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奇異星 | 第三個量子效應是由於希格斯玻色子的出現，相關的連鎖反應導致真空極化效應的出現，奇異物質團並非如同原有理論所稱的具備極長久的穩定性，一部分連鎖反應當可成為夸克輕子的轉換來源，解釋了夸克星當中的弱作用力輕子生成源，而虛粒子對的出現使得夸克星某些局部可以進入比奇異物質更高密度的物質狀態，因此邁向更高的星體密度，向下一步更高密度星體演化的理論可以得到完善的解釋，亦即比夸克星更高密度的星體應當是理論上可以存在的。
希格斯玻色子應當也是夸克星星震的主要貢獻來源之一。
### 弦理論的推論
弦理論是另外一個極為有效的推論方法，但是由於弦理論的預測至今為止沒有任何一項被驗證，因此學界普遍不採信。這包含使用快子場論、量子蟲洞、弦場論（Torsion Field Theory a.k.a. Einstein–Cartan–Sciama–Kibble theory）及引力磁性進行分析的學派，學界一概不採信，因為物理學是以實驗為基礎的科學，而這些理論並不符合必備條件，只能被視為非主流詮釋，可信度還有很高的爭議。
#### 廣義相對論導致大批矛盾
廣義相對論在夸克星模型上導致大批矛盾及錯誤，學界目前都採用半古典修正來避開問題，在夸克星模型研究方面，假定廣義相對論是一個錯誤的理論已經是完全必要，特別是勞倫茲變換在高能階區域的處理方式，產生了無數的爭議，研究夸克星模型必須尋求廣義相對論替代方案來解決矛盾問題，目前最有希望解決矛盾問題的理論是霍拉瓦重力。
### 理論不完備，涉及未知的物理
夸克星相關理論發展並不完備，涉及未知的物理，實驗目前無法驗證理論，需要使用尚未驗證的理論進行推導，「夸克星」性質依然有待更多研究探討其機制，並且逐步以實驗驗證每項物理反應機制的真實性來確認其真實可能存在性。
依循漸進的物理驗證法，首先必須由高能物理證實 H 雙重子的真實存在性，其後還必須驗證奇異物質的奇異性真實存在性，此二者直接關係夸克星理論是否確立無誤，這兩項環節目前都還屬於當前高能物理當中的重大未解課題，目前的實驗都還不支持其真實性。此外，還必須透過實驗觀測多夸克物質的相變問題，多夸克物理當中，除去「奇異態」以外，至少還存在「孤子態」，兩種狀態理論上而言會發生相變，此外還有多夸克物質形成複合玻色子所產生的量子統計規律問題，缺失環節大約有三十多項以上尚未驗證的重要物理預測，技術力與財務限制短期內還無法實現這些實驗，多項變因從而影響夸克星的物態方程，進而全面修正過往發展出來的夸克星理論。 | [
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奇異星 | 因此，夸克星是否存在需要這些實驗確認以後，才能透過觀測來逐步推敲及驗證，討論夸克星是否真實存在，這些實驗完成驗證之前，言之尚早。
目前夸克星機制當中，為方便建立模型而忽略太多重要的細節，大量物理環節都僅僅停留在猜想階段，實驗目前不能驗證其真實性，存在太多變因使過往發展的夸克星模型必須進行修正，此為主要由實驗物理學家所持有保守懷疑態度的堅強反對理據。
## 夸克物質形成機制理論
### 十重態群以下、四夸克以上狀態的粒子
實驗室當中發現的十重態群以下、四夸克以上狀態的粒子：
* Jaffe 1977，建議四夸克狀態的粒子存在，其奇異狀態為（qsqs）。
* Jaffe 1977，建議 H dibaryon，六夸克狀態有相同數量的上夸克、下夸克、奇夸克（表達為 uuddss 或 udsuds）。
* 重夸克束縛多夸克系統（QQqq）。
* 1987 年，具有反魅夸克的五夸克狀態首次被提出（qqqsc）。
* 具有反奇夸克及四個由上下夸克的輕夸克所組成的五夸克狀態（qqqqs）。
* 輕五夸克群凝聚成十重態群（十個夸克一群，antidecuplet），最輕的候選者為，Ө+，LEPS 日本春天八號於 2003 年發現。
* 由 Jaffe 及 Wilczek（QCD）所提出的雙夸克模型亦可解釋這個現象。
* Ө++ 及反粒子Ө−−。
* 雙奇五夸克態
Φ−−
（ssddu），十重態群的一員。
* 2003 年，X (3872) 四夸克態由日本 Belle Collaboration 發現。
* 2004 年，帶反魅夸克的五夸克態Ө0
c(3100)，(uuddc) H1 collaboration 偵測到。
* 2004 年，DSJ(2632) 四夸克態候選由費米實驗室 SELEX 發現。
* 2007 年，Z+(4430) 四夸克態由日本 KEK 的 Belle 實驗室公布發現，其最簡單的夸克結構是四夸克 ccud。
* 2007 年，Y (4660) 四夸克態由日本 Belle Collaboration 發現。
* 2009 年，Y (4140) 四夸克態候選由費米實驗室發現。
* 2010 年，兩名來自 DESY 的科學家及一名來自 Quaid-i-Azam University（烏爾都語：جامعہ قائداعظم）的科學家重新分析先前有關於 ϒ（5S）介子的資料，發現了一個穩定的四夸克共振態。 | [
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奇異星 | * 2012 年，日本 Belle 實驗室發現 2 個帶電荷的介子態，Z+
b(10610) 和 Z+
b(10650)，這兩個介子最簡單的夸克結構是四夸克 bbud。
* 2013 年 3 月，Zc(3900) 四夸克態由中國北京正負電子對撞機上的 BESIII 合作組公布發現，一周後日本 Belle 實驗室發現了稱為 Z (3895) 的同一種粒子。美國的研究人員採用美國康奈爾大學 CLEO-c 實驗保存的數據證實了。2013 年底，伴隨粒子 Zc(4020) 與 Zc(4025) 四夸克態由 BESIII 公布發現。這種介子態的四夸克結構是 ccud。
* 2014 年初，Z (4430) 四夸克態由 LHCb 成功確認。
* 2014 年，德國於利希研究中心發現可能的雙重子態 d\*(2380)。
* 2015 年，五夸克態 P+
c(4380) 與 P+
c(4450) 由 LHCb 發現。
六夸克態（雙重子）、七夸克態與十重態群至今為止，實驗觀測的置信度都低於 6σ，學界依然在爭議當中，目前發現的潛在多夸克態候選粒子，大部分都被排除，絕大多數都只是混雜態，目前還沒有決定性的證據出現，而 SU (4) 四夸克態依然有待完善化。
### 多於十重態群的物理機制解釋目前並不存在
多於十夸克的物理機制，至 2014 年一月為止，並未有任何實際的驗證研究報告提出。夸克星遠遠超過十個夸克，此外尚不論及引力因素，目前所提出的夸克星模型，均多於十個夸克，對於超過十個夸克的物理機制沒有任何說明，並未考量現在的物理事實。
### 實驗室目前無法找到夸克星的基本成分 H 雙重子
實驗室中尚無法製造出上下夸克轉變成奇夸克的多夸克奇異物質，最小號的奇異物質是六夸克雙重子（H-dibaryon,uuddss），俗稱 H 雙重子，Jaffe 的原始推論認為其能階為 81MeV。然而包含 RHIC 與 LHC 等粒子加速器在內實驗二十多年，投注最少上千名物理學博士，沒有任何一個實驗室在各種能階上面報告它們成功找到六夸克 H 雙重子，能量到 2.41GeV 也沒有找到，RHIC 則在幾乎所有的 Low Bound 能階都無法找到 H 雙重子存在的證據，H 雙重子事實上是粒子物理的熱門粒子。如果找到 H 雙重子，而其性質正如奇異物質假說的一樣，夸克星的假說才有成立的可能性。
## 尚無法驗證的理論物理預測
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奇異星 | 夸克星產生的帶負電荷奇異物質能與正常粒子發生作用。
一般而言，奇異物質（Strangelet）都是帶正電荷的，但是在高能撞擊下，卻能形成帶負電荷奇異物質。夸克星的形成機制，卻正好是高能撞擊，因此，如果奇異物質理論正確，帶負電荷奇異物質在極超新星爆發下，並非是個罕見物質。
帶負電荷奇異物質與正常粒子發生作用是一個極危險的反應，並且將正常原子轉化成奇異物質。如果這個反應不幸的是個連鎖反應（按照地球尚未被毀滅的狀況來推論，應該不會是連鎖反應，但是並沒有實際的實驗數據說明它的性質），那麼極超新星爆發的粒子與地球撞擊，將是一個極度可怕的災難。
至今為止，奇異物質理論都還只是一個假說，許多的爭辯還在進行當中。例如，於所有的射線搜尋當中，都沒有找到奇異物質存在的痕跡。NASA 的月球土壤當中，亦沒有奇異物質的痕跡被找到。而對中子星的思維推論，如果奇異物質會轉化一般物質，那麼中子星最後應該連表面都被轉化到奇異物質，但是，實際從光譜上面的觀測，都是正常的核所發射出的光譜線。
### 色禁閉突破問題
色禁閉突破與強引力同步作用的物理機制目前尚不清楚。
夸克星是透過色禁閉突破（Deconfinement）形成的，形成之時，強引力便已經存在。有研究表明，色禁閉突破在強引力條件下，並非必然成立，強引力的條件下，夸克有可能發生相變，從而使得奇異物質假說在強引力的條件下無法形成，然而這樣的研究超出現有的實際物理能力。
此外，帶色夸克（dressed quark）是否會出現，也會影響夸克星的外部觀測性質。
地球上無法製造這種實驗環境，從而無法檢驗色禁閉突破與重力同步作用下，物理效應確切是什麼，理論亦未曾探討過。
