). Enhancing students' engagement, participation, and democratic values and attitudes. Paper presented at the biennial meeting of the Society for the Psychological Study of Social Issues. Ann Arbor, MI.
Battistich, V., Solomon, D., & Watson, M. (1998, April). Sense of community as a mediating factor in promoting children's social and ethical development. Paper presented at the annual meeting of the American Educational Research Association. San Diego, CA.
Solomon, D., & Battistich, V. (1993, August). Students in caring classroom communities. Paper presented at the annual meeting of the American Psychological Association. Toronto, Ontario, Canada.
Battistich, V., Solomon, D., Kim, D., Watson, M., & Schaps, E. (1995). Schools as communities, poverty levels of student populations, and students' attitudes, motives, and performance: A multilevel analysis. American Education Research Journal, 32 (3), 627–658.
Battistich, V., Solomon, D., Watson, M., & Schaps, E. (1997). Caring school communities. Educational Psychologist, 32(3), 137-151.
Schaps, E., Solomon, D., & Watson, M. (1986). A program that combines character development and academic achievement. Educational Leadership, 43 (4), 32–35.
Schaps, E., Battistich, V., & Solomon, D. (2004). Community in school as key to student growth: Findings from the Child Development Project. In J. E. Zins, R. P. Weissberg, M. C. Wang, & H. J. Walberg (Eds.), Building academic success on social and emotional learning: What does the research say? (pp. 189–205). New York: Teachers College Press.
Solomon, D., & Battistich, V. (1996). Teacher beliefs and practices in schools serving communities that differ in socioeconomic level. Journal of Experimental Education, 64 (4), 327–347.
Schaps, E. (2003). A drug abuse prevention program follow-up study. Final report. Oakland, CA: Developmental Studies Center.
Solomon, D., Watson, M., Battistich, V., Schaps, E., & Delucchi, K. (1992). Creating a caring community: Educational practices that promote children's prosocial development. In F. K. Oser, A. Dick, & J. Patry (Eds.), Effective and responsible teaching: The new synthesis. (pp. 383–396.). San Francisco, CA: Jossey-Bass.
Watson, M., Battistich, V., & Solomon, D. (1997). Enhancing students' social and ethical development in schools: An intervention program and its effects. International Journal of Educational Research, 27 (7), 571–586.
Battistich, V. (2001, April). Effects of an elementary school intervention on students' connectedness to school and social adjustment during middle school. Paper presented at the annual meeting of the American Educational Research Association. Seattle, WA.
Battistich, V., & Hom, A. (1997). The relationship between students' sense of their school community and students' involvement in problem behaviors. American Journal of Public Health, 87, 1997-2001.
Lewis, C., Battistich, V., Schaps, E., Solomon, D., & Watson, M. S. (1998, April). School improvement for academic development and resilience: Findings of the Child Development Project. Paper presented at the annual meeting of the American Educational Research Association. San Diego, CA.
Schaps, E., & Lewis, C. (1991). Extrinsic rewards are education's past, not its future. Educational Researcher, 48 (7), 81.
Solomon, D., Battistich, V., & Hom, A. (1996, April). Teacher beliefs and practices in schools serving communities that differ in socioeconomic level. Paper presented at the annual meeting of the American Educational Research Association. New York.
Solomon, D., Watson, M., Schaps, E., Battistich, V., & Solomon, J. (1990). Cooperative learning as part of a comprehensive classroom program designed to promote prosocial development. In S. Sharan (Ed.), Cooperative learning: Theory and research (pp. 231–260). New York: Praeger.
Battistich, V., Solomon, D., & Delucchi, K. (1990, August). Effects of a program to enhance prosocial development on adjustment. Paper presented at the annual meeting of the American Psychological Association. Boston, MA.
An indicator of the effect of the intervention, the improvement index can be interpreted as the expected change in percentile rank for an average comparison group student if that student had received the intervention.
For more, please see the WWC Glossary entry for improvement index.
An outcome is the knowledge, skills, and attitudes that are attained as a result of an activity. An outcome measures is an instrument, device, or method that provides data on the outcome.
A finding that is included in the effectiveness rating. Excluded findings may include subgroups and subscales.
The sample on which the analysis was conducted.
The group to which the intervention group is compared, which may include a different intervention, business as usual, or no services.
The timing of the post-intervention outcome measure.
The number of students included in the analysis.
The mean score of students in the intervention group.
The mean score of students in the comparison group.
The WWC considers a finding to be statistically significant if the likelihood that the finding is due to chance alone, rather than a real difference, is less than five percent.
The WWC reviews studies for WWC products, Department of Education grant competitions, and IES performance measures.
The name and version of the document used to guide the review of the study.
The version of the WWC design standards used to guide the review of the study.
The result of the WWC assessment of the study. The rating is based on the strength of evidence of the effectiveness of the intervention. Studies are given a rating of Meets WWC Design Standards without Reservations, Meets WWC Design Standards with Reservations, or >Does Not Meet WWC Design Standards.
A related publication that was reviewed alongside the main study of interest.
Study findings for this report.
Based on the direction, magnitude, and statistical significance of the findings within a domain, the WWC characterizes the findings from a study as one of the following: statistically significant positive effects, substantively important positive effects, indeterminate effects, substantively important negative effects, and statistically significant negative effects. For more, please see the WWC Handbook.
The WWC may review studies for multiple purposes, including different reports and re-reviews using updated standards. Each WWC review of this study is listed in the dropdown. Details on any review may be accessed by making a selection from the drop down list.
Tier 1 Strong indicates strong evidence of effectiveness,
Tier 2 Moderate indicates moderate evidence of effectiveness, and
Tier 3 Promising indicates promising evidence of effectiveness, as defined in the non-regulatory guidance for ESSA and the regulations for ED discretionary grants (EDGAR Part 77).
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Italian Prime Minister Matteo Renzi stated during a meeting of European socialist leaders held earlier this week that he would like to see students from the United Kingdom who are studying at universities in the European Union receive EU citizenship.
"If a British student decides to spend two, three, four years in a European university, we are studying now if it is possible to give him a European passport," Matteo Renzi told reporters in Brussels.
The proposal could be popular among British students who are now uncertain about their future options and rights as they study in EU countries after the recent EU referendum.
Jo Shelley of CNN writes that poll taken shortly before the referendum vote found 73% of voters under the age of 25 wanted to remain a part of the EU. The national margin was found to be 52-48% in favor of leaving on the June 23 vote.
"[We will study] if it's possible to give him a European passport – Italian, French or German. But for now, nothing is sure," said Mr Renzi.
While Renzi said a final decision had not been made on the idea, significant legal issues could get in the way. In order to avoid illegal discrimination, EU countries who grant citizenship to UK students could be forced to extend the same offer to students studying from other countries.
Meanwhile, the UK universities minister Jo Johnson has already reassured EU students studying at schools in the UK, telling those already enrolled as well as those set to begin this fall that their funding would be honored as long as they remain enrolled in the school.
In addition, Goldsmiths, University of London has promised that it would freeze tuition rates for EU students up to and including courses beginning in 2018-19. Doing so ensures students that their status will not change as long as they are studying, reports Sean Coughlan for the BBC.
Currently, students from the EU are treated equally to students from the UK and pay the same tuition fees. However, if EU students are to be considered "international" students, their fees would end up being much higher.
UK universities are also looking for a better understanding of how the Brexit vote will affect research funding. Data from the Royal Society finds the UK received €8.8 billion in direct EU research funding between 2007 and 2013 while contributing €5.4 billion. While the UK could pay to become an "associate" member of the EU's research program, it would lose all control over how any research was directed.
"The people who study in European universities, in my view, must have the European citizenship and the European passports, so I'm ready to create some initiatives for the people who come from the UK," Renzi said.


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New Face of Iraq
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USF-Iraq Commanding General
General Lloyd J. Austin III
United States Forces – Iraq
General Lloyd J. Austin Ill, who hails from Thomasville, Georgia, was commissioned as a second lieutenant in the infantry in June 1975 from the United States Military Academy, West Point, New York.
His previous assignments include Rifle and Scout Platoon Leader, 3d Infantry Division, U.S Army Europe and Seventh Army; Company Commander, 2d Battalion, 508th infantry, 82d Airborne Division, Fort Bragg, North Carolina; Company Commander, U.S Army Recruiting Battalion, Indianapolis, Indiana; Company Tactical Officer, United States Military Academy and Executive Officer, 1 infantry Brigade, 10th Mountain Division (Light), Fort Drum, New York from June 1991 until October 1992.
General Austin has served in several command and staff assignments throughout his 35-year career.
At Fort Bragg, he served as Commander, 2d Battalion, 505th Parachute infantry Regiment, 82d Airborne Division and OPERATION SAFE HAVEN, Panama; G-3 for the 82d Airborne Division; and Commander, 3d Brigade, 82d Airborne Division from June 1997 to June 1999. Shortly after brigade command, he was assigned to the Pentagon in Washington, D.C., where he served as Chief, Joint Operations Division, J-3 on the Joint Staff.
Other command assignments include Assistant Division Commander (Maneuver), 3d Infantry Division (Mechanized), Fort Stewart, Georgia and OPERATION IRAQI FREEDOM, Iraq from July 2001 until June 2003; Commanding General, 10th Mountain Division (Light), Fort Drum, New York from September 2003 until August 2005; with duty as Commander, Combined Joint Task Force-ISO, OPERATION ENDURING FREEDOM, Afghanistan. He served as the Chief of Staff, United States Central Command, MacDill Air Force Base, Florida from September 2005 until October 2006.
He assumed duties as the Commanding General, XVIII Airborne Corps and Fort Bragg in December 2006, and deployed to Iraq as Commander, Multi-National Corps – Iraq from January 2008 until April 2009. General Austin served as the Director, Joint Staff at the Pentagon from August 2009 to August 2010.
General Austin assumed command of United States Forces – Iraq on 1 September 2010.
His military education includes the infantry Officer Advanced Course, United States Army Infantry School, Fort Benning, Georgia; United States Army Command and General Staff College, Fort Leavenworth, Kansas and United States Army War College, Carlisle Barracks, Pennsylvania.
He holds a Bachelor of Science Degree from the United States Military Academy, a Master's Degree in Education from Auburn University, and a Master's Degree in Business Management from Webster University.
General Austin's awards and decorations include the Defense Distinguished Service Medal (with Oak Leaf Cluster), the Distinguished Service Medal (with Oak Leaf Cluster), the Silver Star, the Defense Superior Service Medal (with Oak Leaf Cluster), the Legion of Merit (with Oak Leaf Cluster), the Defense Meritorious Service Medal, the Meritorious Service Medal (with Four Oak Leaf Clusters), the Joint Service Commendation Medal, the Army Commendation Medal (with Six Oak Leaf Clusters), Army Achievement Medal (with Oak Leaf Cluster), Combat Action Badge, Expert infantryman Badge, Master Parachutist Badge, the Ranger Tab and the Joint Chiefs of Staff Identification Badge.
(Sunday, December 05, 2010 00:00)
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Alaska sounds amazing!! I am going to try the game you suggested. Big game players around here.
Thanks! I see that you your team love games:) I love the quirkiness of your family – as you put it. Lots of fantasy lovers here. Inspiring to get a window into what it is like with seven children. Looks fulfilling!


