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保罗·埃伦费斯特Paul Ehrenfest根据热力学自由能和其他热力学变量的函数关系对相变进行了分类<ref name="ReferenceA">{{cite journal|last1=Jaeger|first1=Gregg|title=The Ehrenfest Classification of Phase Transitions: Introduction and Evolution|journal=Archive for History of Exact Sciences|date=1 May 1998|volume=53|issue=1|pages=51–81|doi=10.1007/s004070050021~~|s2cid=121525126~~}}</ref> 。根据他的方法，可以按照转变时的不连续自由能最低导数标记相变。'''一阶相变first-order phase transitions'''相对于某些热力学变量，具有自由能的一阶导数不连续性。<ref name = Blundell>{{Cite book | last = Blundell | first = Stephen J. |author2=Katherine M. Blundell | title = Concepts in Thermal Physics | publisher = Oxford University Press | year = 2008 | isbn = 978-0-19-856770-7}}</ref> 我们将各种固/液/气的转变都归为一阶相变，因为它们都涉及到密度的不连续变化——这是自由能相对于压力的一阶导数（一阶导数的逆函数）。而''' 二阶相变second-order phase transitions'''在一阶导数中是连续的（有序参数，即自由能相对于外部场的一阶导数，在整个转变过程中是连续的），但在自由能的二阶导数中表现出不连续性。比如'''铁磁相变ferromagnetic transition '''（发生在铁等材料中），其中磁化强度是自由能相对于施加磁场强度的一阶导数。随着温度降低到居里温度以下，磁化强度将从零开始持续增加。而磁化率，是自由能相对于磁场的二阶导数，它的变化是不连续的。以此类推，按照Ehrenfest的分类方法，原则上可以存在第三，第四甚至更高阶的相变。

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保罗·埃伦费斯特Paul Ehrenfest根据热力学自由能和其他热力学变量的函数关系对相变进行了分类<ref name="ReferenceA">{{cite journal|last1=Jaeger|first1=Gregg|title=The Ehrenfest Classification of Phase Transitions: Introduction and Evolution|journal=Archive for History of Exact Sciences|date=1 May 1998|volume=53|issue=1|pages=51–81|doi=10.1007/s004070050021}}</ref> 。根据他的方法，可以按照转变时的不连续自由能最低导数标记相变。'''一阶相变first-order phase transitions'''相对于某些热力学变量，具有自由能的一阶导数不连续性。<ref name = Blundell>{{Cite book | last = Blundell | first = Stephen J. |author2=Katherine M. Blundell | title = Concepts in Thermal Physics | publisher = Oxford University Press | year = 2008 | isbn = 978-0-19-856770-7}}</ref> 我们将各种固/液/气的转变都归为一阶相变，因为它们都涉及到密度的不连续变化——这是自由能相对于压力的一阶导数（一阶导数的逆函数）。而''' 二阶相变second-order phase transitions'''在一阶导数中是连续的（有序参数，即自由能相对于外部场的一阶导数，在整个转变过程中是连续的），但在自由能的二阶导数中表现出不连续性。比如'''铁磁相变ferromagnetic transition '''（发生在铁等材料中），其中磁化强度是自由能相对于施加磁场强度的一阶导数。随着温度降低到居里温度以下，磁化强度将从零开始持续增加。而磁化率，是自由能相对于磁场的二阶导数，它的变化是不连续的。以此类推，按照Ehrenfest的分类方法，原则上可以存在第三，第四甚至更高阶的相变。