### 孤子波震盪關係
Witten 奇異物質的奇異態與李政道所提出的束縛孤子態之間可能發生的相變，極有可能在夸克星當中發生，然而它會在星體內發生什麼樣的具體效應，對於星體機制有何影響，對於星體穩定機制的貢獻如何，至今為止並沒有系統化完整的研究。
### 小出義夫輕子質量公式問題
在強引力作用下，小出義夫輕子質量公式（Yoshio Koide formula）是否還依然成立，這是一個關係到夸克星能源及質量生成機制的重要疑問，目前沒有任何手段來檢驗。
小出義夫輕子質量公式： | [
0.5045192837715149,
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奇異星 | $Q={\frac {m_{e}+m_{\mu }+m_{\tau }}{({\sqrt {m_{e}}}+{\sqrt {m_{\mu }}}+{\sqrt {m_{\tau }}})^{2}}}\approx {\frac {2}{3}}$ ,
1⁄3 < Q < 1 是極為明顯的。
電子、μ 子及 τ 子的實驗測量值分別為 _m_e = 0.510998910(13) MeV/_c_2, _m_μ = 105.658367(4) MeV/_c_2, 及 _m_τ = 1776.84(17) MeV/_c_2，因而得出 $Q=0.666659\approx {\frac {2}{3}}$ 。
小出義夫輕子質量公式是物理學當中尚未有答案的神祕質量生成機制問題，目前夸克星的研究幾乎都專注於「非輕子過程」，對於輕子的交互作用盡力的避開而不討論，使得夸克星模型產生明顯的空缺地帶。
### 大氣消滅速度
夸克星大氣消滅速度問題，假設夸克星大氣也是由奇異物質（H 雙重子）所組成的，就現行對多夸克態物理的了解，猜測稀薄的 H 雙重子存在的壽命很短，如果夸克星大氣是 H 雙重子，那麼夸克星大氣消滅速度能夠讓夸克星維持多久的星體穩定。
由大氣消滅速度問題所產生，學者提出了「裸奇異星」，也就是夸克星外圍殼層因為 H 雙重子大氣的快速衰變，因而使得內核奇異物質完全裸露出來的夸克星。
### 星體穩定機制
夸克星的星體穩定機制解釋，目前實際並不存在，是一個主要具有高度爭議性的課題。
夸克星的正確星體結構圖目前無人能夠提出不產生自相矛盾的理論，這包含了「密度均勻說」及「弱電星結構說」，兩者均在提出後有論文再證明其不可能實際穩定，提出者及證偽者都具有高度的專業水平，最頂級的專家同行沒有長期仔細檢驗亦無法看出破綻。
目前科普文章介紹的夸克星圖鑑及結構均不是正確的星體穩定結構，目前並沒有人成功提出合格的夸克星穩定結構。 | [
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奇異星 | 「密度均勻說」是理想實驗室下的條件，假定了物質完全不發生衰變的條件下才會發生，它將實際上大約 1.7% 衰變的生成率設定為 0%，在物理上 1.7% 衰變生成率是非常高的數值，這意味夸克星內核經常性會發生粒子衰變，導緻密度均勻性的破壞，因此「密度均勻說」是一個看似合理卻完全錯誤的夸克星說法。會有這種說法的出現，基本上來源自科普文章作者並不讀專業論文或是無法讀懂專業論文，刻意或根本無知地將邊界條件設定去除或忽略，而這些邊界條件是絕對重要而不可去除的，科普文章作者選擇性地將容易吸引讀者目光的內容加入，卻不提及其發生可能性高度接近於零，因而導致將錯誤的說法到處傳播的問題。
宣稱星體可以穩定者多數從某一個角度推算而得到結論（例如只從量子力學或只從引力出發），然而由另一個角度或多個角度進行推算的時候，就會發生該模型的隱藏性陷阱。因此，全面而完整的星體穩定機制，目前並不存在，這包含了單純使用體黏滯性抵抗強引力作為主要論證的星體穩定機制，該機制由其他角度推算時有理論陷阱使之實際不成立（黏滯性破壞的衰變機制）。
不過，高度體黏滯性依然是夸克星的最重要特性之一。
或者，夸克星本身在宇宙中，就不可能長期穩定存在，至少在 SN 1987A 的觀測上，可以體驗到這個現實觀測問題。
目前提出的夸克星模型，均沒有辦法妥善解釋內部結構所造成的能量流向可以導致夸克星星體於太空中維持完整性的機制，各類計算機模型均導致星體結構瓦解的預測。這是由於對量子重力效應尚未有任何可行研究成果，對於所發生的效應無法進行正確預測所導致，過去的模型大多使用半古典力學修正建立，發生了一系列模型無法自圓其說的困難。
證明完整的夸克星星體穩定機制及演化過程是一項難度非常高的理論與實驗的工作，牽涉太多無法取得實驗數據的環節，許多這個領域的頂級專業學者嘗試十餘年均以失敗告終。
### 重子污染困難
大部分的夸克星能量模型使用火球模型進行理論分析時，會發生伽瑪射線爆（GRB, Gamma Ray Burst）因重子過多而無法膨脹達到極端相對論的要求，這就是「重子污染」困難，「重子污染」使得夸克星的能量來源形成解釋上的困難。 | [
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奇異星 | 伽瑪射線爆（GRB）不能含有高含量的「重子」，否則能量無法供應粒子進行高速運動而產生伽瑪射線爆，然而，「夸克星」卻是富含「重子」的星體，中子星與夸克星是幾乎完全由「重子」所構成的星體，如何解釋幾乎完全由「重子」所組成的星體卻會發出重子低含量的伽瑪射線爆，成了星體能源模型理論上的超級困難。中子星因為星體中還可以含有相當數量的輕子，因此解釋上困難度不大。夸克星因為在坍縮時期所造成的輕子含量大幅度降低，解釋夸克星的重子含量比例則成了能量上的超級難題。
「重子污染」主要是因為理論不完備而在理論模型上發生解釋上的困難，並非真正會發生「污染重子現象」。
「重子污染困難」導致學界在對待「高速旋轉中子星」是否為夸克星的認定上，形成了長久的極大保留態度。許多解決方案被提出，例如「奇異星相變」理論。
### 奇異物質的體粘滯性質
由於十重態群以上的奇異物質沒有任何的物理性驗證報告，奇異物質的體粘滯性質雖然推論完整，適用於弱磁場下，但是僅為理論，在夸克膠子影響重力效應的實驗檢驗後（引力係由夸克膠子之間的活動產生的理論，此一模型如果成立，奇異物質的體粘滯性質將可能發生理論困難），需要視狀況進行修正。
奇異物質的體粘滯性比普通核物質高上許多個量級（105），是高密度奇異物質最重要的動力學特徵之一。體粘滯性越強，剛性越大，天體越可以高速旋轉，有研究認為高度的體粘滯性，是夸克星可以達到克普勒極限用以依賴高速旋轉抵抗強引力，而維持星體完整的重要原因之一。
中子星的理論旋轉速度只能達 10ms，而夸克星的最低旋轉週期則為 3ms，據此計算，有些天體物理學家認為，超過 300Hz 高速旋轉中子星，實際都應該是夸克星，加上夸克星與中子星的表層大氣物理現象非常接近，使得夸克星與中子星難以分辨，如此而言，夸克星的數量應當是比想像的更多。另外一種計算則認為中子星的極限旋轉速度能達 2ms，而夸克星的最低旋轉週期可達為 1.65ms，因此高速旋轉中子星必須要超過 500Hz 才能夠被懷疑為夸克星。
夸克星能夠達到高速旋轉主要是由體黏滯性所貢獻。奇異物質的體粘滯性是奇異星性質的關鍵性里程碑級重要研究成果，對其物態方程有絕對的影響，是一項使得關於所有夸克星的研究可以繼續進行的重要理論支柱。
### 真空極化問題 | [
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奇異星 | 重力真空星當中的真空極化機制（quark vacuum polarization），是否會在夸克星內核當中觸發「中微子爆發」，伴隨「夸克星星震」，目前依然決定著夸克星結構機制上的一個重大疑問。夸克星結構至今為止，並未有定論，內核是否有真空極化所產生的空間壁，學術研究上還有疑義。
真空極化現象需要電磁場，這表示夸克星內部必須產生強磁與電場，電磁場來源的機制則必須有來源。
如果夸克星不存在真空極化現象，則演化成重力真空星、模糊球及黑洞的理論全部遭遇理論困難，必須尋求替代解釋方案。
如果夸克星存在真空極化現象，夸克星演化過程當中會產生重力真空星當中的空間壁，內部存在擬視界層，中心為空洞或是存在空間泡，則現有的夸克星模型幾乎全部都是錯誤的。
2013 年三月中左右，CERN 宣布的第二次確認希格斯玻色子 125.3-126.0GeV 能階，置信度約 5.9σ，基本上確立「標準模型」原則上是完全正確的一個近代物理理論，夸克星核心透過「中微子爆發」釋放能量，能量的來源基本上來自於真空極化，虛粒子對的產生，導致「希格斯玻色子」及「時間量子」（chronon）生成的機率升高，並使得內核走向更低能階的真空基態，進而產生進一步坍縮的機制，核心開始轉變成空間壁維度緊化而邁向下一步坍縮機制，時間維度因「時間量子」產生被緊緻化，產生不存在時間的純三維空間壁，因此可推導得夸克星的下一個演化星體，原則上應該是「重力真空星」，這使得全部的物理機製得到相對較完善的解釋。這對「廣義相對論」推導的「經典黑洞理論」是一項重大打擊，「經典黑洞理論」長年在天體物理學界產生爭端，具有重大弊端及嚴重矛盾，許多天體物理學家處理「類黑洞星體」時，喜歡將之替換為中子星來近似處理，因為「廣義相對論」無法建立實際的物態方程，對於處理真正的物態機制是一項極嚴重的問題，此一物理流派持續認定會產生「奇點」及「事件視界」的物理理論就是一個錯誤的物理理論。據此，「廣義相對論」是一個錯誤的理論傾向加大，除非在標量不變的等價延展理論上有新的進展，能夠消解掉「奇點」及「事件視界」，並且證明不存在「量子奇點」（Quantum Singularity），否則輔以對「夸克星」、「重力真空星」及「模糊球」做進一步研究，「經典黑洞理論」被證明是錯誤理論的機率大幅度增高，進而反證得到「廣義相對論」是一個錯誤的物理理論。 | [