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Kevin N. Sheth MD, FAHA, FCCM, FNCS, FAAN, FANA
Dr. Kevin Sheth graduated from Johns Hopkins University and the University of Pennsylvania School of Medicine. He was an intern at Brigham and Women's Hospital and a neurology chief resident at Partners Neurology (Massachusetts General and Brigham & Women's Hospitals). After a fellowship in vascular neurology and neuro-critical care at Harvard, he was appointed the first neurology trained neurointensivist at the R Adams Cowley Shock Trauma Center. He was recruited to Yale as the founding chief of the Division of Neurocritical Care and Emergency Neurology. His interests are in the advancement of therapies of patients with devastating acute neurological syndromes, especially those complicated by brain swelling and hemorrhage. He is a nationally recognized clinical and translational scientist and has directed a number of multicenter studies testing potential novel therapies against brain swelling, stroke, and hemorrhage. He is the winner of the prestigious Robert Siekert Award from the American Heart Association (AHA) and the Derek Denny Brown Award from the American Neurological Association , and his research has been funded by the NIH, American Academy of Neurology, AHA, and the US Army. Dr. Sheth is the author of over 120 publications in critical care neurology and stroke. He serves on six editorial boards including the journals Stroke, Neurocritical Care, Neurology and Neurosurgery and has served on study sections for the NIH, AHA, FDA and NASA. He has formed exciting partnerships with entrepreneurs, pharmaceutical companies, medical device start-ups, and academic collaborators in order to bring forward highly innovative solutions. This effort has resulted in extensive knowledge of FDA pathways, development of phase I-III drug programs, and implementation of new technology into the clinical workspace. His highly collaborative work is dedicated to the improved understanding of neurological disease in the critically ill.
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Disciplines Philosophy Ethics
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Essays on Aristotle's Rhetoric
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About the Book About the Author Table of Contents Related Books
Essays on Aristotle's Rhetoric offers a fresh and comprehensive assessment of a classic work. Aristotle's influence on the practice and theory of rhetoric, as it affects political and legal argumentation, has been continuous and far-reaching. This anthology presents Aristotle's Rhetoric in its original context, providing examples of the kind of oratory whose success Aristotle explains and analyzes.
The contributors—eminent philosophers, classicists, and critics—assess the role and the techniques of rhetorical persuasion in philosophic discourse and in the public sphere. They connect Aristotle's Rhetoric to his other work on ethics and politics, as well as to his ideas on logic, psychology, and philosophy of language. The collection as a whole invites us to reassess the place of rhetoric in intellectual and political life.
Amèlie Oksenberg Rorty is Professor of the Humanities and the History of Ideas at Brandeis University. Among the numerous other volumes she has edited are Essays on Descartes' Meditations (California, 1986), Perspectives on Self-Deception (California, 1988), and (with Martha Nussbaum), Essays on Aristotle'sDe Anima (1992).
Jacques Brunschwig, M. F. Burnyeat, Christopher Carey, John M. Cooper, Troels Engberg-Pedersen, Dorothea Frede, Stephen Halliwell, T. H. Irwin, George A. Kennedy, Stephen R. Leighton, Richard Moran, Martha Craven Nussbaum, C. D. C. Reeve, Paul Ricoeur, Amèlie Oksenberg Rorty, Gisela Striker, Robert Wardy
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CARITAS – MYANMAR
Caritas Myanmar, on a mission for the human and social development of all ethnic groups
The charity's president, Mgr Raymond Sumlut Gam, and director, Win Tun Kyi, spoke to AsiaNews about how the Catholic organisation operates. In the wake of the conflicts that tore at the country, its people need "humanitarian assistance and education".
Rome (AsiaNews) - The conflicts that torn Burma asunder for years are "man-made disasters that require Caritas Myanmar to work towards the development of society, with particular attention to humanitarian assistance, something that is badly needed," said Mgr Raymond Sumlut Gam, bishop of Banmaw.
The prelate, who heads the Catholic charity, is currently in Rome for the General Assembly of Caritas Internationalis. Last Monday, Pope Francis led a Mass in St Peter for its opening session.
"As bishop, celebrating the Eucharist with the Holy Father was a very important moment," said the bishop. "His presence gave me great strength."
Caritas Myanmar, said its director Win Tun Kyi, "is involved in a great many regions of the country. Each of the 16 dioceses have a local Caritas office. I think it is the largest charitable organisation in Myanmar. We work primarily in prevention and recovery during and after natural disasters. Floods, typhoons and epidemics top our agenda."
In addition, "we are very involved in the field of child and adult education. We have to raise awareness as much as possible of the importance of education for the development of society and the country. Another field of action is the protection of children's rights, an area in which the Burmese have to work harder."
Although Myanmar is a predominantly Buddhist country, "Caritas can be a real bridge between religions and ethnicities," he said.
"Our operations are not influenced by religion or ethnic identity. Our charitable programmes have earned us the trust of everyone. This is a real gift, for which we thank the Lord every day."
internationalis
Caritas: priest to help families rebuild their shattered lives
Asian Catholics helping Myanmar cyclone victims
Despite government hindrance, Caritas Uzbekistan helps young people and the poor
Caritas' task is to express God's love whilst striving for social justice, says Pope
Card Tagle visits Rohingya refugee camps in Cox's Bazar, calls the situation an 'international crisis' (photos)


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\section{Introduction}
\label{secIntroduction}

Black holes are now widely believed to be thermodynamic objects assigned standard thermodynamic variables such as the temperature and entropy \cite{Bardeen,Bekenstein}. A lot of attention has been devoted to studying thermodynamic properties for different black holes. Motivated by the AdS/CFT correspondence \cite{Maldacena,Gubser,Witten} that thermodynamics of a black hole in AdS space can be identified with that of dual strongly coupled conformal field theory on the boundary of the AdS space, the thermodynamic properties of black holes in anti-de Sitter (AdS) space are researched recently. In AdS space, the so-called Hawking-Page phase transition can occur between stable large black holes and thermal gas in AdS space \cite{Hawking}, which can well explain the confinement/deconfinement phase transition of gauge field \cite{Hawking,Witten2}.

A decade ago, thermodynamics of the charged black holes in AdS space was studied. It was shown in Refs. \cite{Chamblin,Chamblin2,Lemos} that, in a canonical ensemble the black hole system has a small/larger black hole (SBH/LBH) phase transition. With the increasing of the charge, such phase transition terminates at a critical point. The $Q-\Phi$ (charge-chemical potential) diagram of the charged black holes is quite similar to the $P-v$ (pressure-volume) diagram of the van der Waals fluid \cite{Shen,Kubiznak,Wei,Niu,Tsai,Banerjee,Banerjee3,Lala,Spallucci,Smailagic}. Therefore the black hole system is shown to be analogous to the van der Waals fluid. Although the analogy between them is widely studied, it is somewhat problematic because that $Q$ and $\Phi$ are extensive and intensive quantities, while $P$ and $v$ are intensive and extensive quantities. On the other hand, such phase transition of van der Waals type can be also observed for asymptotically flat or de Sitter charged black holes or black branes placed in a finite cavity \cite{Vaidya,Lu,Roy}.

Very recently, by interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume of the black hole \cite{Dolan,Cvetic,Kastor,Traschen,Castro,Menoufi}, the analogy between the charged black hole in AdS space and van der Waals fluid was further enhanced \cite{Kubiznak}. Both systems were found to have the same critical exponents near the critical point and extremely similar phase diagrams. The problematic between $Q-\Phi$ and $P-v$ was modified. Thus the analogy between the charged AdS black hole and the van der Waals system becomes more complete. This analogy has been generalized to different charged black holes and rotating black holes in AdS space in the extended phase space \cite{Gunasekaran,Hendi,Chen,ZhaoZhao,Altamirano,Cai,AltamiranoKubiznak,XuXu,Mo,zou,MoLiu,Altamirano3,Roychowdhury}.

Among these studied, it was shown that besides the SBH/LBH first-order phase transition reminiscent of the liquid/gas transition of the van der Waals fluid, there also appear some new interesting phenomena analogous to the ``every day thermodynamics" of simple substances, such as reentrant phase transitions of multicomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van der Waals type. It was shown that in all $d\geq 6$ dimensions the single spinning vacuum Kerr-AdS black holes demonstrate the peculiar behavior of LBH/SBH/LBH phase transitions reminiscent of reentrant phase transitions \cite{Altamirano}. Such reentrant phase transition was also found in the four-dimensional Born-Infeld-AdS black hole spacetimes, deep in the nonlinear regime of the Born-Infeld theory. However, for the higher dimensional Born-Infeld AdS black holes, such phase transition was not observed \cite{zou}. More intriguingly, the multiply rotating Kerr-AdS black hole system displays a small/intermediate/large black hole (SBH/IBH/LBH) phase transition with one tricritical and two critical points in some range of the parameters \cite{AltamiranoKubiznak}. It was argued in Refs. \cite{Altamirano,AltamiranoKubiznak} that the new phase structure in the thermodynamics of rotating black holes resembles to binary fluids seen in superfluidity and superconductivity. More recently, the authors in Ref. \cite{Altamirano3} also pointed out that the reentrant phase transitions are observed for the asymptotically flat doubly-spinning Myers-Perry black holes of vacuum Einstein gravity. Hence, neither exotic matter nor a cosmological constant is required for this phenomenon to occur in black hole spacetimes.

The aim of this paper is to search for the possible triple points and SBH/IBH/LBH black hole transitions in the charged Gauss-Bonnet (GB) black holes in AdS space. In fact, in our previous work \cite{Wei}, we studied the $Q-\Phi$ criticality for the charged GB black holes. We ignored a subtle thing that worthwhile to note here: we found that the black hole system undergoes two phase transitions of the van der Waals type at one isotherm in some ranges of the parameter (see Table I in Ref. \cite{Wei} for details). This in other words implies a reentrant phase transition or a triple point. It is the partial aim of this paper to examine such phase transition. Considering the problematic of the analogy between the analogy of $Q-\Phi$ and $P-v$, we will do it in the extended phase space, where the cosmological constant appears as a dynamical pressure of the system and its conjugate quantity is the thermodynamic volume of the black holes. The result indicates that there indeed exists the SBH/IBH/LBH black hole transition in the case of $d=6$ and $Q\in(0.1705, 0.1946)$. And the triple point is also found, at which the SBH, IBH, and LBH can coexist. For the other values of the parameters, the phase diagrams are also obtained, which reveals that the SBH/LBH phase transition is ubiquitous.