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一阶相变是那些涉及潜伏热的相变。在这种相变过程中，系统会吸收或释放固定（通常是大量）的能量。在此过程中，即使热量增加，系统的温度也保持恒定：系统处于“混合相状态”，也就是说某些部分已完成转变，而其他部分尚未完成。<ref>Faghri, A., and Zhang, Y., [https://books.google.com/books?id=bxndY2KSuQsC&printsec=frontcover&dq=Transport+Phenomena+in+Multiphase+Systems&hl=en&ei=JJdqTIikDZLdngfY4fjxAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC8Q6AEwAA#v=onepage&q&f=false ''Transport Phenomena in Multiphase Systems''], Elsevier, Burlington, MA, 2006,</ref><ref>Faghri, A., and Zhang, Y., [https://www.springer.com/gp/book/9783030221362 ''Fundamentals of Multiphase Heat Transfer and Flow''], Springer, New York, NY, 2020</ref>常见的例子是冰的融化或水的沸腾（水不会立即变成蒸气，而是成为液态水和蒸气气泡的湍流混合物）。物理学家伊姆利 Imry和沃迪斯 Wortis研究发现，可以将''' 淬火无序quenched disorder'''视为一阶转变。即在有限的温度范围内完成相变，但是过冷或过热现象仍然存在，并且滞后仍然存在于热循环中。<ref>{{cite journal | last1 = Imry | first1 = Y. | last2 = Wortis | first2 = M. | year = 1979 | title = Influence of quenched impurities on first-order phase transitions| journal = Phys. Rev. B | volume = 19 | issue = 7| pages = 3580–3585 | doi=10.1103/physrevb.19.3580|bibcode = 1979PhRvB..19.3580I }}</ref><ref name="KumarPramanik2006">{{cite journal|last1=Kumar|first1=Kranti|last2=Pramanik|first2=A. K.|last3=Banerjee|first3=A.|last4=Chaddah|first4=P.|last5=Roy|first5=S. B.|last6=Park|first6=S.|last7=Zhang|first7=C. L.|last8=Cheong|first8=S.-W.|title=Relating supercooling and glass-like arrest of kinetics for phase separated systems: DopedCeFe2and(La,Pr,Ca)MnO3|journal=Physical Review B|volume=73|issue=18|pages=184435|year=2006|issn=1098-0121|doi=10.1103/PhysRevB.73.184435|arxiv = cond-mat/0602627 |bibcode = 2006PhRvB..73r4435K ~~|s2cid=117080049~~}}</ref><ref name="PasquiniDaroca2008">{{cite journal|last1=Pasquini|first1=G.|last2=Daroca|first2=D. Pérez|last3=Chiliotte|first3=C.|last4=Lozano|first4=G. S.|last5=Bekeris|first5=V.|title=Ordered, Disordered, and Coexistent Stable Vortex Lattices inNbSe2Single Crystals|journal=Physical Review Letters|volume=100|issue=24|pages=247003|year=2008|issn=0031-9007|doi=10.1103/PhysRevLett.100.247003|pmid=18643617|bibcode=2008PhRvL.100x7003P|arxiv=0803.0307}}</ref>

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一阶相变是那些涉及潜伏热的相变。在这种相变过程中，系统会吸收或释放固定（通常是大量）的能量。在此过程中，即使热量增加，系统的温度也保持恒定：系统处于“混合相状态”，也就是说某些部分已完成转变，而其他部分尚未完成。<ref>Faghri, A., and Zhang, Y., [https://books.google.com/books?id=bxndY2KSuQsC&printsec=frontcover&dq=Transport+Phenomena+in+Multiphase+Systems&hl=en&ei=JJdqTIikDZLdngfY4fjxAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC8Q6AEwAA#v=onepage&q&f=false ''Transport Phenomena in Multiphase Systems''], Elsevier, Burlington, MA, 2006,</ref><ref>Faghri, A., and Zhang, Y., [https://www.springer.com/gp/book/9783030221362 ''Fundamentals of Multiphase Heat Transfer and Flow''], Springer, New York, NY, 2020</ref>常见的例子是冰的融化或水的沸腾（水不会立即变成蒸气，而是成为液态水和蒸气气泡的湍流混合物）。物理学家伊姆利 Imry和沃迪斯 Wortis研究发现，可以将''' 淬火无序quenched disorder'''视为一阶转变。即在有限的温度范围内完成相变，但是过冷或过热现象仍然存在，并且滞后仍然存在于热循环中。<ref>{{cite journal | last1 = Imry | first1 = Y. | last2 = Wortis | first2 = M. | year = 1979 | title = Influence of quenched impurities on first-order phase transitions| journal = Phys. Rev. B | volume = 19 | issue = 7| pages = 3580–3585 | doi=10.1103/physrevb.19.3580|bibcode = 1979PhRvB..19.3580I }}</ref><ref name="KumarPramanik2006">{{cite journal|last1=Kumar|first1=Kranti|last2=Pramanik|first2=A. K.|last3=Banerjee|first3=A.|last4=Chaddah|first4=P.|last5=Roy|first5=S. B.|last6=Park|first6=S.|last7=Zhang|first7=C. L.|last8=Cheong|first8=S.-W.|title=Relating supercooling and glass-like arrest of kinetics for phase separated systems: DopedCeFe2and(La,Pr,Ca)MnO3|journal=Physical Review B|volume=73|issue=18|pages=184435|year=2006|issn=1098-0121|doi=10.1103/PhysRevB.73.184435|arxiv = cond-mat/0602627 |bibcode = 2006PhRvB..73r4435K }}</ref><ref name="PasquiniDaroca2008">{{cite journal|last1=Pasquini|first1=G.|last2=Daroca|first2=D. Pérez|last3=Chiliotte|first3=C.|last4=Lozano|first4=G. S.|last5=Bekeris|first5=V.|title=Ordered, Disordered, and Coexistent Stable Vortex Lattices inNbSe2Single Crystals|journal=Physical Review Letters|volume=100|issue=24|pages=247003|year=2008|issn=0031-9007|doi=10.1103/PhysRevLett.100.247003|pmid=18643617|bibcode=2008PhRvL.100x7003P|arxiv=0803.0307}}</ref>