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奇異星 | 使用「時間量子理論」（Caldirola-Montaldi Chronon Model）推導可得相對比較正確的結論，原始的 Caldirola-Montaldi 時間量子方程如下：
: $\theta _{0}={\frac {1}{6\pi \epsilon _{0}}}{\frac {e^{2}}{m_{0}c^{3}}}\ $
由於 m0 會導致無限大的出現，此一原始方程實際不具備正確的物理含意，物理含意無法正確詮釋，修正為如下方程：
: $\theta _{0}m_{0}={\frac {1}{6\pi \epsilon _{0}}}{\frac {e^{2}}{c^{3}}}\ $
則得到「時間量子與質量的量子纏繞態」，含意即為時間與質量具備纏繞關係，使用「時間晶格」分析手段，質量越小，時間量子由於相對增大，這會導致量子力學的效應越明顯，而質量越大，時間量子由於逼近於零，使得我們的觀測技術下，物理效應逼近於傳統的「廣義相對論」。轉換到量子物理上的詮釋，即為希格斯玻色子與時間量子的量子纏繞態造成近似「廣義相對論」的幾何拓撲詮釋方式，這使得「量子引力理論」與「時間場力學」（一種假設時間本身也具有場及力的交互作用理論）的發展成為真實可能，不過這種詮釋方式則留下希格斯玻色子、時間量子及引力子如何完成這樣的量子纏繞程序的疑問。
2014 年二月份的新研究顯示，給定上下界進行量子化以後，由於「量子退相干」效應，該方程直接使得「廣義相對論」在中子星與夸克星的適用性失效，必須改用霍拉瓦重力推導，才能得到正確的推論結果。此理論若可得到證實，則是「廣義相對論」證偽的重要理論，主要的關鍵即為勞倫茲變換在高能階的處理方式導致重整化的經常性失敗，「廣義相對論」做了錯誤的假設，在凝聚態物理當中已經發現「廣義相對論」的失誤，在夸克星模型當中事實上也發現了「廣義相對論」的重大缺失，該理論也同時指出「經典黑洞」不太可能真實存在，這使得研究夸克星對未來物理學發展的意義越來越重大。有越來越多的跡象顯示「廣義相對論」在高能階處理方式上有誤，夸克星模型則是其中之一，對夸克星的精細量子效應的研究將成為新物理學的重要突破口。 | [
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奇異星 | 如果第三方實驗數據確認發現希格斯玻色子，則夸克星的引力機制及質量生成機制，涉及使用「廣義相對論」的部份，必須全面修改，做更進一步深入研究，則可檢驗「廣義相對論」是否是一個錯誤的物理理論有很大的幫助。目前可替代「廣義相對論」的新理論則有霍拉瓦重力及 Scalar invariant 系列的等價延展理論，霍拉瓦重力對勞倫茲變換在高能區域做了新的處理方式，等價延展理論則去除掉「廣義相對論」的一些不太恰當的假設，如果成功結合 Unparticle 理論，全系列的物理理論都會相對比較協調，消解掉「廣義相對論」所造成的長年爭議。以目前而言，採用「廣義相對論」對夸克星推導，都持續產生矛盾的推論結果，採用新的量子引力理論來推導夸克星機制，會有比較新觀點的正確推論結果出現。
這個議題在學術研究上還有很大爭議。
### γ- 模不穩定性
γ- 模不穩定性及窗口在區分中子星和夸克星時，有相當大的幫助。γ- 模的不穩定性會導致緻密星的旋轉速度改變。
星體物質的體黏滯性會直接影響星體的 γ- 模不穩定性，中子星和夸克星的 γ- 模不穩定性窗口在溫度上有極大的差異，是分辨中子星和夸克星的重要依據。
### 色超導與超固態
如果夸克星中存在色超導狀態（Color Superconductivity），則會對夸克星造成一系列的影響，包含了冷卻性質、轉動不穩定性、禁閉突破密度與磁場。適當條件下則會產生超固態。
### 引力輻射的抑制問題
CFS 機制導致引力輻射的不穩定，結果會導致發生坍縮成黑洞、再度星爆、星震或伽瑪射線爆。
### 奇異物質低密度聲速物性行為完全相反
奇異物質於高密度及高速的狀態下，性質是穩定的。然而，理論推算則是認為奇異物質於低密度的聲速狀態下，物性行為完全相反。這個推論至今為止無法用實驗來驗證。
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奇異星 | 以目前來說，**夸克星**仍只是理論上的假設星體，尚未獲得證實，目前所觀測的潛在夸克星僅為臆測。至 2011 年一月為止，有七個天體被懷疑是**夸克星**，分別為 RX J1856.5-3754、3C58、XTE J1739-285、SN 1987A、SN 2005gj、SN 2005ap 及 SN 2006gy。1998 年左右，兩顆由華人天體物理學家所提出的高速旋轉中子星可能是夸克星，但是因為提不出數據驗證，其他組的天體物理學家給出不同的分析結果，因而未被列入候選星當中。如果將超過 300Hz 的高速旋轉中子星及超強磁中子星也列入夸克星候選星，則可能是夸克星的天體大約達三十餘顆，例如 SAX J1808.4-3658（401Hz）及 PSR B0943+10。
錢德拉 X 射線天文台在 2002 年 4 月 10 日觀測到的兩個星體 RXJ1856 和 3C58 被認為可能是**夸克星**。在這之前，這兩個星體一直被認為是中子星。
天文學家在 2008 年則新發現三顆可能的夸克星：SN 2005gj、SN 2005ap、SN 2006gy。其中 SN 2005ap、SN 2006gy 分別為有史以來最強大的超新星爆發，SN 2006gy 亮度甚至超過周圍的整個銀河系星系核（NGC 1260）數倍，SN 2005ap 亮度更是 SN 2006gy 的兩倍。目前最有希望實際觀測到夸克星性質的應該是這兩顆超新星。但是這三顆的爆發後的觀測時間較短，並沒有後續報告來強力支持其為夸克星，只能說明它們可能不是中子星。
### RX J1856.5-3754
根據已知的物理定律，如果它們確實是中子星，那麼距離太陽大約 400 光年的 RX J1856.5-3754 似乎過小，錢德拉 X 射線天文台報告顯示它的直徑僅有七公里。以此粗略地推論這兩個星體應該是由比中子星密度更高的物質所構成。另一則由 Timothy M. Braje 及 Roger W. Romani 等人提出的報告則認為 RX J1856.5-3754 的直徑應有二十七公里，依然是一顆中子星。
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奇異星 | 錢德拉 X 射線天文台 ACIS S3 所得數據顯示距離太陽 10,000 光年左右的 3C58 則是過冷，無法用標準理論解釋中子星形成機制。但是所使用的模型是 QGP「夸克膠子電漿模型」，並非「夸克 - 膠子 - 引力機制」，RHIC 及 LHC 報告 hadronic gas、mixed phase、QGP 三者有相當程度的不同，使用 QGP 解釋夸克星非常不適當，推測它是夸克星，也有很大的觀測困難，對其是否屬於夸克星並沒有很大幫助，只能說明它可能不是中子星，尚沒有辦法斷定它是哪一種星體。
### XTE J1739-285
愛荷華大學教授認為距離 39,000 光年的 XTE J1739-285 可能是潛在的**夸克星**，它是一顆快速旋轉的星體（1,122Hz，高速旋轉中子星），並沒有詳盡的報告相關的數據，亦沒有磁場強度報告。多組天體物理學家的資料分析則給出不同的結果，目前普遍依然認為它是一顆中子星，學界對此星體看法較為保守。
### SN 1987A
2009 年，由於長達二十多年沒有發現理論上預測的中子星，天文愛好者認為 SN 1987A 可能是潛在的夸克星，爆發總能量約為 1046 焦耳，99% 以中微子的形式釋出，SN 1987A 由於找不到理論上預測應該存在的星體，除了留下周圍的迴光現象（light echoes），並沒有觀測到任何理論所預期的星體，目前無法解釋其整個超新星爆發的過程及發生機制。
質量遺失問題（SN 1987A mass loss process）二十多年來導致許多學術爭議，除非使用未經驗證的理論，例如中子星量子蟲洞機制，否則無法充分解釋 SN 1987A 現象，但是量子蟲洞機制本身卻存在理論不自洽的問題，實驗精度置信度都很低，不為學界共同承認，以夸克星做解釋也有理論盲點導致不自洽的問題，形成「先子球」的說法雖然可以解釋大部分現象，但是「先子」理論卻是個尚未證實的理論。SN 1987A 目前為止依然是個科學懸念，至今為止，許多頗負盛名的天體物理學家進行的努力、全部理論解釋均告失敗，沒有關鍵性的觀測證據支持任何一個理論。
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奇異星 | SN 2005gj 離地球約為 8 億 6400 萬光年，由史隆尋天計畫於 2005 年 9 月 29 日所發現的，光度超過 SN 1991T 的三倍，能量遠超過正常超新星爆發。由於距離地球太遠，實際的觀測技術無法了解具體細節，僅能以理論推斷其爆發結果所產生的星體並非中子星或黑洞，唯一合理的「推論」是爆發結果產生了夸克星。