The outline of our paper is as follows. In Sec. \ref{vanderWaalsfluid}, we review the van der Waals fluid. Through rewriting the equation of state in the form of $T=T(v, P)$, we show that the critical behavior and the phase diagram can be equivalently obtained. In Sec. \ref{CGB}, we give some thermodynamic quantities of the charged GB black holes in AdS space, and we clearly show that the information of the phase transition is encoded in the $T-r_{h}$ line with fixed $P$. The divergence and sign of the heat capacity are also encoded in it. In Secs. \ref{Phase1}, \ref{Phase2}, and \ref{Phase3}, we respectively discuss the phase diagrams of $d=5$, $d=6$, and $d\geq 7$ in detail. Section \ref{Conclusion} is devoted to the conclusions and discussions.

\section{Review of van der Waals fluid}
\label{vanderWaalsfluid}

Compared to the ideal gas, the van der Waals fluid approaches the real fluid for considering the nonzero size of molecules and the attraction between them. The equation of state and the Gibbs free energy read
\begin{eqnarray}
 T&=&\bigg(P+\frac{a}{v^{2}}\bigg)(v-b),\label{pt0}\\
 G&=&-T\bigg(1+\ln (v-b)T^{3/2}/\Phi_{0}\bigg)-\frac{a}{v}+Pv,
\end{eqnarray}
where we have set the Boltzmann constant $k=1$. $v=V/N$, $P$, $T$, and $\Phi_{0}$ are the specific volume of the fluid, pressure, temperature, and a (dimensionful) constant characterizing the gas, respectively. The parameter $a$ measures the attraction between the molecules, and $b$ describes the nonzero size of the molecules.

\begin{figure}
\includegraphics[width=8cm]{vanPV_1.eps}
\caption{$P-v$ diagram of van der Waals fluid for fixed $T$. The parameters $a=1$, $b=0.15$. The temperature $T=2.2$, $T_{c}$, 1.9, and 1.8 from top to bottom. Here the critical temperature $T_{c}=1.9753$.}\label{Pvanpv}
\end{figure}

\begin{figure}
\center{\subfigure[]{\label{PCriticalPb}
\includegraphics[width=8cm,height=6cm]{CriticalPb_2a.eps}}
\subfigure[]{\label{Ppvt}
\includegraphics[width=8cm,height=6cm]{PpvT_2b.eps}}}
\caption{The parameter $a=1$. (a) The critical point in $P-b$ plane. The equation $(\partial_{v}T)_{P}=0$ has two roots in the range I, one root at the line, and no root in the range II. (b) Roots of equation $(\partial_{v}T)_{P}=0$ with $b=0.15$.}
\end{figure}

\begin{figure}
\center{\subfigure[]{\label{Ptv}
\includegraphics[width=8cm,height=6cm]{Tv_3a.eps}}
\subfigure[]{\label{Pgt}
\includegraphics[width=8cm,height=6cm]{GT_3b.eps}}}
\caption{(a) The temperature $T$ as a function of the volume $v$. (b) The Gibbs free energy as a function of the temperature $T$ with $\Phi_{0}=1$. The parameters $a=1$, $b=0.15$. And the pressure $P=2$, $P_{c}$, 1, and 0.8 from top to bottom, with $P_{c}=1.6461$. Black arrows indicate increasing $v$.}\label{Pv2}
\end{figure}

The state function (\ref{pt0}) (reexpressed in the form $P=P(v, T)$) is often used to describe the basic qualitative features of the liquid/gas phase transition. The qualitative behavior of isotherms in the $P-v$ diagram for the van der Waals fluid can be found in Fig. \ref{Pvanpv}. From it, we clear see that there exists a critical case at $T=T_{c}$. For $T>T_{c}$, it describes the ``ideal gas''. And for $T<T_{c}$, there is a classical behavior of isotherms for the van der Waals fluid. There are three branches in such case, two have negative slope described by the solid lines. While the third one has positive slope described in the dashed line, which is unphysical and should be replaced by the horizontal line determined by the Maxwell's equal area law. Such behavior generally implies a first-order liquid/gas phase transition. The critical point is an inflection point, which is determined by
\begin{eqnarray}
 \bigg(\frac{\partial P}{\partial v}\bigg)_{T}=0,\quad
 \bigg(\frac{\partial^{2} P}{\partial v^{2}}\bigg)_{T}=0.
\end{eqnarray}
The index means the temperature $T$ is fixed. Solving this equation, we obtain the critical point
\begin{eqnarray}
 T_{c}=\frac{8a}{27b},\quad v_{c}=3b,\quad
 P_{c}=\frac{a}{27b^{2}}.\label{vdwcp}
\end{eqnarray}
And at the critical point
\begin{eqnarray}
 P_{c}v_{c}/T_{c}=3/8,
\end{eqnarray}
which is a universal number for all fluids. On the other hand, the equation of state (\ref{pt0}) can be regarded as $T=T(v, P)$. And since $(\partial_{P}T)_{v}=(v-b)>0$, the critical point can be equivalently determined by
\begin{eqnarray}
 \bigg(\frac{\partial T}{\partial v}\bigg)_{P}=0,\quad
 \bigg(\frac{\partial^{2} T}{\partial v^{2}}\bigg)_{P}=0.\label{vdwcp2}
\end{eqnarray}
Solving it, the critical point (\ref{vdwcp}) will be obtained. The critical point satisfying (\ref{vdwcp2}) is depicted in Fig. \ref{PCriticalPb} with $a=1$. After a simple calculation, the equation $(\partial_{v}T)_{P}=0$ has two roots in the range I, one root at the line, and no root in the range II. For an example, we plot the roots of $(\partial_{v}T)_{P}=0$ in Fig. \ref{Ppvt}, for $a=1$ and $b=0.15$. From it, we see that there exists a maximal value of $P$ corresponding to the critical pressure $P_{c}$, which can be found in Fig. \ref{PCriticalPb} with $b=0.15$. For small $P<P_{c}$, there are two roots. And the two roots meet each other at $P=P_{c}$. With $P$ further increasing, no root can be found. The corresponding behavior of $T$ as a function of $v$ with fixed $P$ is given in Fig. \ref{Ptv}. For $P>P_{c}$, $T$ monotonically increases with $v$, which denotes the ``ideal gas" branch. However, when $P<P_{c}$, the two roots showed in Fig. \ref{Ppvt}, i.e., the points $A$ and $B$, divide one line into three branches. Two of the branches in the solid lines have positive slope, while the other one in the dashed line has negative slope. The Gibbs free energy is depicted in Fig. \ref{Pgt} with the same parameters. A detailed study shows that the three branches in the $P-v$ line with constant $P<P_{c}$ are consistent with the three branches in the $G-T$ line with the same constant $P$. The discontinuous points in $G-T$ line are the roots of $(\partial_{v}T)_{P}=0$ (see the points $A$ and $B$ shown in Fig. \ref{Ppvt} and Fig. \ref{Pv2}). It is clear that the characteristic swallow tail behavior of the Gibbs free energy appears for $P<P_{c}$ implies a first-order phase transition. The coexistence line of liquid and gas phases of the van der Waals fluid can be determined by the intersection of the two branches of the same $G-T$ line shown in solid line, and we do not show it here.

Now to summarize the result. The appearance of the swallow tail behavior implies a first-order phase transition. And the behavior needs at least three branches on one $G-T$ line, which is bounded with two discontinuous points determined by $(\partial_{v}T)_{P}=0$. So if there is no or only one discontinuous point, then there is no swallow tail behavior, and therefore there is no the liquid/gas phase transition of van der Waals type. In general, such phase transition is studied with the state function $P=P(v, T)$. However, we show in the above that with the state function expressed in $T=T(v, P)$, the phase transition can also be equivalently obtained. In the following, we will show that for a black hole system, $T=T(v, P)$ is a natural and convenient choice. Under such choice, we can get some information of the phase transition through examining the behavior of the temperature $T$ vs. pressure $v$. Moreover, we will show in the next section that the discontinuous point in the $G-T$ line has a local extremum in the $T-v$ ($T-r_{h}$) line. And the points is related to a divergence of the heat capacity. Adopting such choice of the equation of state, we will investigate the phase diagram of the charged GB black holes in AdS space.

\section{Thermodynamics of charged Gauss-Bonnet black holes in AdS space}
\label{CGB}