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在有限的温度范围内，胶体粒子的凝胶化转变显示为bA紊乱-扩展的一阶转变。随着温度降低，'''低温平衡相low-temperature equilibrium phase'''的分数从零增加到一（100％）。随温度变化而变化的馏分共存创造了许多有趣的可能性。比如在冷却时，一些液体会逐渐玻璃化，而不是转变为'''平衡晶相equilibrium crystal phase'''。这种情况往往发生在冷却速率比临界冷却速率快的时候——分子运动变得十分缓慢，以至于分子无法重新排列到晶体位置。<ref>{{cite journal | year = 1995 | title = Metallic Glasses| journal = Science | volume = 267 | issue = 5206| pages = 1947–1953 |bibcode = 1995Sci...267.1947G |doi = 10.1126/science.267.5206.1947 | pmid = 17770105| last1 = Greer| first1 = A. L.~~| s2cid = 220105648~~}}</ref>分子运动的减速通常发生在气温降至玻璃的形成温度'T''g以下时——这可能需要外部施加压力来实现。<ref name="J. Non-Cryst 2013"/><ref>{{cite journal | last1 = Tarjus | first1 = G. | year = 2007 | title = Materials science: Metal turned to glass| journal = Nature | volume = 448 | issue = 7155| pages = 758–759 | doi=10.1038/448758a| pmid = 17700684 |bibcode = 2007Natur.448..758T ~~| s2cid = 4410586~~ }}</ref> 如果'T''g落在该一阶冻结相变发生的特定温度范围内，一种有趣的现象就会发生，即当转变不完整时该转变会停止。同理可以考虑在低温下被阻止的一阶磁相变，我们可以观察到不完全的磁相变，即两个磁相同时存在直至到达最低温度。自关于铁磁到反铁磁相变的报道首次公开以来，<ref name="ManekarChaudhary2001">{{cite journal |last1=Manekar |first1=M. A. |last2=Chaudhary |first2=S. |last3=Chattopadhyay |first3=M. K. |last4=Singh |first4=K. J. |last5=Roy |first5=S. B. |last6=Chaddah |first6=P. |title=First-order transition from antiferromagnetism to ferromagnetism inCe(Fe0.96Al0.04)2 |journal=Physical Review B |volume=64 |issue=10 |pages=104416 |year=2001 |issn=0163-1829 |doi=10.1103/PhysRevB.64.104416 |arxiv=cond-mat/0012472 |bibcode=2001PhRvB..64j4416M~~|s2cid=16851501~~ }}</ref> 人们逐渐发现了各种一阶磁相变的持久相共存现象。包括'''庞磁电阻锰矿材料colossal-magnetoresistance manganite materials'''<ref>{{cite journal|doi=10.1088/0953-8984/18/49/L02|arxiv = cond-mat/0611152 |bibcode = 2006JPCM...18L.605B |title = Coexisting tunable fractions of glassy and equilibrium long-range-order phases in manganites |journal = Journal of Physics: Condensed Matter |volume = 18 |issue = 49 |pages = L605 |year = 2006 |last1 = Banerjee |first1 = A. |last2 = Pramanik |first2 = A. K. |last3 = Kumar |first3 = Kranti |last4 = Chaddah |first4 = P. |~~s2cid = 98145553~~ }}</ref><ref>{{cite journal |authors = Wu W., Israel C., Hur N., Park S., Cheong S. W., de Lozanne A. | year = 2006 | title = Magnetic imaging of a supercooling glass transition in a weakly disordered ferromagnet| journal = Nature Materials | volume = 5 | issue = 11| pages = 881–886 |bibcode = 2006NatMa...5..881W |doi = 10.1038/nmat1743 | pmid = 17028576 ~~| s2cid = 9036412~~ }}</ref>、'''磁制冷材料magnetocaloric materials'''<ref name="RoyChattopadhyay2006">{{cite journal |last1=Roy |first1=S. B. |last2=Chattopadhyay |first2=M. K. |last3=Chaddah |first3=P. |last4=Moore |first4=J. D. |last5=Perkins |first5=G. K. |last6=Cohen |first6=L. F. |last7=Gschneidner |first7=K. A. |last8=Pecharsky |first8=V. K. |title=Evidence of a magnetic glass state in the magnetocaloric material Gd5Ge4 |journal=Physical Review B |volume=74 |issue=1 |pages=012403 |year=2006 |issn=1098-0121 |doi=10.1103/PhysRevB.74.012403 |bibcode = 2006PhRvB..74a2403R }}</ref> 、'''磁性形状记忆材料magnetic shape memory materials'''<ref name="LakhaniBanerjee2012">{{cite journal |last1=Lakhani |first1=Archana |last2=Banerjee |first2=A. |last3=Chaddah |first3=P. |last4=Chen |first4=X. |last5=Ramanujan |first5=R. V. |title=Magnetic glass in shape memory alloy: Ni45Co5Mn38Sn12 |journal=Journal of Physics: Condensed Matter |volume=24 |issue=38 |year=2012 |pages=386004 |issn=0953-8984 |doi=10.1088/0953-8984/24/38/386004 |pmid=22927562 |arxiv = 1206.2024 |bibcode = 2012JPCM...24L6004L ~~|s2cid=206037831~~ }}</ref> and other materials.<ref name="KushwahaLakhani2009">{{cite journal |last1=Kushwaha |first1=Pallavi |last2=Lakhani |first2=Archana |last3=Rawat |first3=R. |last4=Chaddah |first4=P. |title=Low-temperature study of field-induced antiferromagnetic-ferromagnetic transition in Pd-doped Fe-Rh |journal=Physical Review B |volume=80 |issue=17 |pages=174413 |year=2009 |issn=1098-0121 |doi=10.1103/PhysRevB.80.174413 |arxiv=0911.4552 |bibcode=2009PhRvB..80q4413K~~|s2cid=119165221~~ }}</ref>等。当''T''g 落在相变发生的温度范围内时，观测结果非常有趣，其一阶磁相变受到了磁场的影响——就像结构相变会受到压力影响一样。不过与压力相比，控制磁场相对容易，这大大提高了研究者们运用穷举法研究''T''g 和''T''c之间相互作用的成功率。一阶磁相变的相位共存将有助于解决和玻璃有关的一系列突出问题。