### SN 2005ap
SN 2005ap 是 ROTSE-IIIb 於 2005 年四月三日發現，距離地球大約 47 億光年，光度超過 SN 2006gy 大約兩倍，是目前發現的超新星爆發規模最大的一個，是正規 II 型超新星爆發的 300 倍左右。此等規模的超新星爆發，理論預測應該會產生夸克星，但是目前沒有能力去判定是否確實有星體處於爆發位置。
### SN 2006gy
SN 2006gy 由 Robert Quimby 及 P. Mondol 於 2006 年 9 月 18 日所發現，距離地球兩億三千八百萬光年，它是當時記錄最大的超新星爆發，僅次於後來驗證發現的 SN 2005ap。觀測的結果沒有發現任何黑洞特徵的跡象，推測產生了夸克星，目前的觀測能力沒有辦法驗證這項猜測。
### 沒有直接證據說明夸克星真實存在
由於相關研究存在多項理論環節缺失，許多的形成機制假設性過高，未驗證或無法驗證的環節數量甚多，疑義及爭議過多，多數天體物理學者則普遍認為這些觀察結果仍不足以採信，但均以樂觀的態度來審視這些新發現。
## 未來可能發展成夸克星的鄰近星體
### 「海山二」形成夸克星對地球的威脅
目前離地球最接近，最有可能轉變成夸克星的星體是高光度藍變星「海山二」（船底座 η），離地球約 7500 光年，質量一百五十倍太陽，星等為五等。由於質量相當的大，又是個多星系統，其星爆威力預計將會極強，爆發時不見得會形成黑洞，有可能形成星雲、黑洞、模糊球、重力真空星、中子星或夸克星。海山二的爆發威力相當於 SN 2005ap 等級，威脅極強，足以發生銀河系內的生物大滅絕。 | [
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奇異星 | 由於對於「奇異物質」的物理性質了解甚少，學術上的爭辯依然在進行當中，尚不足以完成精確的計算，目前連 LHC 的能量等級均不足以製造並找到「奇異物質」，因此上述數據出入數字甚大，是否會發生上述所稱的災難，高能天體物理學界看法兩極化，目前僅知因為帶負電荷奇異物質能夠與正常的粒子發生作用，只要極少量的（少於十顆）帶負電荷奇異物質（Negatively Charged Strangelet）撞擊地球，就能給地球帶來四億五千萬年前奧陶紀與志留紀時期毀滅性的結果。天體物理學界對於「海山二」的興趣極高，花費極大量的資源、金錢及太空天文望遠鏡專門投注於海山二研究，2003 年甚至曾經動用全球的望遠鏡來觀測海山二的 X 光蝕，可見其十分受天體物理學家的關注，投入的精力僅次於 SN 1987A，就其對地球的潛在威脅而言，不難看出天體物理學家們心中的憂慮。
### 其他可能發展成夸克星的星體
其他具有威脅力的星體是：
* LBV 1806-20（30,000 光年，約一百五十倍太陽）
* 半人馬座 V766（12,000 光年，HR 5171 A，黃特超巨星）
* 手槍星（25,000 光年，約一百五十倍太陽）
* 大犬座 VY（5000 光年，約四十倍太陽）
均隨時可能引發超新星爆發。
## 中子星與夸克星的區分方式
由於體黏滯性差別甚遠，某些高速旋轉中子星被認為應該是夸克星。
### 質量與半徑的關係
以冷星處理，使用 Tolman Oppenheimer Volkoff 方程計算所得的中子星及夸克星半徑關係：
中子星的質量與半徑的關係是 $M\propto {\frac {1}{R^{3}}}$ 。
夸克星的質量與半徑的關係是 $M\propto R^{3}$ ，上限大約是 10,000 公尺。
中子星的半徑有下限，否則不可能形成中子星，有趣的是夸克星半徑有上限而並幾乎無下限，最小可以接近零，幾滴落魄的「奇異滴」在太空中遊盪，理論上也可被稱為「夸克星」。一種看法是這樣的現象非常可能發生，如此也補齊了宇宙間的暗物質來源解釋。
物態方程的改變會直接改變夸克星半徑的預測值，以目前 H 雙重子預期的穩定島處於 H（2220）到 H（2250）之間的數值估算，所有關於夸克星的宣稱均需要大幅度修改。
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奇異星 | 據某些推論分析，「超強磁中子星」應該是夸克星，磁層輻射應當與典型中子星有磁場等級上的極大區分，然而，這些理論在輕子含量比例、電荷分離、真空極化、漂移子脈衝、熱輻射機制及非熱輻射機制等等解釋上均遭遇重大矛盾。因此，目前學界依然將超強磁中子星歸類在中子星裡面。
### 星爆特徵
中子星與夸克星的星爆與星震由於基本組成物質特性大不相同，伽瑪射線爆所造成的現象有所區分。
### 時空場差異
由於中子星與夸克星的兩項主要差異「旋轉速度」及「質量半徑關係」所造成的物理現象，中子星與夸克星的時空場有所區別，係為極佳的區分方式。
### 表面物理
中子星表面一般有碳大氣層及電子殼層，夸克星的表面理論上應當是 H 雙重子與碳大氣層混和，有時則因為外圍物質全部拋出而成為夸克固態，變成裸奇異星，這兩者有一定程度上的區分。碳的來源是由前身星核融合燃燒後，轉變成的碳元素，恆星爆發的時候，一般富含碳元素，中子星與夸克星在爆發時期，已經形成強引力，碳元素來不及逃逸，因而被捕獲而形成碳大氣層。而夸克星則會由星核向外溢出 H 雙重子，因此可以產生中子星與夸克星不同之處。
結合參考系拖拽所造成的時空場差異，中子星與夸克星外圍物質的不同，可透過光譜及物質頻譜紅移進行區分，是極為有效的觀測區分手段。
## 相關條目
* 量子色動力學
* 禁閉突破
* 托爾曼 - 奧本海默 - 沃爾可夫極限（中子星質量的上下限）
* 奇異原子
* 緻密星
* 夸克新星
* 重力真空星
* 奇特星
* 奇異星
* Q 星（灰洞）
* 中子星 - 中子物質 - 中子簡併物質 - 中子
* 脈衝星
* 恆星黑洞
* 白洞
* 蟲洞
* 模糊球
* 磁星
* 暗物質星
* 白矮星
* 玻色星（Boson Star）
* 先子星（Preon Star）
* 孤子星（Soliton Star）
* 簡併物質
* 量子色動力學物質
* 夸克 - 膠子電漿體
* 奇異夸克團
* 奇異物質
* 夸克物質
* 多中子核物質
* 先子物質
## 註釋
## 參考
## 外部連結
* 中子星和夸克星組成圖（頁面存檔備份，存於網際網路檔案館）
* SN 1987A 的迴光現象（頁面存檔備份，存於網際網路檔案館） | [
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] |
奇異星 | * Perhaps a Stable Dihyperon Subscription or purchase required
* Neutron Star/Quark Star Interior (image to print)（頁面存檔備份，存於網際網路檔案館）
* Quark star glimmers（頁面存檔備份，存於網際網路檔案館）, _Nature_, April 11, 2002.
* Debate sparked on quark stars（頁面存檔備份，存於網際網路檔案館）, _CERN Courier_ **42**, #5.
* Wish Upon a Quark Star, Paul Beck, _Popular Science_, June 2002.
* Is RX J185635-375 a Quark Star?, J. J. Drake et al., arXiv, astro-ph/0204159.
* Curious About Astronomy: What process would bring about a quark star?（頁面存檔備份，存於網際網路檔案館）
* Quarks or Quirky Neutron Stars?（頁面存檔備份，存於網際網路檔案館）, Mark K. Anderson, _Wired News_, April 19, 2002.
* Strange Quark Stars（頁面存檔備份，存於網際網路檔案館）, Ask an Astrophysicist, question submitted April 12, 2002.
* Seeing "Strange" Stars（頁面存檔備份，存於網際網路檔案館）, physorg.com, February 8, 2006.
* Quark Stars Could Produce Biggest Bang（頁面存檔備份，存於網際網路檔案館）, spacedaily.com, June 7, 2006.
* Meissner Effect in Strange Quark Stars, Brian Niebergal, web page, University of Calgary. | [
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奇異星 | * Strange Exotic States and Compact Stars, Irina Sagert, Mirjam Wietoska, and Jurgen Schaffner-Bielich; arXiv, astro-ph/0608317.