The action of the Einstein-Gauss-Bonnet gravity in $d$-dimensional spacetime with a negative cosmological constant reads as
\begin{eqnarray}
 S=\int d^{d}x\sqrt{-g}
 \Big[\frac{1}{16\pi G_{d}}(\mathcal{R}
    -2\Lambda
    +\alpha_{\text{GB}}\mathcal{L}_{\text{GB}})
  -\mathcal{L}_{\text{matter}}\Big], \label{action}
\end{eqnarray}
where $\alpha_{\text{GB}}$ is the GB coupling constant with dimension $(\text{length})^{2}$ and it is regarded as the inverse string tension with positive value. The GB Lagrangian $\mathcal{L}_{\text{GB}}$ and the electromagnetic Lagrangian $\mathcal{L}_{matter}$ are
\begin{eqnarray}
&&\mathcal{L}_{\text{GB}}=\mathcal{R}_{\mu\nu\gamma\delta}\mathcal{R}^{\mu\nu\gamma\delta}
                    -4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}+\mathcal{R}^{2},\\
&&\mathcal{L}_{\text{matter}}=4\pi \mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}.
\end{eqnarray}
The Maxwell field strength is defined as
$\mathcal{F}_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}$
with $\mathcal{A}_{\mu}$ the vector potential. The solution of the $d$-dimensional static charged GB-AdS black hole for the action (\ref{action}) is
\begin{eqnarray}
 ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}+\cos^{2}\theta d\Omega^{2}_{d-4}),\label{metric}
\end{eqnarray}
with the metric function given by \cite{Boulware,Cai2,Wiltshire,Cvetic2}
\begin{eqnarray}
 f(r)=1+\frac{r^{2}}{2\alpha}
   \left[1-\sqrt{1 + \frac{2\alpha}{d-2} \left(\frac{32\pi M}{\Sigma_{d-2} r^{d-1}}-\frac{4 Q^{2}}{(d-3)r^{2d-4}}+\frac{4\Lambda}{d-1}\right)}\right],
\end{eqnarray}
where $\alpha=(d-3)(d-4)\alpha_{\text{GB}}$. $\Sigma_{d-2}$ is the area of a unite $(d-2)$-dimensional sphere, and we set $\Sigma_{d-2}=1$ for simplicity. Recent development on the thermodynamics of black holes in extended phase space shows that the cosmological constant can be interpreted as the thermodynamic pressure and treated as a thermodynamic variable in its own right,
\begin{eqnarray}
 P=-\frac{1}{8\pi}\Lambda.
\end{eqnarray}
It is shown that the differential form holds \cite{Cai}
\begin{eqnarray}
 dH=TdS+\Phi dQ+vdP+\mathcal{A}d\alpha,
\end{eqnarray}
where $H\equiv M$ is the enthalpy of the gravitational system \cite{KastorRay}. The thermodynamic volume $v$ is the thermodynamic quantity conjugating to the pressure $P$. And $\mathcal{A}$ is the conjugate quantity conjugating to the GB coefficient $\alpha$. We list the thermodynamic quantities
\begin{eqnarray}
 H&=&\frac{(d-2)r_{h}^{d-3}}{16\pi}\bigg(1+\frac{\alpha}{r_{h}^{2}}+
     \frac{16\pi Pr_{h}^{2}}{(d-1)(d-2)}\bigg)+\frac{Q^{2}}{8\pi(d-3)r_{h}^{d-3}},\\
 T&=&\frac{16\pi Pr_{h}^{4}/(d-2)+(d-3)r_{h}^{2}+(d-5)\alpha-2Q^{2}r_{h}^{8-2d}/(d-2)}
    {4\pi r_{h}(r_{h}^{2}+2\alpha)},\\
 S&=&\frac{r_{h}^{d-2}}{4}\left(1+\frac{2(d-2)\alpha}{(d-4)r_{h}^{2}}\right),
     \quad v=\frac{r_{h}^{d-1}}{d-1},\quad \Phi=\frac{Q}{4\pi(d-3)r_{h}^{d-3}},\\
 \mathcal{A}&=&\frac{(d-2)}{16\pi}r_{h}^{d-5}-\frac{(d-2)T}{2(d-4)}r_{h}^{d-4},\\
 16\pi G&=&(d-2) r_{h}^{d-3}
    \bigg(1+\frac{1}{r_{h}^2}+\frac{16 \pi  P r_{h}^2}{d^2-3d+2}\bigg)
    +\frac{2 Q^2 r_{h}^{3-d}}{d-3}\nonumber\\
   &&-\frac{r_{h}^{d-3}}{2+r_{h}^2}
   \bigg(1+\frac{2(d-2)}{(d-4)r_{h}^2}\bigg)
   \bigg(d-5+(d-3) r_{h}^2+\frac{16 \pi  P r_{h}^4-2 Q^2 r_{h}^{8-2
   d}}{d-2}\bigg).\label{quantities}
\end{eqnarray}
$r_{h}$ is the horizon radius of the black hole and it is determined by the largest real root of the equation $f(r_{h})=0$. The range of $T<0$ represents the non-black hole case. So, we will only focus on the range of $T>0$ and $S>0$ in this paper.

The heat capacity $C_{P}$ at fixed $P$ measuring the local thermodynamical stability is defined as
\begin{eqnarray}
 C_{P}=T\bigg(\frac{\partial S}{\partial T}\bigg)_{P}.
\end{eqnarray}
Using the thermodynamic quantities, $C_{P}$ can be reexpressed as
\begin{eqnarray}
 C_{P}=T\bigg(\frac{\partial_{r_{h}} S}{\partial_{r_{h}} T}\bigg)_{P}
       \propto (\partial_{r_{h}} T)^{-1}_{P}.\label{heat}
\end{eqnarray}
The second step is guaranteed with $(\partial_{r_{h}} S)_{P}>0$ for $d\geq5$ and positive temperature for a black hole system. Thus it is qualitatively clear that the heat capacity $C_{P}$ diverges at $(\partial_{r_{h}} T)_{P}=0$. And positive (negative) $(\partial_{r_{h}} T)^{-1}_{P}$ relates with positive (negative) $C_{P}$. Therefore, in the $T-r_{h}$ plane, the black hole branch of (negative) positive slope relate to the thermodynamically (unstable) stable phase. Since $(\partial_{P}T)_{v}>0$ for positive $\alpha$, the critical point can be determined by Eq. (\ref{vdwcp2}). On the other hand we have $v\propto r_{h}^{d-1}$, so the critical condition (\ref{vdwcp2}) for the black hole can be expressed as
\begin{eqnarray}
 \bigg(\frac{\partial T}{\partial r_{h}}\bigg)_{P}=0,\quad
 \bigg(\frac{\partial^{2} T}{\partial r_{h}^{2}}\bigg)_{P}=0.\label{vdwcp3}
\end{eqnarray}
In the following discussion, without loss of generality we adopt $\alpha=1$. Several remarks on the $T-r_{h}$ line and phase transitions are in order:

(i) The extremal points determined by $(\partial_{r_{h}}T)_{P}=0$ divide one $T-r_{h}$ line into several black hole branches. The extremal points correspond to these discontinuous points in the $G-T$ line.

(ii) At the extremal points, the heat capacity diverges.

(iii) The black hole branch with (negative) positive slope in the $T-r_{h}$ plane is thermodynamically (unstable) stable.

(iv) The critical point occurs at the multiple root of $(\partial_{r_{h}}T)_{P}=0$.

(v) The number of the extremal points could tell us the number of the characteristic swallow tail behavior of the Gibbs free energy, and the possible existence of the phase transition of ``everyday thermodynamics". Swallow tail behavior and phase transition will not occur if there is no or one extremal point, while two extremal points generally produce one swallow tail behavior and the liquid/gas phase transitions of the van der Waals type. More extremal points will potentially give the reentrant phase transitions.

\section{The phase diagram when $d=5$}
\label{Phase1}

For a five-dimensional static charged GB-AdS black hole ($d=5$), the temperature, Gibbs free energy, and heat capacity are
\begin{eqnarray}
 T&=&\frac{8 \pi Pr_{h}^6+3r_{h}^4-Q^2}{6\pi r_{h}^5+12\pi r_{h}^3},\\
 G&=&\frac{-4\pi Pr_{h}^8+(3-72\pi P)
   r_{h}^6+Q^2\left(5r_{h}^2+18\right)-9r_{h}^4+18r_{h}^2}{48\pi r_{h}^2
   \left(r_{h}^2+2\right)},\\
 C_{P}&=&\frac{3 r_{h}\left(r_{h}^2+2\right)^2
                \left(8\pi Pr_{h}^6-Q^2+3 r_{h}^4\right)}
              {4\left[r_{h}^4 \left(8\pi Pr_{h}^4
                +(48\pi P-3)r_{h}^2+6\right)+Q^2 \left(5r_{h}^2+6\right)\right]}.
\end{eqnarray}
The critical point can be found with the condition (\ref{vdwcp3}). In the small charge limit, we have
\begin{eqnarray}
 \frac{P_{c}r_{hc}}{T_{c}}=\frac{1}{4}+\frac{1}{72}Q^{2}+\mathcal{O}(Q^{4}).
\end{eqnarray}
This result is different from that of the van der Waals fluid, for which the right-hand side of the corresponding equation is $8/3$. When the charge $Q$=0, our result recovers that shown in Ref. \cite{Cai}. The behavior of the critical point is displayed in the $P-Q$ plane in Fig. \ref{Pd5PQ}. It behaves very similar to that of van der Waals fluid shown in Fig. \ref{PCriticalPb}. In the region I, $(\partial_{r_{h}}T)_{P}=0$ has two roots, one is the local maximum and another is the local minimum. On the line, the two roots meet each other. And in the region II, there exists no root. However, different from the van der Waals fluid, the critical pressure $P_{c}$ here has a maximum $1/(48\pi)$ at $Q=0$. The heat capacity is plotted in Fig. \ref{Pd5CP} for $Q=0.18$. When $P=0.0060<P_{c}$, $(\partial_{r_{h}}T)_{P}=0$ (described by the blue solid line) has two roots, at which $C_{P}$ diverges. It is also clear that $C_{P}$ changes its sign at the roots. When $P=P_{c}\approx0.0066$, the two roots coincide with each other, resulting only one divergent behavior of $C_{P}$ described by the green dashed line, and the range of negative $C_{P}$ is squeezed out. Further increase the pressure to $P=0.0075$, $C_{P}$ will be positive and continuous for any $r_{h}$. For the three constant values of the pressure $P$, i.e., $P$=0.0060, 0.0066, and 0.0075, the behavior of the temperature $T$ is depicted in Fig. \ref{Pd5Trh}. When $P<P_{c}$, there are three branches on one $T-r_{h}$ line with fixed $P$. The two branches, i.e., the small and larger black hole branches, have positive slope, and thus have positive heat capacity $C_{P}$, which indicates these branches are thermodynamically stable. The slope of the other branch, i.e., the intermediate black hole branch, is negative, and this leads to negative $C_{P}$. This result is consistent with that shown in Fig. \ref{Pd5CP}. As analyzed in Section \ref{vanderWaalsfluid}, the three branches on one $T-r_{h}$ line with $P<P_{c}$ will construct a swallow tail behavior, and the first-order phase transition will occur. For clarity, the Gibbs free energy is shown in Fig. \ref{Pd5GT} for fixed $P$. It is clear that the swallow tail behavior appears when $P<P_{c}$. It is also worth noting that in Fig. \ref{PPrhgt}, the dashed lines correspond to negative $C_{P}$ implying a local thermodynamically unstable branch, while the solid ones correspond to positive $C_{P}$. The horizon radius $r_{h}$ increases from left to right along the $G-T$ line with fixed $P$.

Now let us focus on the $G-T$ line with $P=0.0060<P_{c}$. If we start increasing the temperature from, say $T=0.062$, the system follows the small stable black hole branch (with smaller Gibbs free energy) until it approaches the intersection with the larger stable black hole branch, which corresponds to a SBH/LBH phase transition of first order at the intersection. Further increasing $T$, the system will follow along the larger stable black hole branch. So we observe the SBH/LBH phase transition. The situation is clearly illustrated in the $P-T$ diagram in Fig. \ref{Pd5PT}. There is a SBH/LBH line of coexistence, and it corresponds to the liquid/gas line of the van der Waals fluid. With the increasing of $T$, the coexistence line terminates at a critical point, at which the system undergoes a second-order phase transition.