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在有限的温度范围内，胶体粒子的凝胶化转变显示为bA紊乱-扩展的一阶转变。随着温度降低，'''低温平衡相low-temperature equilibrium phase'''的分数从零增加到一（100％）。随温度变化而变化的馏分共存创造了许多有趣的可能性。比如在冷却时，一些液体会逐渐玻璃化，而不是转变为'''平衡晶相equilibrium crystal phase'''。这种情况往往发生在冷却速率比临界冷却速率快的时候——分子运动变得十分缓慢，以至于分子无法重新排列到晶体位置。<ref>{{cite journal | year = 1995 | title = Metallic Glasses| journal = Science | volume = 267 | issue = 5206| pages = 1947–1953 |bibcode = 1995Sci...267.1947G |doi = 10.1126/science.267.5206.1947 | pmid = 17770105| last1 = Greer| first1 = A. L.}}</ref>分子运动的减速通常发生在气温降至玻璃的形成温度'T''g以下时——这可能需要外部施加压力来实现。<ref name="J. Non-Cryst 2013"/><ref>{{cite journal | last1 = Tarjus | first1 = G. | year = 2007 | title = Materials science: Metal turned to glass| journal = Nature | volume = 448 | issue = 7155| pages = 758–759 | doi=10.1038/448758a| pmid = 17700684 |bibcode = 2007Natur.448..758T }}</ref> 如果'T''g落在该一阶冻结相变发生的特定温度范围内，一种有趣的现象就会发生，即当转变不完整时该转变会停止。同理可以考虑在低温下被阻止的一阶磁相变，我们可以观察到不完全的磁相变，即两个磁相同时存在直至到达最低温度。自关于铁磁到反铁磁相变的报道首次公开以来，<ref name="ManekarChaudhary2001">{{cite journal |last1=Manekar |first1=M. A. |last2=Chaudhary |first2=S. |last3=Chattopadhyay |first3=M. K. |last4=Singh |first4=K. J. |last5=Roy |first5=S. B. |last6=Chaddah |first6=P. |title=First-order transition from antiferromagnetism to ferromagnetism inCe(Fe0.96Al0.04)2 |journal=Physical Review B |volume=64 |issue=10 |pages=104416 |year=2001 |issn=0163-1829 |doi=10.1103/PhysRevB.64.104416 |arxiv=cond-mat/0012472 |bibcode=2001PhRvB..64j4416M}}</ref> 人们逐渐发现了各种一阶磁相变的持久相共存现象。包括'''庞磁电阻锰矿材料colossal-magnetoresistance manganite materials'''<ref>{{cite journal|doi=10.1088/0953-8984/18/49/L02|arxiv = cond-mat/0611152 |bibcode = 2006JPCM...18L.605B |title = Coexisting tunable fractions of glassy and equilibrium long-range-order phases in manganites |journal = Journal of Physics: Condensed Matter |volume = 18 |issue = 49 |pages = L605 |year = 2006 |last1 = Banerjee |first1 = A. |last2 = Pramanik |first2 = A. K. |last3 = Kumar |first3 = Kranti |last4 = Chaddah |first4 = P. |}}</ref><ref>{{cite journal |authors = Wu W., Israel C., Hur N., Park S., Cheong S. W., de Lozanne A. | year = 2006 | title = Magnetic imaging of a supercooling glass transition in a weakly disordered ferromagnet| journal = Nature Materials | volume = 5 | issue = 11| pages = 881–886 |bibcode = 2006NatMa...5..881W |doi = 10.1038/nmat1743 | pmid = 17028576 }}</ref>、'''磁制冷材料magnetocaloric materials'''<ref name="RoyChattopadhyay2006">{{cite journal |last1=Roy |first1=S. B. |last2=Chattopadhyay |first2=M. K. |last3=Chaddah |first3=P. |last4=Moore |first4=J. D. |last5=Perkins |first5=G. K. |last6=Cohen |first6=L. F. |last7=Gschneidner |first7=K. A. |last8=Pecharsky |first8=V. K. |title=Evidence of a magnetic glass state in the magnetocaloric material Gd5Ge4 |journal=Physical Review B |volume=74 |issue=1 |pages=012403 |year=2006 |issn=1098-0121 |doi=10.1103/PhysRevB.74.012403 |bibcode = 2006PhRvB..74a2403R }}</ref> 、'''磁性形状记忆材料magnetic shape memory materials'''<ref name="LakhaniBanerjee2012">{{cite journal |last1=Lakhani |first1=Archana |last2=Banerjee |first2=A. |last3=Chaddah |first3=P. |last4=Chen |first4=X. |last5=Ramanujan |first5=R. V. |title=Magnetic glass in shape memory alloy: Ni45Co5Mn38Sn12 |journal=Journal of Physics: Condensed Matter |volume=24 |issue=38 |year=2012 |pages=386004 |issn=0953-8984 |doi=10.1088/0953-8984/24/38/386004 |pmid=22927562 |arxiv = 1206.2024 |bibcode = 2012JPCM...24L6004L }}</ref> and other materials.<ref name="KushwahaLakhani2009">{{cite journal |last1=Kushwaha |first1=Pallavi |last2=Lakhani |first2=Archana |last3=Rawat |first3=R. |last4=Chaddah |first4=P. |title=Low-temperature study of field-induced antiferromagnetic-ferromagnetic transition in Pd-doped Fe-Rh |journal=Physical Review B |volume=80 |issue=17 |pages=174413 |year=2009 |issn=1098-0121 |doi=10.1103/PhysRevB.80.174413 |arxiv=0911.4552 |bibcode=2009PhRvB..80q4413K }}</ref>等。当''T''g 落在相变发生的温度范围内时，观测结果非常有趣，其一阶磁相变受到了磁场的影响——就像结构相变会受到压力影响一样。不过与压力相比，控制磁场相对容易，这大大提高了研究者们运用穷举法研究''T''g 和''T''c之间相互作用的成功率。一阶磁相变的相位共存将有助于解决和玻璃有关的一系列突出问题。