* Quark Stars Involved in New Theory of Brightest Supernovae（頁面存檔備份，存於網際網路檔案館） - The first-ever evidence of a neutron star collapsing into a quark star is announced, _Space.com_, 3 June 2008
* Quark Stars, Alternate View Column AV-114（頁面存檔備份，存於網際網路檔案館）, John G. Cramer, Published in the November-2002 issue of Analog Science Fiction & Fact Magazine
* 反眛五夸克重子態物質發現的新證據，H1 Collaboration, Phys.Lett.B588:17,2004
* http://prl.aps.org/abstract/PRL/v106/i16/e162001
* http://iopscience.iop.org/0954-3899/28/7/323
* http://theorie.physik.uni-giessen.de/~hees/publ/cup.pdf（頁面存檔備份，存於網際網路檔案館）
* http://www.sciencedirect.com/science/article/pii/S0920563203015317
* http://www-panda.gsi.de/db/papersDB/pochodzalla\_josef\_proc\_leap03.pdf
* http://psroc.phys.ntu.edu.tw/bimonth/download.php?d=1&cpid=128&did=6 （頁面存檔備份，存於網際網路檔案館）
* 量子蟲洞與形標量場，上海天文台，1991,2002 年
* 超引力狀態 N=1 下的量子蟲洞態？The Smithsonian/NASA Astrophysics Data System | [
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] |
奇異星 | * 時空蟲洞與量子蟲洞在平坦的 Λ 衰變下的宇宙學，2006, 亞塞拜然大學 （頁面存檔備份，存於網際網路檔案館）
* 量子蟲洞及其動力學，Ewha Womans University, 南韓
* Jordan-Brans-Dicke 量子蟲洞及 Coleman 機制
* Quark star on arxiv.org（頁面存檔備份，存於網際網路檔案館）
* Johan Hansson and Fredrik Sandin, _Preon stars: a new class of cosmic compact objects_. Phys. Lett. B 616, 1 (2005).
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* Johan Hansson, _A hierarchy of cosmic compact objects - without black holes（頁面存檔備份，存於網際網路檔案館）_ . Acta Phys.Polon. B38, 91 (2007). _PDF_
* Johan Hansson and Fredrik Sandin, _The observational legacy of preon stars - probing new physics beyond the LHC_.
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* Article in NatureNews : _Splitting the quark（頁面存檔備份，存於網際網路檔案館）_ . (Nov. 2007) | [
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奇異星 | * Jaffe, R. Perhaps a Stable Dihyperon. Physical Review Letters. 1977, **38** (5): 195–198. Bibcode:1977PhRvL..38..195J. doi:10.1103/PhysRevLett.38.195.
* Neutron Star/Quark Star Interior (image to print)（頁面存檔備份，存於網際網路檔案館）
* Quark star glimmers（頁面存檔備份，存於網際網路檔案館）, _Nature_, April 11, 2002.
* Debate sparked on quark stars（頁面存檔備份，存於網際網路檔案館）, _CERN Courier_ **42**, #5.
* Wish Upon a Quark Star, Paul Beck, _Popular Science_, June 2002.
* Curious About Astronomy: What process would bring about a quark star?（頁面存檔備份，存於網際網路檔案館）
* RX J185635-375: Candidate Quark Star（頁面存檔備份，存於網際網路檔案館）, Astronomy Picture of the Day, April 14, 2002.
* Quarks or Quirky Neutron Stars?（頁面存檔備份，存於網際網路檔案館）, Mark K. Anderson, _Wired News_, April 19, 2002.
* Strange Quark Stars（頁面存檔備份，存於網際網路檔案館）, Ask an Astrophysicist, question submitted April 12, 2002.
* Seeing "Strange" Stars（頁面存檔備份，存於網際網路檔案館）, physorg.com, February 8, 2006.
* Quark Stars Could Produce Biggest Bang（頁面存檔備份，存於網際網路檔案館）, spacedaily.com, June 7, 2006. | [
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奇異星 | * Meissner Effect in Strange Quark Stars, Brian Niebergal, web page, University of Calgary.
* Irina Sagert; Mirjam Wietoska; Jurgen Schaffner-Bielich. Strange Exotic States and Compact Stars. Journal of Physics G. 2006, **32** (12): S241–S249. Bibcode:2006JPhG...32S.241S. arXiv:astro-ph/0608317 . doi:10.1088/0954-3899/32/12/S30.
* Quark Stars Involved in New Theory of Brightest Supernovae（頁面存檔備份，存於網際網路檔案館） - The first-ever evidence of a neutron star collapsing into a quark star is announced, _Space.com_, 3 June 2008
* Quark Stars, Alternate View Column AV-114（頁面存檔備份，存於網際網路檔案館）, John G. Cramer, Published in the November-2002 issue of Analog Science Fiction & Fact Magazine
* New Scientist issue 2643, "Could preon stars reveal a hidden reality?"（頁面存檔備份，存於網際網路檔案館）, 6 February 2008
* New Scientist issue 2472, "Micro-stars may manage to avoid black-hole fate"（頁面存檔備份，存於網際網路檔案館）, 6 November 2008 | [
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积分 | **積分**（英語：Integral）是微積分學與數學分析裡的一個核心概念。通常分為**定積分**和**不定積分**兩種。直觀地說，對於一個給定的正實值函數 $f(x)$ ， $f(x)$ 在一個實數區間 $[a,b]$ 上的定積分
:
可以在數值上理解為在 $\textstyle Oxy$ 坐標平面上，由曲線 $(x,f(x))$ （ $x\in [a,b]$ ），直線 $x=a$ ， $x=b$ 以及 $x$ 軸圍成的曲邊梯形的面積值。
$f(x)$ 的不定積分（或原函數）是指任何滿足導數是函數 $f(x)$ 的函數 $F(x)$ 。一個函數 $f(x)$ 的不定積分不是唯一的：只要 $F(x)$ 是 $f(x)$ 的不定積分，那麼與之相差一個常數的函數 $F(x)+C$ 也是 $f$ 的不定積分。
微積分基本定理是微積分學中的一條重要定理，由艾薩克・牛頓和戈特弗里德・威廉・萊布尼茨在十七世紀分別獨立發現。微積分基本定理將積分與微分建立聯繫，通過找出一個函數的原函數，即可方便地計算它在一個區間上的積分。積分和導數已成為高等數學中最基本的工具，並在自然科學和工程學中得到廣泛運用。
積分的一個嚴格的數學定義由波恩哈德・黎曼給出，因此習慣上我們常見的積分也稱為「黎曼積分」。黎曼的定義運用了極限的概念，把曲邊梯形設想為一系列矩形組合的極限。從十九世紀起，更高級的積分定義逐漸出現，有了對各種積分區間上的各種類型的函數的積分。對微分形式的積分是微分幾何中的基本概念。
對積分概念的推廣來自於物理學的需要，並體現在許多重要的物理定律中，尤其是電動力學。現代的積分概念基於測度論，主要是由昂利・勒貝格建立的勒貝格積分。
## 簡介
積分發展的動力源自實際應用中的需求。實際操作中，有時候可以用粗略的方式進行估算一些未知量，但隨著科技的發展，很多時候需要知道精確的數值。要求簡單幾何形體的面積或體積，可以套用已知的公式。但如果游泳池是卵形、拋物型或更加不規則的形狀，就需要用積分來求出容積。物理學中，常常需要知道一個物理量（比如位移）對另一個物理量（比如力）的累積效果，這時也需要用到積分。
我們以下面這個問題作為介紹積分概念的開始：
: 考慮平方根函數 $f:\,x\mapsto {\sqrt {x}}$ ，其中 $x\in [0,\,1]$ 。在區間 [0,1] 上，函數 $f$ 「下方」的面積是多少？ | [