We close this section by noting that there exists a SBH/LBH phase transition for a five-dimensional charged GB-AdS black hole, which is reminiscent of the liquid/gas phase transition of van der Waals fluid.

\begin{figure}
\center{\subfigure[]{\label{Pd5PQ}
\includegraphics[width=8cm,height=6cm]{d5PQ_4a.eps}}
\subfigure[]{\label{Pd5CP}
\includegraphics[width=8cm,height=6cm]{d5CP_4b.eps}}}
\caption{In five-dimensional spacetime. (a) The critical point displayed in $P-Q$ plane. (b) The heat capacity $C_{P}$ as a function of $r_{h}$ with fixed $Q$=0.18. The blue solid line, green dashed line, and red dot dashed lines are for $P$=0.0060, 0.0066, and 0.0075, respectively.}
\end{figure}

\begin{figure}
\center{\subfigure[]{\label{Pd5Trh}
\includegraphics[width=8cm,height=6cm]{d5Trh_5a.eps}}
\subfigure[]{\label{Pd5GT}
\includegraphics[width=8cm,height=6cm]{d5GT_5b.eps}}}
\caption{In five-dimensional spacetime and $Q=0.18$, $P=$0.0060, 0.0066, and 0.0075 from bottom to top. The branches described by the dashed line have negative heat capacity $C_{P}$ indicating locally thermodynamically unstable. (a) Behavior of $T$ as a function of $r_{h}$. For $P<P_{c}$, there are one unstable and two stable branches. (b) The Gibbs free energy as a function of $T$. The characteristic swallow tail behavior appears for $P<P_{c}$. The horizon radius $r_{h}$ increases from left to right along the $G-T$ line.}\label{PPrhgt}
\end{figure}

\begin{figure}
\center{\includegraphics[width=8cm,height=6cm]{d5PT_6.eps}}
\caption{Phase diagram for the five-dimensional charged GB black hole with $Q=0.18$. The other value of $Q$ shares the same behavior of the phase diagram.}\label{Pd5PT}
\end{figure}

\section{The triple points and phase diagram when $d=6$}
\label{Phase2}

\begin{figure}
\center{\includegraphics[width=8cm,height=6cm]{d6PQ_7.eps}}
\caption{The critical point displayed in $P-Q$ plane in six-dimensional spacetime. The points $A$, $B$, $C$, and $D$ have coordinates (0.1914, 0.0197), (0.2018, 0.0188), (0.1705, 0.01976), and (0.1946, 0.0194). The blue, green, and red lines correspond to different critical points. These lines divide the $P-Q$ plane into four regions: I, II, III, and IV, in which the equation $(\partial_{r_{h}}T)_{P}=0$ has two roots, no root, two roots, and four roots. The black dashed line segment $CD$ denotes the triple point for different values of the charge $Q$, which is discussed in the text. The black arrows indicate increasing $r_{h}$.}
\label{Pd6PQ}
\end{figure}

\begin{figure}
\center{\subfigure[]{\label{Pd6q0gta}
\includegraphics[width=8cm,height=6cm]{d6q0gta_8a.eps}}
\subfigure[]{\label{Pd6q0gtb}
\includegraphics[width=8cm,height=6cm]{d6q0gtb_8b.eps}}}\\
\center{\subfigure[]{\label{Pd6q0gtc}
\includegraphics[width=8cm,height=6cm]{d6q0gtc_8c.eps}}
\subfigure[]{\label{Pd6q0gtd}
\includegraphics[width=6cm,height=5cm]{d6q0gtd_8d.eps}}}
\caption{Behavior of the Gibbs free energy $G$ in six-dimensional spacetime as a function of $T$ for the case of $Q=0.15<Q_{C}$.  The fixed pressure $P$ is taken as 0.0185, 0.0197, 0.03, and 0.035, respectively. The branches described by the black dashed lines are thermodynamically unstable. And these described by the blue, red, and green solid lines are stable ones. The critical points have the pressure $P_{c}=(0.01956, 0.01978, 0.03213)$. Other thermodynamic quantities at the critical points can be found in Table \ref{criticalparameters}. The horizon radius $r_{h}$ increases from left to right along the $G-T$ line.}\label{Pd6q0gtt}
\end{figure}

For a six-dimensional black hole, the temperature, Gibbs free energy, and heat capacity are
\begin{eqnarray}
 T&=&\frac{8\pi Pr_{h}^8+6r_{h}^6+2r_{h}^4-Q^2}{8\pi r_{h}^7+16\pi r_{h}^5},\\
 G&=&\frac{5Q^2\left(7r_{h}^2+20\right)-6r_{h}^4\left(4\pi Pr_{h}^6
     +(48\pi P-5) r_{h}^4+5r_{h}^2-20\right)}{480\pi r_{h}^3 \left(r_{h}^2+2\right)},\\
 C_{P}&=&\frac{r_{h}^2 \left(r_{h}^2+2\right)^2 \left(8\pi Pr_{h}^8-Q^2+6r_{h}^6
      +2r_{h}^4\right)}{2r_{h}^4\left(4\pi Pr_{h}^6+3(8\pi P-1)r_{h}^4
       +3r_{h}^2-2\right)+Q^2 \left(7 r_{h}^2+10\right)}.
\end{eqnarray}
For this case, the critical point can also be found with the condition (\ref{vdwcp3}). However, the situation is more subtle. In an appropriate range of the parameters, there are three critical points corresponding to SBH, IBH, and LBH, respectively. In the small charge limit, the universal relation for these critical points becomes,
\begin{eqnarray}
 \frac{P^{S}_{c}r^{S}_{hc}}{T^{S}_{c}}&=&\frac{\sqrt{5}}{16Q}-\frac{79}{48}
               +\frac{95839}{4608\sqrt{5}}Q-\frac{18057}{512}Q^{2}+\mathcal{O}(Q^{3}),\\
 \frac{P^{I}_{c}r^{I}_{hc}}{T^{I}_{c}}&=&\frac{1}{4}-\sqrt{\frac{11}{384}}Q
               -\frac{21}{64}Q^{2}+\mathcal{O}(Q^{3}),\\
  \frac{P^{L}_{c}r^{L}_{hc}}{T^{L}_{c}}&=&\frac{1}{4}+\sqrt{\frac{11}{384}}Q
               -\frac{21}{64}Q^{2}+\mathcal{O}(Q^{3}).
\end{eqnarray}
The behavior of the critical point is displayed in the $P-Q$ plane in Fig. \ref{Pd6PQ}. It behaves very different from that of the van der Waals fluid and the five-dimensional black hole case. There are four regions in the parameter space: I, II, III, and IV. In the regions I and III, $(\partial_{r_{h}}T)_{P}=0$ has two roots. In the region II, the equation has no root. And in the region IV, four roots will be obtained, which will result in a rich phase transition phenomena.

\begin{figure}
\center{\subfigure[]{\label{Pd6q0PQ}
\includegraphics[width=8cm,height=6cm]{d6q0PPT_9a.eps}}
\subfigure[]{\label{Pd6Trh}
\includegraphics[width=8cm,height=6cm]{d6Trh_9b.eps}}}
\caption{Phase diagram and behavior of $T$ as a function of $r_{h}$ for the six-dimensional charged GB-AdS black holes. (a) Phase diagram for the case $Q=0.15<Q_{C}$. The critical point is located at $(P_{c3}, T_{c3})=(0.03213, 0.11868)$, which can also be found in Table \ref{criticalparameters}. (b) Behavior of $T$ as a function of $r_{h}$ for fixed $Q_{C}<Q=0.18<Q_{D}$ and $P$=0.0185, 0.0195, 0.02, 0.022 from bottom to top. The black dashed lines have negative slopes and describe unstable branches. For small pressure, we clearly see there are stable SBH and LBH branches, and unstable IBH branch. With the increasing of the pressure, there appears a new stable IBH branch. As we further increase the pressure, this branch disappears.}
\end{figure}

\begin{figure}
\centerline{\subfigure[$P=0.0185$]{\label{Pd6gta}
\includegraphics[width=8cm,height=6cm]{d6gta_10a.eps}}
\subfigure[$P=0.0195$]{\label{Pd6gtb}
\includegraphics[width=8cm,height=6cm]{d6gtb_10b.eps}}}
\centerline{\subfigure[$P=0.019603$]{\label{Pd6gtc}
\includegraphics[width=8cm,height=6cm]{d6gtc_10c.eps}}
\subfigure[$P=0.01964$]{\label{Pd6gtd}
\includegraphics[width=8cm,height=6cm]{d6gtd_10d.eps}}}
\centerline{\subfigure[$P=0.02$]{\label{Pd6gte}
\includegraphics[width=8cm,height=6cm]{d6gte_10e.eps}}
\subfigure[$P=0.022$]{\label{Pd6gtf}
\includegraphics[width=8cm,height=6cm]{d6gtf_10f.eps}}}
\caption{Behavior of the Gibbs free energy $G$ as a function of $T$ for the six-dimensional charged GB-AdS black holes. The charge $Q$ is fixed as $Q_{C}<Q=0.18<Q_{D}$. The pressure $P$ is fixed as $P$=0.0185, 0.0195, 0.019603, 0.01964, 0.02, and 0.022. The unstable branches are described by black dashed lines. When $P<P_{t}$, there only exists the SBH/LBH phase transition. While when $P_{t}<P<P_{c2}$, there is the SBH/IBH/LBH phase transition. Thus $P=P_{t}$ is a triple point, at which the SBH, IBH, and LBH branches can coexist. The critical pressures are $(P_{c1},P_{c2},P_{c3})=(0.01927, 0.01974, 0.02155)$. And the pressure at the triple point is $P_{t}=0.01960$. The horizon radius $r_{h}$ increases from left to right along the $G-T$ line.}\label{Pd6gtt}
\end{figure}

According to the structure of the critical point displayed in Fig. \ref{Pd6PQ}, the following discussion will be restricted to fixed charges, $Q<Q_{C}$, $Q_{C}<Q<Q_{D}$, $Q_{D}<Q<Q_{B}$, and $Q_{B}<Q$.

\subsection{\textbf{Case 1}: \quad $Q=0.15<Q_{C}$}

First, we study the small charge $Q<Q_{C}$ case. Without loss of generality, we take $Q=0.15$. In this case, there are three critical points at $P_{c}$=(0.01956, 0.01978, 0.03213). The Gibbs free energy $G$ is plotted in Fig. \ref{Pd6q0gtt}. For $P<P_{c1}=0.01956$, there are three BH branches. The SBH and LBH branches (solid lines) are stable, while the IBH branch (dashed line) is unstable. There is one swallow tail, which implies that, with the increasing of $T$, the system will undergo a SBH/LBH phase transition with fixed $P$. When the pressure crosses $P_{c1}$, a new swallow tail behavior appears for the appearance of a new stable IBH branch described by the green solid line in Fig. \ref{Pd6q0gtb}. Since such branch has a larger value of $G$, it does not affect the phase transition. Thus there is still the SBH/LBH phase transition. Further increase the pressure such that $P_{c2}<P<P_{c3}$, the second swallow tail behavior disappears. And the SBH/LBH phase transition ends when the pressure $P$ approaches $P_{c3}$, at which the first-order phase transition turns to be a second-order one. The phase diagram is clearly shown in Fig. \ref{Pd6q0PQ}, and the result is consistent with the above analysis we given. We also clear that when the parameter locates in the region IV shown in Fig. \ref{Pd6PQ}, there are two swallow tails. Although the second one does not participate in the phase transition, it shows a rich structure of the $G-T$ line. And for the case 2, we will show that this two-swallow tail behavior will lead to a rich structure of phase transition, i.e., there exists a triple point and the SBH/IBH/LBH phase transition will occur in the proper range of the parameters.