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人们曾普遍认为，不管温度在临界温度之上还是临界温度之下，临界指数保持不变。但是现实证明这不一定正确：当连续对称属性因不相关的'''各向异性anisotropies'''（在'''重整化群理论renormalization group'''意义上）而分解为离散对称属性时，某些指数（例如<math>\gamma </math>，''' 磁化率指数Exponent of the susceptibility'''）会有所不同。<ref>{{cite journal |last1=Leonard |first1=F. |last2=Delamotte |first2=B. |year = 2015 |title=Critical exponents can be different on the two sides of a transition | journal = Phys. Rev. Lett. | volume = 115 | issue = 20| page = 200601 | arxiv = 1508.07852 |bibcode = 2015PhRvL.115t0601L | doi = 10.1103/PhysRevLett.115.200601 |pmid=26613426~~|s2cid=22181730~~ }}</ref>

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人们曾普遍认为，不管温度在临界温度之上还是临界温度之下，临界指数保持不变。但是现实证明这不一定正确：当连续对称属性因不相关的'''各向异性anisotropies'''（在'''重整化群理论renormalization group'''意义上）而分解为离散对称属性时，某些指数（例如<math>\gamma </math>，''' 磁化率指数Exponent of the susceptibility'''）会有所不同。<ref>{{cite journal |last1=Leonard |first1=F. |last2=Delamotte |first2=B. |year = 2015 |title=Critical exponents can be different on the two sides of a transition | journal = Phys. Rev. Lett. | volume = 115 | issue = 20| page = 200601 | arxiv = 1508.07852 |bibcode = 2015PhRvL.115t0601L | doi = 10.1103/PhysRevLett.115.200601 |pmid=26613426 }}</ref>

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当<math>-1<α<0</math>时，热容在相变温度下具有“扭结”性质。这是'''液氦liquid helium'''从正常状态到超流体状态的<math>\gamma </math>相变行为，在此实验中发现{{mvar|α}}= -0.013±0.003。为最小化样品中的压力差，至少有一次实验在轨道卫星的零重力条件下进行<ref>{{cite journal | doi=10.1103/PhysRevB.68.174518 | title=Specific heat of liquid helium in zero gravity very near the lambda point | year=2003 | last1=Lipa | first1=J. | last2=Nissen | first2=J. | last3=Stricker | first3=D. | last4=Swanson | first4=D. | last5=Chui | first5=T. | journal=Physical Review B | volume=68 | issue=17| pages=174518 |arxiv = cond-mat/0310163 |bibcode = 2003PhRvB..68q4518L ~~| s2cid=55646571~~ }}</ref> 。α的这个实验值与基于'''变分微扰理论variational perturbation theory'''的预测值相符。<ref>{{cite journal | doi=10.1103/PhysRevD.60.085001 | title=Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions | year=1999 | last1=Kleinert | first1=Hagen | journal=Physical Review D | volume=60 | issue=8| pages=085001 |arxiv = hep-th/9812197 |bibcode = 1999PhRvD..60h5001K }}</ref>