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积分 | 問題中的「下方」面積，是指函數 $y=f(x)$ 的圖象與 x 軸之間的部分的面積 $S$ （見右圖）。我們把這個面積稱為函數 $f$ 在區間 [0,1] 上的積分，寫作：
: $S=\int _{0}^{1}{\sqrt {x}}\,\mathrm {d} x\,\!.$
其中的 $\mathrm {d} x$ 稱為**積分變量**，表示要求面積的範圍是用坐標軸橫軸的刻度計算； $\int _{0}^{1}$ 則表示從 0 開始算起，到 1 為止，稱為**積分範圍**或**積分域**，其中 0 稱為**積分下界**，1 稱為**積分上界**， $\int $ 叫做**積分號**，是從拉長的字母 S 演變過來的。函數 ${\sqrt {x}}$ 寫在中間，稱為**被積函數**。
改進的方法是用更多的小方框來將函數圖象「覆蓋」，如右圖中的做法，就是將坐標軸橫軸 [0,1] 等分成 5 個部分：[0,0.2)、[0.2,0.4)、[0.4,0.6)、[0.6,0.8)、[0.8,1]，然後每一部分上放一個黃色的長方形（見右圖■）。這 5 個長方形的高度分別是函數在每個部分的極大值（也就是最右側的值）： ${\sqrt {0.2}}$ 、 ${\sqrt {0.4}}$ 、 ${\sqrt {0.6}}$ 、 ${\sqrt {0.8}}$ 、 $1$ 。這樣函數下方的部分就被 5 個黃色長方形覆蓋了，所以面積 $S$ 小於 5 個黃色長方形面積之和：
: ${\sqrt {0.2}}\left(0.2-0\right)+{\sqrt {0.4}}\left(0.4-0.2\right)+{\sqrt {0.6}}\left(0.6-0.4\right)+{\sqrt {0.8}}\left(0.8-0.6\right)+{\sqrt {1}}\left(1-0.8\right)\approx 0.7497.\,\!$
求出了 $S$ 的上限之後，用類似的方法可以求 $S$ 的下限。同樣是將坐標軸等分成若干部分，然後在每個部分放上長方形，不過這時候長方形的高度需要是函數在這個部分的最小值，也就是最左側的值。比如，如果將橫軸等分成 12 個部分，然後按照以上的方法放上綠色長方形（如右圖■），那麼從圖中可以看出， $S$ 必定大於綠色長方形面積之和： | [
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] |
积分 | : ${\sqrt {\frac {0}{12}}}\left({\frac {1}{12}}-0\right)+{\sqrt {\frac {1}{12}}}\left({\frac {2}{12}}-{\frac {1}{12}}\right)+\cdots +{\sqrt {\frac {11}{12}}}\left(1-{\frac {11}{12}}\right)\approx 0.6203.\,\!$
於是，面積 $S$ 的取值介於 0.6203 和 0.7497 之間。要取得更加精確的估計，可以將橫軸細分成更多的部分，並按照同樣的方法放置長方形，計算長方形的面積之和。隨著長方形越來越多，每個長方形越來越「細」，計算出的 $S$ 的範圍會越來越窄，最後得出 $S$ 的精確值。
以上的方法可能出現的「漏洞」，是所謂的「取值範圍」不一定會越來越小，最後聚集到同一個值上。雖然直觀上來說，由於函數下方的圖形面積是確定的，只要不斷地用相似的形狀「逼近」，最後總會趨向函數下方圖形的**真實面積**。然而，對於某些「病態」的函數，以上的方法是無法得到確定的數值的。十九世紀的數學家波恩哈德・黎曼證明了，對於滿足某些條件的良態函數，以上的方法一定能求出函數下方的面積。現代的數學家將這種方法求出的面積稱為**黎曼積分**，並給出了嚴格的定義（見 #嚴格定義一節）。對於那些無法用黎曼的方法定義「函數下方圖形面積」的函數，黎曼之後的數學家發展出了一些更寬泛的定義，讓這些函數也能定義積分。
### 術語和標記
如果一個函數的積分存在，並且有限，就說這個函數是**可積的**。一般來說，被積函數不一定只有一個變量，積分域也可以是不同維度的空間，甚至是沒有直觀幾何意義的抽象空間。如同上面介紹的，對於只有一個變量 $x$ 的實值函數 $f$ ， $f$ 在閉區間 $[a,b]$ 上的積分記作
: $\int _{a}^{b}f(x)\,\mathrm {d} x.$
其中的 $\mathrm {d} x$ 除了表示 $x$ 是 $f$ 中要進行積分的那個變量（**積分變量**）之外，還可以表示不同的含義。在黎曼積分中， $\mathrm {d} x$ 表示分割區間的標記；在勒貝格積分中，表示一個測度；或僅僅表示一個獨立的量（微分形式）。一般的區間或者積分範圍 $J$ ， $J$ 上的積分可以記作 $\int _{J}f(x)\,\mathrm {d} x.$ | [
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积分 | 如果變量不只一個，比如說在二重積分中，函數 $f(x,y)\,\!$ 在區域 D 上的積分記作
: $\iint _{D}f(x,y)\,\!\,\mathrm {d} \sigma $ 或者 $\iint _{D}f(x,y)\,\!\,\mathrm {d} x\mathrm {d} y$
其中 $\mathrm {d} \sigma $ 與區域 D 對應，是相應積分域中的微分元。
## 嚴格定義
定義積分的方法不止一種，各種定義之間也不是完全等價的。其中的差別主要是在定義某些特殊的函數：在某些積分的定義下這些函數不可積分，但在另一些定義之下它們的積分存在。然而有時也會因為教學的原因造成定義上的差別。最常見的積分定義是黎曼積分和勒貝格積分。
### 黎曼積分
黎曼積分得名於德國數學家波恩哈德・黎曼，建立在函數在區間取樣分割後的黎曼和之上。設有閉區間 $[a,b]$ ，那麼 $[a,b]$ 的一個**分割**是指在此區間中取一個有限的點列 $a=x_{0}<x_{1}<x_{2}<\ldots <x_{n}=b$ 。每個閉區間 $[x_{i},x_{i+1}]$ 叫做一個子區間。定義 $\lambda $ 為這些子區間長度的最大值： $\lambda =\max(x_{i+1}-x_{i})$ ，其中 $0\leq i\leq n-1$ 。而閉區間 $[a,b]$ 上的一個**取樣分割**是指在進行分割 $a=x_{0}<x_{1}<x_{2}<\ldots <x_{n}=b$ 後，於每一個子區間中 $[x_{i},x_{i+1}]$ 取出一點 $x_{i}\leq t_{i}\leq x_{i+1}$ 。
對一個在閉區間 $[a,b]$ 有定義的實值函數 $f$ ， $f$ 關於取樣分割 $x_{0},\ldots ,x_{n}$ 、 $t_{0},\ldots ,t_{n-1}$ 的**黎曼和**定義為以下和式：
$\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})$
和式中的每一項是子區間長度 $x_{i+1}-x_{i}$ 與在 $t_{i}$ 處的函數值 $f(t_{i})$ 的乘積。直觀地說，就是以標記點 $t_{i}$ 到 X 軸的距離為高，以分割的子區間為長的矩形的面積。 | [
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积分 | 最簡單的取樣分割方法是將區間均勻地分成若干個長度相等的子區間，然後在每個子區間上按相同的準則取得標記點。例如取每個子區間右端 $t_{i}=x_{i+1}$ （見左圖左上角）或者取每個子區間上函數的極大值對應的 $t_{i}$ （左圖左下角）等等。不同的取樣分割方式得到的黎曼和一般都不相同，而如果當 $\lambda $ 足夠小的時候，所有的黎曼和都趨於某個極限，那麼這個極限就叫做函數 $f$ 在閉區間 $[a,b]$ 上的黎曼積分。即， $S$ 是函數 $f$ 在閉區間 $[a,b]$ 上的黎曼積分，若且唯若對於任意的 $\epsilon >0$ ，都存在 $\delta >0$ ，使得對於任意的取樣分割 $x_{0},\ldots ,x_{n}$ 、 $t_{0},\ldots ,t_{n-1}$ ，只要它的子區間長度最大值 $\lambda \leq \delta $ ，就有：
$\left|\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-S\right|<\epsilon .\,$
也就是說，對於一個函數 $f$ ，如果在閉區間 $[a,b]$ 上，無論怎樣進行取樣分割，只要它的子區間長度最大值足夠小，函數 $f$ 的黎曼和都會趨向於一個確定的值 $S$ ，那麼 $f$ 在閉區間 $[a,b]$ 上的黎曼積分存在，並且定義為黎曼和的極限 $S$ 。這時候稱函數 $f$ 為**黎曼可積**的。將 $f$ 在閉區間 $[a,b]$ 上的黎曼積分記作：
$\int _{a}^{b}f(x)\mathrm {d} x.$
### 勒貝格積分
勒貝格積分的出現源於概率論等理論中對更為不規則的函數的處理需要。黎曼積分無法處理這些函數的積分問題。因此，需要更為廣義上的積分概念，使得更多的函數能夠定義積分。同時，對於黎曼可積的函數，新積分的定義不應當與之衝突。勒貝格積分就是這樣的一種積分。 黎曼積分對初等函數和分段連續的函數定義了積分的概念，勒貝格積分則將積分的定義推廣到測度空間里。:Intro.2-3 | [
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积分 | 勒貝格積分的概念定義在測度的概念上。測度是日常概念中測量長度、面積的推廣，將其以公理化的方式定義。黎曼積分實際可以看成是用一系列矩形來儘可能鋪滿函數曲線下方的圖形，而每個矩形的面積是長乘寬，或者說是兩個區間之長度的乘積。測度為更一般的空間中的集合定義了類似長度的概念，從而能夠「測量」更不規則的函數曲線下方圖形的面積，從而定義積分。在一維實空間中，一個區間 _A_ = [_a_, _b_] 的勒貝格測度 μ(_A_) 是區間的右端值減去左端值， _b_ − _a_。這使得勒貝格積分和正常意義上的黎曼積分相兼容。在更複雜的情況下，積分的集合可以更加複雜，不再是區間，甚至不再是區間的交集或併集，其「長度」則由測度來給出。:Intro.3
給定一個集合 $\Omega $ 上的 $\sigma -$ 代數 ${\mathcal {F}}$ 以及 ${\mathcal {F}}$ 上的一個測度 $\mu $ ，那麼對於 ${\mathcal {F}}$ 中的一個元素 $A\subset \Omega $ ，定義指示函數 $1_{A}$ 關於測度 $\mu $ 的積分為：
: $\int 1_{A}\,d\mu =\mu (A)$
再定義可測的非負**簡單函數** $f=\sum _{i=1}^{n}a_{i}1_{A_{i}}$ （其中 $A_{i}\in {\mathcal {F}},\,\,a_{i}\geqslant 0$ ）的積分為：