\subsection{\textbf{Case 2}: \quad $Q_{C}<Q=0.18<Q_{D}$}

For the case of $Q=0.18$, which satisfies $Q_{C}<Q<Q_{D}$, the pressure $P$ has three critical points, i.e.,$(P_{c1},P_{c2},P_{c3})=(0.01927, 0.01974, 0.02155)$. We plot the temperature $T$ as a function of $r_{h}$ for $P=$0.0185, 0.0195, 0.02, and 0.022 (from bottom to top) in Fig. \ref{Pd6Trh}. The locally thermodynamically unstable branch is described by the dashed line. When $P<P_{c1}$, there are three branches, the stable SBH and LBH branches, and the unstable IBH branch. This situation is similar to the $d=5$ case, which implies that there exists a characteristic swallow tail behavior in the $G-T$ diagram, and a SBH/LBH phase transition will occur. Further increasing $P$ such that $P_{c1}<P<P_{c2}$, there appears a new stable IBH branch, which splits the unstable IBH branch into two, thus there are five branches along a $T-r_{h}$ line. Further increasing $P$, the stable IBH branch will disappear when $P_{2}<P<P_{3}$, and only one stable branch survives when $P>P_{c3}$. The corresponding Gibbs free energy is illustrated in Fig. \ref{Pd6gtt}. In the ranges $P<P_{c1}$ and $P_{c2}<P<P_{c3}$, it displays one characteristic swallow tail behavior. When $P>P_{3}$, there is no such behavior, and no phase transition occurs. However, in the range $P_{c1}<P<P_{c2}$ shown in Figs. \ref{Pd6gtc}-\ref{Pd6gtf}, the situation becomes more subtle. There appear three stable branches and two characteristic swallow tails, which imply a rich structure of the phase transition. For fixed $P$ satisfying $P_{c1}<P<P_{z}$, there is a SBH/LBH phase transition with the increasing of the temperature. As $P$ approaches $P_{z}$, we see that the three stable branches intersect in a unique point. At that point, three black hole phases (i.e., small, large and intermediate black holes) coexist together. Therefore, we observe a triple point characterized by $(P_{t}, T_{t})=(0.01960, 0.11226)$ at $Q=0.18$. Slight above this pressure, the system will emerge a standard SBH/IBH/LBH phase transition with the increase of $T$. And such phase transition disappears when $P_{c2}$ is approached. One thing worthwhile to note is that when the pressure $Q=Q_{A}$, the Gibbs free energy will encounter one swallow tail and two swallow tails with the increasing of $P$. And the two swallow tails simultaneously disappear at $P=P_{A}$.

The interesting phase diagram is displayed in Fig. \ref{Pd6PPTQ=0.18}. It is clear that there is a triple point, at which the small, large and intermediate black holes can coexist together. Below this pressure $P_{t}$, the system will undergo a SBH/LBH phase transition of first order with the increasing of $T$. Above $P_{t}$ and below $P_{c2}$, there will be a SBH/IBH/LBH phase first-order IBH/LBH phase transition ends at $P_{c2}$, at which the phase transition becomes the second order. And above the pressure $P_{t}$, the first-order SBH/IBH phase transition emerges and it terminates at critical $P_{c3}$. It is worthwhile to note that the critical point $(P_{c1}, T_{c1})$ is absent from this phase diagram. This is because that the first critical point measures the appearance of the second swallow tail, which does not participate in phase transition. Varying the charge $Q$, the triple point will be got, and it is shown in Fig. \ref{Pd6PQ} with the black dashed line $CD$. It is clear that the triple point is limited in a small range. The thermodynamic quantities at the triple point of different values of the charge $Q$ are listed in Table \ref{Triple}.

\begin{table}[h]
\begin{center}
\caption{The thermodynamic quantities at the triple point for different values of the charge $Q$.}\label{Triple}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
  \hline
  \hline

   $Q$ &$P$ & $r_{h}$ & $v$ & $H$ & $T$&$S$&$\Phi$&$G$ \\
  \hline
   0.173 &0.01971&1.39476&1.05566&0.34787&0.11236&2.89145&0.00169&0.02299\\
   0.178 &0.01963&1.39484&1.05597&0.34784&0.11230&2.89190&0.00174&0.02308\\
   0.183 &0.01956&1.39492&1.05628&0.34782&0.11224&2.89234&0.00179&0.02317\\
   0.188 &0.01948&1.39500&1.05658&0.34779&0.11219&2.89278&0.00184&0.02326\\
   0.193 &0.01940&1.39508&1.05687&0.34777&0.11213&2.89321&0.00189&0.02336\\
  \hline\hline
\end{tabular}
\end{center}
\end{table}

There are some notes for the critical point and the triple point. The critical point implies that the black hole branch and the swallow tail behavior appear or disappear in the $T-r_{h}$ line or $G-T$ line. It is a multiple root of $(\partial_{r_{h}}T)_{P}=0$. Thus, the heat capacity $C_{P}$ diverges at that point. However, the triple point shown here is not consistent with the root of $(\partial_{r_{h}}T)_{P}=0$. So at the triple point, the heat capacity $C_{P}$ is finite and positive.

\begin{figure}
\center{\includegraphics[width=8cm,height=6cm]{d6PPT_11.eps}}
\caption{Phase diagram for the six-dimensional charged GB-AdS black hole with $Q_{C}<Q=0.18<Q_{D}$. Another value of $Q\in(Q_{C}, Q_{D})$ shares the same behavior of the phase diagram.}\label{Pd6PPTQ=0.18}
\end{figure}

\subsection{\textbf{Case 3}: \quad $Q_{D}<Q=0.195<Q_{B}$}

For the case of $Q=0.195$ satisfying $Q_{D}<Q<Q_{B}$, the pressure $P$ also has three critical points, i.e., $(P_{c1},P_{c2},P_{c3})=(0.01900, 0.01933, 0.01972)$. The behavior of the temperature $T$ is similar to the case of $Q=0.18$. However, the behavior of the Gibbs free energy is different, which is depicted in Fig. \ref{Pd6q2gt}. When $P<P_{c1}$, there exists three black hole branches. The SBH and LBH branches are thermodynamically stable, while IBH branch is unstable. There exists a characteristic swallow tail for such case, indicating a SBH/LBH phase transition. When the pressure $P$ crosses the critical point $P_{c1}$, a new IBH branch appears, and there appears a second characteristic swallow tail. Further increasing the pressure such that $P_{c2}<P<P_{c3}$, the second swallow tail disappears. Finally, the first swallow tail fades out at $P=P_{c3}$. Since the stable IBH branch has a larger Gibbs free energy than the LBH branch at the same $P$ and $T$, it can be concluded that, in the range $0<P<P_{c3}$, we only observe the SBH/LBH phase transition with the increasing of $T$.

The phase diagram is illustrated in Fig. \ref{Pd6PPT}. At the coexistence line, the SBH and LBH phases have the same Gibbs free energy. Crossing this line with the increasing of $T$, the system will undergo a SBH/LBH phase transition of first order. The coexistence line terminates at a critical point $(T_{c3}, P_{c3})$, where the second-order phase transition occurs.

\begin{figure}
\center{\subfigure[]{\label{Pd6q2gta}
\includegraphics[width=8cm,height=6cm]{d6q2gta_12a.eps}}
\subfigure[]{\label{Pd6q2gtb}
\includegraphics[width=8cm,height=6cm]{d6q2gtb_12b.eps}}}\\
\subfigure[]{\label{Pd6q2gtc}
\includegraphics[width=8cm,height=6cm]{d6q2gtc_12c.eps}}
\subfigure[]{\label{Pd6q2gtd}
\includegraphics[width=8cm,height=6cm]{d6q2gtd_12d.eps}}
\caption{Behavior of the Gibbs free energy $G$ of the six-dimensional charged GB-AdS black hole as a function of $T$ for the case of $Q_{D}<Q=0.195<Q_{B}$. The pressure $P$ is fixed as $P$=0.0185, 0.0192, 0.0196, and 0.02. The unstable BH branches are described by the black dashed lines. For such case, only the SBH/LBH phase transition can be observed. The critical pressure is $(P_{c1},P_{c2},P_{c3})=(0.01900, 0.01933, 0.01972)$ . The horizon radius $r_{h}$ increases from left to right along the $G-T$ line.}\label{Pd6q2gt}
\end{figure}

\begin{figure}
\center{\includegraphics[width=8cm,height=6cm]{d6q2PPT_13.eps}}
\caption{Phase diagram for the six-dimensional charged GB-AdS black hole with $Q_{D}<Q=0.195<Q_{B}$. Other values of $Q$ in the range $Q\in(Q_{D}, Q_{B})$ share the same behavior of the phase diagram.}\label{Pd6PPT}
\end{figure}

\subsection{\textbf{Case 4}: \quad $Q_{B}<Q=0.3$}

There is only one critical point with $P_{c}=0.01956$ for the case of $Q=0.3(<Q_{B})$. It is quite similar to the van der Waals fluid. The SBH/LBH phase transition occurs for $P<P_{c}$ and it ends at $(T_{c}, P_{c})$, where the first-order phase transition becomes the second-order one, and the phase transition fades out for $P>P{c}$. The Gibbs free energy and the phase diagram are shown in Fig. \ref{Pd6q3gt}. This case is analogous to that of the van der Waals fluid.

\begin{figure}
\center{\subfigure[]{\label{Pd6q3gta}
\includegraphics[width=8cm,height=6cm]{d6q3gta_14a.eps}}
\subfigure[]{\label{Pd6q3PPT_12b}
\includegraphics[width=8cm,height=6cm]{d6q3PPT_14b.eps}}}
\caption{Gibbs free energy and phase diagram for the six-dimensional charged GB-AdS black hole with $Q_{B}<Q=0.3$. (a) Behavior of the Gibbs free energy for fixed $P$=0.0185, 0.01956, and 0.025 form left to right. The horizon radius $r_{h}$ increases from left to right along the $G-T$ line. (b) Phase diagram. Other values of $Q$ in the range $Q_{B}<Q$ share the same behavior of the phase diagram.}\label{Pd6q3gt}
\end{figure}

\begin{figure}
\center{\subfigure[]{\label{Pd7pq}
\includegraphics[width=8cm,height=6cm]{d7pq_15a.eps}}
\subfigure[]{\label{Pd7pt}
\includegraphics[width=8cm,height=6cm]{d7pt_15b.eps}}}
\caption{(a) The critical point displayed in $P-Q$ plane for $d$=7, 8, 9, and 10 from bottom to top. (b) The phase diagram for the charged GB-AdS black hole with $d$=7, 8, 9, and 10 from left to right with fixed $Q=0.18$. It is clear that they have the similar behavior of the critical points and phase diagram.}\label{Pd77}
\end{figure}

\section{The phase diagram when $d\geq7$}
\label{Phase3}

When the dimension of the black hole is larger than 7, one naturally expects a richer structure of the phase diagram. However, it is not the case. We plot the critical point in the $P-Q$ plane in Fig. \ref{Pd7pq} for $d$=7-10. For fixed $d$, $(\partial_{r_h}T)_{P}=0$ has two roots below the line. While above the line, it has no root. Thus, it is clear that the behavior is quiet similar to the van der Waals fluid. And after some detailed study, it shows that there only the SBH/LBH phase transition. The triple point is not observed. The phase diagram is clearly illustrated in Fig. \ref{Pd7pt}, where only the SBH/LBH phase transition occurs. The values of these critical points are displayed in Table \ref{criticalparameters}.