+

当<math>-1<α<0</math>时，热容在相变温度下具有“扭结”性质。这是'''液氦liquid helium'''从正常状态到超流体状态的<math>\gamma </math>相变行为，在此实验中发现{{mvar|α}}= -0.013±0.003。为最小化样品中的压力差，至少有一次实验在轨道卫星的零重力条件下进行<ref>{{cite journal | doi=10.1103/PhysRevB.68.174518 | title=Specific heat of liquid helium in zero gravity very near the lambda point | year=2003 | last1=Lipa | first1=J. | last2=Nissen | first2=J. | last3=Stricker | first3=D. | last4=Swanson | first4=D. | last5=Chui | first5=T. | journal=Physical Review B | volume=68 | issue=17| pages=174518 |arxiv = cond-mat/0310163 |bibcode = 2003PhRvB..68q4518L }}</ref> 。α的这个实验值与基于'''变分微扰理论variational perturbation theory'''的预测值相符。<ref>{{cite journal | doi=10.1103/PhysRevD.60.085001 | title=Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions | year=1999 | last1=Kleinert | first1=Hagen | journal=Physical Review D | volume=60 | issue=8| pages=085001 |arxiv = hep-th/9812197 |bibcode = 1999PhRvD..60h5001K }}</ref>

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相变在生物系统中也具有重要的作用。比如''' 脂质双层lipid bilayer'''的形成，''' 蛋白质折叠protein folding'''和''' DNA解链DNA melting'''过程中的''' 坍塌转变coil–globule transition'''，''' DNA缩合DNA condensation'''过程中的液晶转变，以及具有相变特征的配体与DNA和蛋白质的结合。<ref>{{cite journal | doi=10.1080/07391102.2000.10506578 | pmid=10798534 | title=Long-range interactions between ligands bound to a DNA molecule give rise to adsorption with the character of phase transition of the first kind | year=2000| author = D.Y. Lando and V.B. Teif| journal=J. Biomol. Struct. Dynam. | volume=17 | issue=5 | pages=903–911 ~~| s2cid=23837885~~ }}</ref>

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相变在生物系统中也具有重要的作用。比如''' 脂质双层lipid bilayer'''的形成，''' 蛋白质折叠protein folding'''和''' DNA解链DNA melting'''过程中的''' 坍塌转变coil–globule transition'''，''' DNA缩合DNA condensation'''过程中的液晶转变，以及具有相变特征的配体与DNA和蛋白质的结合。<ref>{{cite journal | doi=10.1080/07391102.2000.10506578 | pmid=10798534 | title=Long-range interactions between ligands bound to a DNA molecule give rise to adsorption with the character of phase transition of the first kind | year=2000| author = D.Y. Lando and V.B. Teif| journal=J. Biomol. Struct. Dynam. | volume=17 | issue=5 | pages=903–911 }}</ref>

第256行：第256行： −

曾经有观点认为生物系统可能位于临界点附近。例如蝾螈视网膜中的神经网络<ref>{{cite arXiv|eprint=1407.5946|last1=Tkacik|first1=Gasper|title=Thermodynamics for a network of neurons: Signatures of criticality|last2=Mora|first2=Thierry|last3=Marre|first3=Olivier|last4=Amodei|first4=Dario|last5= Berry II|first5=Michael J.|last6=Bialek|first6=William|class=q-bio.NC|year=2014}}</ref> bird flocks<ref>{{cite journal|last1=Bialek|first1=W|last2=Cavagna|first2=A|last3=Giardina|first3=I|title = Social interactions dominate speed control in poising natural flocks near criticality|journal=PNAS|volume=111|issue=20|pages=7212–7217|year = 2014|bibcode=2014PNAS..111.7212B|doi=10.1073/pnas.1324045111|pmid=24785504|pmc=4034227|arxiv=1307.5563}}</ref>，果蝇中的鸟群基因表达网络和'''<ref>{{cite journal | last1=Krotov |first1=D|last2=Dubuis|first2=J O|last3=Gregor|first3=T|last4=Bialek|first4=W|title = Morphogenesis at criticality|journal = PNAS|year = 2014|doi=10.1073/pnas.1324186111|pmid=24516161|pmc=3956198|volume=111|issue=10|pages=3683–3688|arxiv=1309.2614|bibcode=2014PNAS..111.3683K}}</ref> 蛋白质折叠protein folding'''<ref>{{cite journal|last1=Mora|first1=Thierry|last2=Bialek|first2=William|title = Are biological systems poised at criticality?|journal = Journal of Statistical Physics|volume=144|issue=2|pages=268–302|year = 2011 |arxiv=1012.2242 |doi= 10.1007/s10955-011-0229-4|bibcode=2011JSP...144..268M~~|s2cid=703231~~}}</ref> 。但是，尚不清楚替代原因是否可以解释某些现象来支持关键性论证。另一个观点认为，生物有机体具有两个重要的相变特性：宏观行为的变化和系统在临界点的一致性。<ref>{{cite journal |last1=Schwab|first1=David J|last2=Nemenman|first2=Ilya|last3=Mehta|first3=Pankaj|title = Zipf's law and criticality in multivariate data without fine-tuning|journal = Physical Review Letters|volume=113|issue=6|pages=068102|year = 2014 |arxiv=1310.0448 |bibcode= 2014PhRvL.113f8102S|doi= 10.1103/PhysRevLett.113.068102|pmid=25148352|pmc=5142845}}</ref> It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point.<ref>{{Cite journal|last1=Longo|first1=G.|last2=Montévil|first2=M.|date=2011-08-01|title=From physics to biology by extending criticality and symmetry breakings|url=https://www.academia.edu/23155991|journal=Progress in Biophysics and Molecular Biology|series=Systems Biology and Cancer|volume=106|issue=2|pages=340–347|doi=10.1016/j.pbiomolbio.2011.03.005|pmid=21419157|arxiv=1103.1833~~|s2cid=723820~~}}</ref>