: $\int f\,d\mu =\int \left(\sum _{i=1}^{n}a_{i}1_{A_{i}}\right)\,d\mu =\sum _{i=1}^{n}a_{i}\int 1_{A_{i}}\,d\mu =\sum _{i=1}^{n}a_{i}\mu (A_{i})$ :28
對於一般的函數 $f:\Omega \rightarrow \mathbb {R} $ ，如果對每個區間 $(a,b]$ ，都滿足 $f^{-1}\left((a,b]\right)\in {\mathcal {F}}$ ，那麼測度論中定義 $f$ 是可測函數。對於一個**非負的可測函數** $f$ ，它的積分定義為：
: $\int f\,d\mu =\sup {\bigg \{}g,\quad g$ 為簡單函數，並且 $f-g$ 恆大於零 $.\,{\bigg \}}$ :30 | [
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积分 | 這個積分可以用以下的方式逼近：
: $\int f\,d\mu =\lim _{n\to +\infty }\left[\sum _{k=0}^{n2^{n}-1}{\frac {k}{2^{n}}}\mu \left({\frac {k}{2^{n}}}\leqslant f<{\frac {k+1}{2^{n}}}\right)+n\mu (f\geqslant n)\right]=\lim _{n\to +\infty }\left[{\frac {1}{2^{n}}}\sum _{k=0}^{n2^{n}-1}\mu \left({\frac {k}{2^{n}}}\leqslant f\right)\right]$ :344
直觀上，這種逼近方式是將 $f$ 的值域分割成等寬的區段，再考察每段的「長度」，用其測度表示，再乘以區段所在的高度。其覆蓋之處如右圖中的紅色區域所示。佛蘭德（Folland）總結說，「黎曼積分是把定義域區間 [_a_, _b_] 劃分為子區間」，而勒貝格積分則是「劃分 $f$ 的值域」。
至於一般的（有正有負的）**可測函數** $f$ ，它的積分是函數曲線在 x 軸上方「圍出」的面積，減去曲線在 x 軸下方「圍出」的面積。嚴格定義需要引進「正部函數」和「負部函數」的概念：
: $f^{+}:$ 如果 $f(x)\geqslant 0,$ 則 $f^{+}(x)=f(x),$ 否則 $f^{+}(x)=0.$
: $f^{-}:$ 如果 $f(x)\leqslant 0,$ 則 $f^{-}(x)=-f(x),$ 否則 $f^{-}(x)=0.$
可以驗證，總有 $f(x)=f^{+}(x)-f^{-}(x).$ 而 $f$ 的積分定義為： $\int f\,d\mu =\int f^{+}\,d\mu -\int f^{-}\,d\mu $ :41-42:345
以上定義有意義僅當 $\int f^{+}\,d\mu $ 和 $\int f^{-}\,d\mu $ 中至少有一個的值是有限的（否則會出現無窮大減無窮大的情況），這時稱 $f$ 的勒貝格**積分存在**或**積分有意義**。如果 $\int f^{+}\,d\mu $ 和 $\int f^{-}\,d\mu $ 都是有限的，那麼稱 $f$ **可積**。:42-45:345 | [
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积分 | 給定一個可測集合 $A$ ，可以定義可積函數在 $A$ 上的積分為：
: $\int _{A}f\,d\mu =\int f1_{A}\,d\mu .$ :345
### 其他定義
除了黎曼積分和勒貝格積分以外，還有若干不同的積分定義，適用於不同種類的函數。
* 達布積分：等價於黎曼積分的一種定義，比黎曼積分更加簡單，可用來幫助定義黎曼積分。
* 黎曼－斯蒂爾傑斯積分：黎曼積分的推廣，用一般的函數 g (x) 代替 x 作為積分變量，也就是將黎曼和中的 $(x_{i+1}-x_{i})$ 推廣為 $(g(x_{i+1})-g(x_{i}))$ 。
* 勒貝格－斯蒂爾傑斯積分：勒貝格積分的推廣，推廣方式類似於黎曼－斯蒂爾傑斯積分，用有界變差函數 g 代替測度 $\mu $ 。
* 哈爾積分：由阿爾弗雷德・哈爾於 1933 年引入，用來處理局部緊拓撲群上的可測函數的積分，參見哈爾測度。
* 伊藤積分：由伊藤清於二十世紀五十年代引入，用於計算包含隨機過程如維納過程或半鞅的函數的積分。
## 性質
通常意義上的積分都滿足一些基本的性質。以下的 ${\mathcal {I}}$ 在黎曼積分意義上表示一個區間，在勒貝格積分意義下表示一個可測集合。
### 線性
積分是線性的。如果一個函數 $f$ 可積，那麼它乘以一個常數後仍然可積。如果函數 $f$ 和 $g$ 可積，那麼它們的和與差也可積。
: $\int _{\mathcal {I}}(\alpha f+\beta g)=\alpha \int _{\mathcal {I}}f+\beta \int _{\mathcal {I}}g\,$
所有在 ${\mathcal {I}}$ 上可積的函數構成了一個線性空間。黎曼積分的意義上，所有區間 [_a_, _b_] 上黎曼可積的函數 $f$ 和 $g$ 都滿足：
: $\int _{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.\,$
所有在可測集合 ${\mathcal {I}}$ 上勒貝格可積的函數 $f$ 和 $g$ 都滿足： | [
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积分 | : $\int _{\mathcal {I}}(\alpha f+\beta g)\,d\mu =\alpha \int _{\mathcal {I}}f\,d\mu +\beta \int _{\mathcal {I}}g\,d\mu .$
在積分區域上，積分有可加性。黎曼積分意義上，如果一個函數 $f$ 在某區間上黎曼可積，那麼對於區間內的三個實數 a, b, c，有
: $\int _{a}^{c}f(x)\,dx=\int _{a}^{b}f(x)\,dx+\int _{b}^{c}f(x)\,dx\,$
如果函數 $f$ 在兩個不相交的可測集 ${\mathcal {I}}$ 和 ${\mathcal {J}}$ 上勒貝格可積，那麼
: $\int _{{\mathcal {I}}\cup {\mathcal {J}}}f\,d\mu =\int _{\mathcal {I}}f\,d\mu +\int _{\mathcal {J}}f\,d\mu .$
如果函數 $f$ 勒貝格可積，那麼對任意 $\epsilon >0$ ，都存在 $\delta $ ，使得 ${\mathcal {F}}$ 中任意的元素 $A$ ，只要 $\mu (A)<\delta $ ，就有 $\int _{A}\left|f\right|\,d\mu <\epsilon $
### 保號性
如果一個函數 $f$ 在某個區間上黎曼可積，並且在此區間上大於等於零。那麼它在這個區間上的積分也大於等於零。如果 $f$ 勒貝格可積並且幾乎總是大於等於零，那麼它的勒貝格積分也大於等於零。作為推論，如果兩個 ${\mathcal {I}}$ 上的可積函數 $f$ 和 $g$ 相比， $f$ （幾乎）總是小於等於 $g$ ，那麼 $f$ 的（勒貝格）積分也小於等於 $g$ 的（勒貝格）積分。
如果黎曼可積的非負函數 $f$ 在 ${\mathcal {I}}$ 上的積分等於 0，那麼除了有限個點以外， $f=0$ 。如果勒貝格可積的非負函數 $f$ 在 ${\mathcal {I}}$ 上的積分等於 0，那麼 $f$ 幾乎處處為 0。如果 ${\mathcal {F}}$ 中元素 $A$ 的測度 $\mu (A)$ 等於 0，那麼任何可積函數在 $A$ 上的積分等於 0。 | [
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积分 | 函數的積分表示了函數在某個區域上的整體性質，改變函數某點的取值不會改變它的積分值。對於黎曼可積的函數，改變有限個點的取值，其積分不變。對于勒貝格可積的函數，某個測度為 0 的集合上的函數值改變，不會影響它的積分值。如果兩個函數幾乎處處相同，那麼它們的積分相同。如果對 ${\mathcal {F}}$ 中任意元素 $A$ ，可積函數 $f$ 在 $A$ 上的積分總等於（大於等於）可積函數 $g$ 在 $A$ 上的積分，那麼 $f$ 幾乎處處等於（大於等於） $g$ 。
### 介值性質
如果 $f$ 在 ${\mathcal {I}}$ 上可積， $M$ 和 $m$ 分別是 $f$ 在 ${\mathcal {I}}$ 上的最大值和最小值，那麼：
: $mL({\mathcal {I}})\leqslant \int _{\mathcal {I}}f\leqslant ML({\mathcal {I}})$
其中的 $L({\mathcal {I}})$ 在黎曼積分中表示區間 ${\mathcal {I}}$ 的長度，在勒貝格積分中表示 ${\mathcal {I}}$ 的測度。
### 絕對連續性
積分的絕對連續性表明，如果函數在某區間或集合上可積，那麼當積分區域是近乎全區域的時候，積分的值也會逼近在全區域上的積分值。如果函數 $f$ 在某區間 ${\mathcal {I}}$ 上黎曼可積，那麼對於滿足 ${\mathcal {I}}_{n}\subset {\mathcal {I}}_{n+1}$ ， $\lim _{n\to \infty }{\mathcal {I}}_{n}={\mathcal {I}}$ 的區間序列 $\left({\mathcal {I}}_{n}\right)_{n\in \mathbb {N} }$ ，有
: $\lim _{n\to \infty }\int _{{\mathcal {I}}_{n}}f(x)\,dx=\int _{\mathcal {I}}f(x)\,dx$
### 積分不等式
涉及積分的基本不等式可以看作是一些離散不等式的類比。如柯西不等式的積分版本：假如有函數 $f$ 和 $g$ 使得 $fg$ 、 $f^{2}$ 、 $g^{2}$ 都在區間 ${\mathcal {I}}$ 上黎曼可積，那麼 | [
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积分 | : $\left(\int _{\mathcal {I}}(fg)(x)\,dx\right)^{2}\leq \left(\int _{\mathcal {I}}f(x)^{2}\,dx\right)\left(\int _{\mathcal {I}}g(x)^{2}\,dx\right).$