The results show that the triple point only exists for $Q_{C}<Q<Q_{D}$ in $d=6$ spacetime. So it seems that $d=6$ is an exception. However, it is worthwhile to note that we have set the coupling constant $\alpha=1$. If we let it freely vary, one may find that the triple point could exist in $d\neq 6$ spacetime. However, in our opinion, the expectation of the triple point with $d\neq 6$ is still to some extent extremely weak.

\begin{table}[h]
\begin{center}
\caption{The values of these critical parameters shown in this paper. Note that all the critical points shown here have positive temperature, so these cases are corresponded to black hole system rather a non-black hole system.}\label{criticalparameters}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
  \hline
  \hline

   $d$&$Q$ &$P$ & $r_{h}$ & $v$ & $H$ & $T$&$S$&$\Phi$&$G$ \\
  \hline
   5 &0.18 &0.00661&2.45821&9.12880&0.48082&0.06492&7.40091&0.00119&0.00037\\
      \hline
   6 &0.15 &0.01956&1.21313&0.52549&0.24906&0.11228&2.01315&0.00223&0.02303\\
     &     &0.01978&1.53448&1.70151&0.44343&0.11241&3.74069&0.00143&0.02295\\
     &     &0.03213&0.67641&0.52549&0.25578&0.11868&2.01315&0.00290&0.01687\\
   ~ &0.18 &0.01927&1.13426&0.37549&0.21392&0.11208&1.70034&0.00327&0.02333\\
     &     &0.01974&1.55453&1.81563&0.45861&0.11238&3.87652&0.00127&0.02297\\
     &     &0.02155&0.79278&0.06263&0.10495&0.11271&0.72726&0.00958&0.02298\\
   ~ &0.195&0.01900&1.06666&0.27617&0.18712&0.11193&1.46140&0.00426&0.02354\\
     &     &0.01933&0.87761&0.10412&0.12639&0.11203&0.91850&0.00765&0.02348\\
     &     &0.01972&1.56420&1.87280&0.46610&0.11236&3.94334&0.00135&0.02304\\
   ~ &0.3  &0.01956&1.62636&2.27567&0.51655&0.11217&4.39409&0.00185&0.02365\\
   \hline
   7 &0.18 &0.13225&0.73919&0.02719&0.08872&0.19038&0.39175&0.01199&0.01414\\
   \hline
   8 &0.18 &0.25874&0.77359&0.02368&0.09539&0.26462&0.32217&0.01034&0.01013\\ \hline
   9 &0.18 &0.40182&0.80394&0.02181&0.10533&0.33668&0.28934&0.00884&0.00791\\
    \hline
   10&0.18 &0.56449&0.82804&0.02033&0.11660&0.40754&0.27014&0.00767&0.00651\\
  \hline\hline
\end{tabular}
\end{center}
\end{table}

\section{Discussions and conclusions}
\label{Conclusion}

In this paper, we investigated the triple points and phase diagrams of the charged GB black holes in $d$-dimensional AdS space in the canonical (fixed charge $Q$) ensemble. We discussed this issue in the extended phase space with the cosmological constant treated as the pressure of the thermodynamic system and its conjugate quantity as the thermodynamic volume of the black holes.

We first dealt with the van der Waals liquid-gas system. By rewriting the equation of state into $T=T(v, P)$, we showed that the critical behavior and phase transition can also be equivalently determined by the $T-v$ line for fixed $P$. Then, by employing such treatment, we studied the phase diagram of the charged GB black holes in AdS space. It was shown that much information about the phase transition is encoded in the $T-r_{h}$ line, i.e., the divergence and sign of the heat capacity, critical point, number of the swallow tails of the Gibbs free energy, as well as the possible ranges where the SBH/LBH, SBH/IBH/LBH phase transitions take place. Thus, the $T-r_{h}$ line with fixed $P$ was shown to be a powerful tool to study the phase diagram of the black hole.

In this paper, we aimed to study the triple points and possible reentrant phase transitions in the charged GB-AdS black holes. And the method adopted here can be easily extended to other black holes. We summarize it as follows. First, plot the critical points determined by the condition (\ref{vdwcp3}) in the parameter space. Second, examine the number of the roots of $(\partial_{r_{h}}T)_{P}=0$ in each range of the parameter space. Using the number of the roots, we can get the possible type of the phase transition. For example, if there is only one root or no root, then where is no swallow tail behavior, and so no phase transition exists. And if there are two roots in some range of the parameter space, while no root in other range, then there exists the phase transition of the SBH/LBH type. If the number is larger than three, then there exists rich structure of the phase transition in some ranges of the parameter space, where the triple points and possible reentrant phase transitions may exist. Finally, combined with the behavior of the Gibbs free energy, the phase diagram will be obtained.

Using this method, we studied the phase diagram for the full range of the parameter charge $Q$. It was shown that the SBH/LBH phase transition exists in any dimension of $d$. However, in six-dimensional spacetime, the phase diagram is more subtle. In some range of the parameters, there are three critical points for a fixed charge $Q$, and $(\partial_{r_{h}}T)_{P}=0$ has four roots. In such range, the Gibbs free energy displays the behavior of two swallow tails. However, the triple point does not always exist in that range (i.e., in the range IV shown in Fig. \ref{Pd6PQ}). The triple point and the SBH/IBH/LBH phase transition are limited in the black dashed line $CD$ in Fig. \ref{Pd6PQ}. Thus we observe triple point and reentrant SBH/IBH/LBH phase transition in the charged GB-AdS black holes. However it seems that they are limited in a narrow range in parameter space.

Before closing the discussion, we need to stress another one issue. In this paper we only dealt with the case with fixed charge $Q$. One may wonder what happens when the pressure $P$ is fixed. In fact the situation is similar. This can be clearly interpreted from Fig. \ref{Pd6PQ} or other figures for the critical points displayed in $P-Q$ plane. The black hole system will encounter the similar structure of critical point for fixed $Q$ with the increasing of $P$ or for fixed $P$ with the increasing of $Q$. So the result is similar to the case with fixed charge $Q$.

We remark that in the charged GB-AdS black holes, we observed the existence of the triple point and the multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van der Waals type, analogous to the ``every day thermodynamics".

\section*{Acknowledgements}
This work was supported by the National Natural Science Foundation of China (Grants No. 11205074 and No. 11375075), and the Fundamental Research Funds for the Central Universities (Grants No. lzujbky-2013-18 and No. lzujbky-2013-21).

redpajama_set_name: RedPajamaArXiv


text:
Behind the Art of the Ten x Ten x Ten Collection
Ten X Ten X Ten
Tag Archives: abstract
"Cerulean Alpha Centauri", Contemporary Modern Series, No. 40
I returned to New York City a month later after landing a May solo show at the famed gallery "Pop International" in SOHO. I wanted to revisit the city not only to reconnect to the curator/owner over there, but to continue the inspiration in my head that seemed to be exploding back and forth like a pinball in rapture. I had already completed "Mr. Busy Head", the 2nd of Metropolitan series about New York City, but this time my visit was to focus on the Contemporary Modern series by revisiting the Whitney to see Steve Wheeler's work, as well as the Guggeheim's collection of Kandinsky and Picasso.
While in my friend Michael's apartment on the upper west side on Central Park West and 75th, I sat there with a partially finished abstract expressionist piece focusing on Steve Wheeler's "Laughing Boy Rolling", and became slightly bored with the fact that everything had already been drafted and color coded. I felt doing any more work on the piece, now that it had been laid out, would have just been filler. I decided to roll up the canvas and store it to finish in my studio in Los Feliz (Los Angeles, CA). I grabbed a new canvas and started staring blankly at the piece. I wanted to create something spontaneous…. something stream of consciousness that would be completely done in New York City.
I grabbed colors here and there and just started working on the piece which rested on a side table in the living room. I started thinking beyond just geometric shapes… (which is what I will ALWAYS initially gravitate towards). After laying a coat of warm colors down on the face, I decided to work on mixing the matte acrylic down and using the tip of my brush to flicker paint on the face.
At the same time I was doing this, I was hearing on NPR about the recent news about "Alpha Centauri" and this discovery of a new planet in its binary star system. While the broadcaster was talking about this, I began to parallel how stars in the sky always looked like Jackson Pollock masterpieces. I decided to tape up the Mickey and do a night sky with stars splattered against its surface in relation to the news. The sun rose and set in the New York City sky that day, and I don't think I ever left the apartment. While the taxi's honked and the people flittered about walking and talking below me in the city streets, I focused on creating a busy star splattered piece about Mickey Mouse.
This piece is titled "Cerulean Alpha Centauri" based off the galaxy like splatters, was primarily done with Holbein Matte Acrylic labeled "Cerulean Blue". This is number 45 in the collection, and falls under the "Contemporary Modern Series" for its Pollock like reference. Here is a little video I look about the process in New York City over this creation.
Tags: abstract, alpha centauri, arts, binary star system, disney, disney fine art, inspiration in my head, jackson pollack, mickeymouse, pop art, splatter, splatter art, video
Categories Contemporary Modern
"Abstract Space Invaders", Contemporary Modern Series, No. 39
While viewing the Yayoi Kusama retrospective and the "Fireflies on the Water" exhibit at The Whitney Museum in NYC, I had a chance to view their 2nd floor collection entitled "Signs and Symbols". In this room I got to see one of my favorite art pieces of all time called "Laughing Boy Rolling" by Steve Wheeler. Mesmerized, I spent 45 minutes walking back in forth in front of this painting, while simultaneously gazing here and there at work of Ted Faiers that hung nearby. I was in heaven primarily because I was seeing for the first time the original works of a rare art movement.
Steve Wheeler was an abstract expressionist that belonged to the sub group called the 'Indian Space Painters'. While the term is contemporarily deemed politically incorrect, and rightfully so, this subgroup actually paid a positive homage to the cultural style and heritage of Native American art. Steve Wheeler, while born in Slovakia, immigrated to America as a young infant by his parents who were searching for a better life. They resided in New Salem, Pennsylvania, where his father worked as a coal miner in the town. At 16, Wheeler found himself working in his father's footsteps in the same mines, and claims to have found his calling deep in the dark tunnels of his toiling labor. It was here that Wheeler heard the voice of an oracle deep in the mines telling him of his destiny to be an artist. Steve, now determined to reach this lofty goal, continued to work in the daytime as a coal miner, and educated himself at night as an artist in his small attic bedroom. He then later moved to New York City, and dedicated the first half of his life to being a painter. Wheeler's executions were geometrically sound and flawless, often going above and beyond in complexity of most cubist abstract expressionists at that time. His attention to detail combined with his clever merging of his influence of Northwest Native American art aesthetic made him a rising star within the small group of Indian Space Painters. He was known as an eccentric, often labeled as arrogant and sometimes even belligerent, but nonetheless a genuis. Wheeler spent the 2nd half of his life as a recluse and for 40 years until his death, he did not paint.
I'm gathering that my obsession over Steve Wheeler had half to do with his incredibly complex and detail oriented work, and half to do with his life and his behavior. The 45 minutes I spent grazing like an obsessed buffoon in front of his work was mostly due to my imaginings of him creating his pieces. I wondered what was going on in his head as he was creating this particular painting. and more importantly, how he felt when he completed it.
I flew back home to my studio in Los Feliz, and mulled around my drafting table for hours. I spent days drafting and redrafting my direction of this movement, consistently focusing and refocusing on my attempt to infuse Northwest Native American elements into an abstract expressionist piece. This is what emerged.
This piece is titled "Abstract Space Invaders" in homage to Steve Wheeler and what I'd like to politically correctly called the "Native American Space Painters". This is number 44 out of the collection and ties into the Contemporary Modern Series.
Tags: abstract, abstract art, abstract expressionism, abstract expressionist, abstract expressionists, art, arts, indian space painters, indian space painting, mickey mouse, new salem pennsylvania, painting, science, steve wheeler, symbols and signs, ted faiers, wheeler art, whitney museum
"Everything Is Going To Be Okay", Sketchwave Series, No. 14
My hands haven't been able to work on such an intimate and intricate level like this in almost a decade. I could feel it happen on No. 12, I could feel myself lose control on No. 13, and then on No.14, when the pattern erupted on my fingertips in the words "Here is Love, Love is Here", in that fluorescent lit 4th floor office in Echo Park at 4pm, … thats where this piece and the next few pieces began.
Every single drawing did not take place in my studio. Everything you see here was done outside of my normal working element. Each panel is a story written about the time that was happening at that very moment, what was in my head, and what was going on around me. Some of these panels are astral garbage working its way through my hands, some of these are thought out nostalgic executions, but the one thing they have in common, was that there was NO PLAN in how they were going to appear. This is pure stream of consciousness art forms carried through the diners, coffee shops, park benches, meeting places, restaurants, and libraries of Los Angeles.
This piece focuses on the following:
The Eye of Providence
Building it Higher
Cold Confectionary
Lightning Storms
Elementary French Lessons
Late Night Diner Atmosphere
The title "Everything Is Going To Be Okay" references the celebration of thought, no matter how intricate, overwhelming, and uncontrollable as it seems. It crosses the ideas of sight into touch, sound, taste, and scent.
Tags: abstract, diner, mickey mouse, stream of consciousness
Categories Sketchwave
Series and Content
Children of the 1980's