+

曾经有观点认为生物系统可能位于临界点附近。例如蝾螈视网膜中的神经网络<ref>{{cite arXiv|eprint=1407.5946|last1=Tkacik|first1=Gasper|title=Thermodynamics for a network of neurons: Signatures of criticality|last2=Mora|first2=Thierry|last3=Marre|first3=Olivier|last4=Amodei|first4=Dario|last5= Berry II|first5=Michael J.|last6=Bialek|first6=William|class=q-bio.NC|year=2014}}</ref> bird flocks<ref>{{cite journal|last1=Bialek|first1=W|last2=Cavagna|first2=A|last3=Giardina|first3=I|title = Social interactions dominate speed control in poising natural flocks near criticality|journal=PNAS|volume=111|issue=20|pages=7212–7217|year = 2014|bibcode=2014PNAS..111.7212B|doi=10.1073/pnas.1324045111|pmid=24785504|pmc=4034227|arxiv=1307.5563}}</ref>，果蝇中的鸟群基因表达网络和'''<ref>{{cite journal | last1=Krotov |first1=D|last2=Dubuis|first2=J O|last3=Gregor|first3=T|last4=Bialek|first4=W|title = Morphogenesis at criticality|journal = PNAS|year = 2014|doi=10.1073/pnas.1324186111|pmid=24516161|pmc=3956198|volume=111|issue=10|pages=3683–3688|arxiv=1309.2614|bibcode=2014PNAS..111.3683K}}</ref> 蛋白质折叠protein folding'''<ref>{{cite journal|last1=Mora|first1=Thierry|last2=Bialek|first2=William|title = Are biological systems poised at criticality?|journal = Journal of Statistical Physics|volume=144|issue=2|pages=268–302|year = 2011 |arxiv=1012.2242 |doi= 10.1007/s10955-011-0229-4|bibcode=2011JSP...144..268M}}</ref> 。但是，尚不清楚替代原因是否可以解释某些现象来支持关键性论证。另一个观点认为，生物有机体具有两个重要的相变特性：宏观行为的变化和系统在临界点的一致性。<ref>{{cite journal |last1=Schwab|first1=David J|last2=Nemenman|first2=Ilya|last3=Mehta|first3=Pankaj|title = Zipf's law and criticality in multivariate data without fine-tuning|journal = Physical Review Letters|volume=113|issue=6|pages=068102|year = 2014 |arxiv=1310.0448 |bibcode= 2014PhRvL.113f8102S|doi= 10.1103/PhysRevLett.113.068102|pmid=25148352|pmc=5142845}}</ref> It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point.<ref>{{Cite journal|last1=Longo|first1=G.|last2=Montévil|first2=M.|date=2011-08-01|title=From physics to biology by extending criticality and symmetry breakings|url=https://www.academia.edu/23155991|journal=Progress in Biophysics and Molecular Biology|series=Systems Biology and Cancer|volume=106|issue=2|pages=340–347|doi=10.1016/j.pbiomolbio.2011.03.005|pmid=21419157|arxiv=1103.1833}}</ref>

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