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text:
Numbers 5:2
"Command the Israelites to expel 1 from the camp every leper, 2 everyone who has a discharge, 3 and whoever becomes defiled by a corpse. 4
"Command the Israelites to send away from the camp anyone who has an infectious skin disease or a discharge of any kind, or who is ceremonially unclean because of a dead body.
"Command the sons of Israel that they send away from the camp every leper and everyone having a discharge and everyone who is unclean because of a dead person.
"Command the people of Israel to remove anyone from the camp who has a contagious skin disease or a discharge, or who has been defiled by touching a dead person.
"Command the People of Israel to ban from the camp anyone who has an infectious skin disease, anyone who has a discharge, and anyone who is ritually unclean from contact with a dead body.
Give orders to the children of Israel to put outside the tent-circle every leper, and anyone who has any sort of flow from his body, and anyone who is unclean from the touch of the dead;
Command the Israelites to put out of the camp everyone who is leprous, or has a discharge, and everyone who is unclean through contact with a corpse;
"Command the children of Israel that they put out of the camp every leper, everyone who has a discharge, and whoever becomes defiled by a corpse.
that they put out
of the camp
every leper
and every one that hath an issue
and whosoever is defiled
by the dead
"Command
the sons
that they send
from the camp
leper
and everyone
having a discharge
who is unclean
because of a dead person
spnl
ewru
hnxmh
wxlsyw
ynb
wu (5:2)
prostaxon
V-AAD-2S
toiv
T-DPM
uioiv
N-DPM
exaposteilatwsan
V-AAD-3P
T-GSF
parembolhv {N-GSF} panta
lepron
gonorruh {A-ASM} kai
akayarton
N-DSF
the Israelites
to expel
, everyone
who has a discharge
, and whoever
becomes defiled
by a corpse
1 tn The construction uses the Piel imperative followed by this Piel imperfect/jussive form; it is here subordinated to the preceding volitive, providing the content of the command. The verb שָׁלַח (shalakh) in this verbal stem is a strong word, meaning "expel, put out, send away, or release" (as in "let my people go").
2 sn The word צָרוּעַ (tsarua'), although translated "leper," does not primarily refer to leprosy proper (i.e., Hansen's disease). The RSV and the NASB continued the KJV tradition of using "leper" and "leprosy." More recent studies have concluded that the Hebrew word is a generic term covering all infectious skin diseases (including leprosy when that actually showed up). True leprosy was known and feared certainly by the time of Amos (ca. 760 b.c.). There is evidence that the disease was known in Egypt by 1500 b.c. So this term would include that disease in all probability. But in view of the diagnosis and healing described in Leviticus 13 and 14, the term must be broader. The whole basis for the laws of separation may be found in the book of Leviticus. The holiness of the Lord who dwelt among his people meant that a high standard was imposed on them for their living arrangements as well as access to the sanctuary. Anything that was corrupted, diseased, dying, or contaminated was simply not compatible with the holiness of God and was therefore excluded. This is not to say that it was treated as sin, or the afflicted as sinners. It simply was revealing – and safeguarding – the holiness of the Lord. It thus provided a revelation for all time that in the world to come nothing unclean will enter into the heavenly sanctuary. As the Apostle Paul says, we will all be changed from this corruptible body into one that is incorruptible (1 Cor 15:53). So while the laws of purity and holiness were practical for the immediate audience, they have far-reaching implications for theology. The purity regulations have been done away with in Christ – the problem is dealt with differently in the new covenant. There is no earthly temple, and so the separation laws are not in force. Wisdom would instruct someone with an infectious disease to isolate, however. But just because the procedure is fulfilled in Christ does not mean that believers today are fit for glory just as they are. On the contrary, they must be changed before going into his presence. In like manner the sacrifices have been done away in Christ – not what they covered. Sin is still sin, even though it is dealt with differently on this side of the cross. But the ritual and the regulations of the old covenant at Sinai have been fulfilled in Christ.
3 sn The rules of discharge (Lev 12 and 15) include everything from menstruation to chronic diseases (see G. Wyper, ISBE 1:947, as well as R. K. Harrison, Leviticus (TOTC), 158-66, and G. J. Wenham, Leviticus (NICOT), 217-25.
4 tn The word is נֶפֶשׁ (nefesh), which usually simply means "[whole] life," i.e., the soul in the body, the person. But here it must mean the corpse, the dead person, since that is what will defile (although it was also possible to become unclean by touching certain diseased people, such as a leper).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ALL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
TIP #19: Use the Study Dictionary to learn and to research all aspects of 20,000+ terms/words. [ALL]


text:
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'Better Call Saul' Fans Think One Scene Went Too Far
by Victoria Santiago December 8, 2021
written by Victoria Santiago December 8, 2021
A scene in Season 3, Episode 9 of "Better Call Saul" is being labeled as unnecessarily cruel and heartbreaking.
Jimmy McGill, played by Bob Odenkirk, is arguably one of the more interesting characters featured on "Breaking Bad." When the pilot episode for "Better Call Saul" opened with Odenkirk's character managing a Cinnabon, expectations for the show may have been low… but whatever preconceptions anyone had about the show were quickly thrown out the window.
We know Jimmy McGill for his seemingly spur-of-the-moment and often sleazy schemes. And his incredible ability to talk his way out of or into anything. We immediately dive into the consistently tragic backstory of Jimmy McGill, but should we root for him? Or against him? Watching Jimmy evolve into Saul Goodman has been a whirlwind rollercoaster filled with emotions, good and bad.
Jimmy McGill's Evil, Opportunistic Actions
Although Odenkirk's character is known for being quite evil at times, one scene has viewers in awe of his bad behavior. Season 3, Episode 9 is very hard to watch, viewers say. Titled "Fall," the episode focuses on what happens as the Sandpiper class-action lawsuit begins to sputter. Jimmy, desperate for money, begins to manipulate the representative of the Sandpiper lawsuit, an older woman named Irene. He sneaks his way into Irene's life and turns everyone against her.
Jimmy begins by finding Irene's walking group at the mall and gifts her a pair of expensive walking shoes. Irene, happy to have new shoes, is oblivious to the lies Jimmy begins to tell her friends. He convinces her friends that her new shoes are the result of the settlement money she is hiding from them. Of course, we know that there is no settlement money yet, but her friends don't.
Fast forward to a game of bingo – Jimmy offers to call out the numbers. Irene is there, as are her friends. Undoubtedly mad about the lies Jimmy told them earlier, they choose not to sit with Irene. Jimmy secretly gives Irene the winning bingo card. She wins, of course, and no one claps for her. Obviously upset and confused about why her friends are icing her out, Irene begins to cry and leaves the room.
A Win for Jimmy, a Loss for Those With Hearts
All of these bad things are a win for Jimmy. He follows Irene back to her room and tells her that if she takes the Sandpiper settlement, her friends will want to be around her again. After convincing her to take the money, Jimmy goes home to celebrate. What's more, Jimmy is so busy winning he doesn't even stop to think about how he has ruined Irene's life. Jimmy's actions show a breakdown of his morals and foreshadow his evolution into Saul Goodman.


After a career in video production, editing shows that aired nationwide on the SPEED Channel, I made the decision to go into the world of design. I knew I wanted to make products easier to use before I ever heard the term "user experience."
My journey took me to the Institute of Design, where I graduated with a Master's Degree in Design focusing on Research and Strategy.
Since then, I've been a UX Researcher and Designer focusing on making digital products and services. I've worked entirely in client services which has given me exposure to working with everybody from startups to Fortune 100 clients. I've worked on consumer-facing and enterprise/B2B applications. I've enjoyed the variety and ability to completely dive into a new topic with every project, immersing myself in an industry I probably didn't know existed previously (I've done 2 projects on pet insurance).
If you'd like to see my résumé, please download a PDF.