而更廣泛的赫爾德不等式也有積分版本。設有正實數 $p$ 和 $q$ ，其倒數和為 1： ${\frac {1}{p}}+{\frac {1}{q}}=1$ ，則對黎曼可積函數 $f$ 和 $g$ ，有以下不等關係（在下式各項有意義的時候）：
: $\left|\int f(x)g(x)\,dx\right|\leqslant \left(\int \left|f(x)\right|^{p}\,dx\right)^{\frac {1}{p}}\left(\int \left|g(x)\right|^{q}\,dx\right)^{\frac {1}{q}}.$
可以看出柯西不等式是赫爾德不等式在 $p=q=2$ 的時候的特例。
此外閔可夫斯基不等式也有積分版本。設有正實數 $p\geqslant 1$ ，則對黎曼可積函數 $f$ 和 $g$ ，有以下不等關係：
: $\left(\int \left|f(x)+g(x)\right|^{p}\,dx\right)^{\frac {1}{p}}\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{\frac {1}{p}}+\left(\int \left|g(x)\right|^{p}\,dx\right)^{\frac {1}{p}}.$
對于勒貝格可積的函數，類似的不等式可以幫助構建 $L^{p}$ 空間。
一個函數 $f$ 可積若且唯若函數 $|f|$ 可積，並且 $f$ 的積分的絕對值，小於等於其絕對值的積分： $\left|\int _{\mathcal {I}}f\right|\leqslant \int _{\mathcal {I}}|f|$ 。如果函數 $f$ 勒貝格可積，那麼 $|f|$ 幾乎處處有限。
## 微積分基本定理
微積分基本定理是將微分運算（求導運算）和積分運算（原函數）聯繫在一起的基本定理。從基本定理可以看出微分和積分運算之間的互逆關係。定理敘述如下：
微積分基本定理的一個實用的直接推論，也被稱為微積分第二基本定理： | [
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积分 | ## 推廣
### 反常積分
狹義的黎曼積分中，被積函數是定義在閉區間（長度有限）上的函數，因此取值也是在有限區間中。反常積分也稱為廣義積分，是對更一般區間上的函數定義的積分，研究在狹義黎曼積分的被積函數條件沒有滿足時，是否能夠有積分的定義。一個基本的情形是，被積函數在半開區間 [_a_, _b_) 上有定義，然而在自變量趨向開區間的某一端（比如說 _b_）時，函數有「瑕點」（函數值趨向無窮或沒有極限）。這時候，考察被積函數在閉區間 [_a_, _b_ - ε] 上的積分值 $I_{\epsilon }$ ，如果當其中的正實數 ε 趨向於 0 的時候，積分值 $I_{\epsilon }$ 趨於一個極限 $I$ ，那麼就稱被積函數在 [_a_, _b_) 上廣義可積，並且稱其為瑕積分 $I$ 。這個定義也可以簡單地記作：
: $\int _{a}^{b}f(x)\,\mathrm {d} x=\lim _{\epsilon \to 0}\int _{a}^{b-\epsilon }f(x)\,\mathrm {d} x$
另一個基本的情形是區間長度為無限大的情形，稱為無窮限廣義積分。比如說被積函數在在閉區間 [_a_, ∞) 上有定義。考慮被積函數在閉區間 [_a_, _b_] 上的積分值 $I_{b}$ ，如果當 _b_ 趨向正無窮大的時候，積分值 $I_{b}$ 趨於一個極限 $I$ ，那麼就稱被積函數在 [_a_, ∞) 上廣義可積，並且稱為無窮限積分 $I$ 。這個定義也可以簡單地記作：
: $\int _{a}^{\infty }f(x)\,\mathrm {d} x=\lim _{b\to \infty }\int _{a}^{b}f(x)\,\mathrm {d} x$
其餘更加複雜的情形包括瑕點在區間內部，或者同時包含了無窮限的情形等等。這些情形都可以拆分為基本情形的組合，然後使用以上的方法探討廣義積分的存在性。比如，考慮函數 $f(x)={\frac {1}{(x+1){\sqrt {x}}}}$ 在正實數區間（0 到正無窮）上的積分（如右圖所示）。這是一個雙重廣義積分。一方面函數在 0 處有瑕點（在 0 附近趨向正無窮），另一方面函數積分區域是無窮限（直到正無窮大）。這時候可以將這個積分分割為兩個部分來考察。比如說以 1 為界限，左右分割為 0 到 1 的積分和 1 到正無窮大的積分。 | [
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积分 | 首先考察 1 到正無窮大的部分，依據上述方法，可以首先考察 $f(x)$ 在閉區間 [1, _t_] 上的積分：
: $I_{t}=\int _{1}^{t}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=2\arctan {\sqrt {t}}-{\frac {\pi }{2}}$
當實數 _t_ 趨於無窮大的時候，上述積分值的極限為 $\lim _{t\to \infty }\left(2\arctan {\sqrt {t}}-{\frac {\pi }{2}}\right)={\frac {\pi }{2}}.$ 所以 $f(x)$ 從 1 到正無窮大的積分可以定義為：
: $\int _{1}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\lim _{t\to \infty }\int _{1}^{t}{\frac {dx}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}$
同樣地，考察從 0 到 1 的部分，可以首先考察 $f(x)$ 在閉區間 [_s_, 1] 上的積分：
: $I_{s}=\int _{s}^{1}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}-2\arctan {\sqrt {s}}$
當正實數 _s_ 趨於 0 的時候，上述積分值的極限為 $\lim _{s\to 0}\left({\frac {\pi }{2}}-2\arctan {\sqrt {s}}\right)={\frac {\pi }{2}}.$ 所以 $f(x)$ 從 0 到 1 的積分可以定義為：
: $\int _{0}^{1}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\lim _{s\to 0}\int _{s}^{1}{\frac {dx}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}$
因此可以定義 $f(x)={\frac {1}{(x+1){\sqrt {x}}}}$ 在正實數區間（0 到正無窮）上的積分為這兩部分的和： | [
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积分 | : $\int _{0}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}=\int _{0}^{1}{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}+\int _{1}^{\infty }{\frac {\mathrm {d} x}{(x+1){\sqrt {x}}}}={\frac {\pi }{2}}+{\frac {\pi }{2}}=\pi $
### 多重積分
狹義積分的積分範圍是實數的一個區間或者可測子集。多重積分將積分範圍擴展到多維空間中的區域或可測子集。比如說二重積分的積分範圍是平面上的一個區域。這時候積分 $\int _{D}f(x)\,\mathrm {d} x$ 中的變量 $x$ 可以是（賦予了拓撲結構的）向量空間裡面的一個向量。富比尼定理證明，在一定條件下，多重積分可以轉換為累次積分。也就是說，在多維空間上的積分可以通過轉化為多個嵌套的一重積分來計算。通常的方法是將多重的積分變量轉變為各個坐標指標上的積分變量。例如，考慮以下二重積分：
: $\int _{C}e^{-x^{2}-y^{2}}\,\mathrm {d} \sigma .$
其中的 $C=\{(x,y)\;|\;x^{2}+y^{2}\leqslant 1\}$ 是一個半徑為 1 的圓盤。這個二重積分可以轉變成：
: $\int _{C}e^{-x^{2}-y^{2}}\,\mathrm {d} \sigma =\int _{-1}^{1}\int _{-{\sqrt {1-y^{2}}}}^{\sqrt {1-y^{2}}}e^{-x^{2}-y^{2}}\,\mathrm {d} x\mathrm {d} y=\int _{0}^{2\pi }\int _{0}^{1}e^{-r^{2}}\,r\mathrm {d} r\mathrm {d} \theta .$
### 路徑積分與曲面積分 | [
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积分 | 路徑積分也稱曲線積分，可以看作是區間上積分的推廣。積分的範圍不是區間（直線段），而是高維空間中的有向曲線。後者稱為積分路徑。路徑積分有很多種類，當積分路徑為閉合曲線時，稱為環路積分或圍道積分。路徑積分的被積函數可以是純量函數（純量場）或向量函數（向量場）。如果被積函數 F 是一個梯度場，那麼 F 的曲線積分與所取的路徑無關，而只與路徑的起點和終點的選取有關。與路徑積分類似，平面區域的二重積分可以推廣為在高維空間中的（有向）曲面上進行積分，稱為曲面積分。路徑積分和曲面積分是物理學中很重要的工具，例如計算電場或重力場中的做功、量子力學中計算粒子出現的概率，會用到路徑積分。流體力學中計算流體的流量、電力學中使用高斯定律計算電場和電荷分布時，會用到曲面積分。
## 種類
* 黎曼積分
* 達布積分
* 勒貝格積分
* 黎曼 - 斯蒂爾吉斯積分
* 數值積分
## 注釋
## 參見
* 微積分基本定理
* 不定積分
* 定積分
* 積分符號
* 積分表
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磁化學 | **磁化學**（magnetic chemistry）是化學的分支，研究化學物質與電磁的關係。物質的磁性的產生，和原子或分子中電子的種種特性有關。
磁化學即研究分子原子中特性與磁學相關的化學問題。
## 磁化率
磁化學中最重要的指標就是磁化率。它衡量物質放在磁場中產生相互作用的強度。**體積磁化率**，用符號 $\chi _{v}$ 表示，由如下關係定義：
: ${\vec {M}}=\chi _{v}{\vec {H}}$
其中， ${\vec {M}}$ 是材料的磁化強度（單位體積中的磁偶極矩），用安培每米（國際單位制）表示； ${\vec {H}}$ 是磁場強度，也用安培每米表示。磁化率是無量綱量。在化學中，**摩爾磁化率**（χmol）是更常用的物理量。它用 m3·mol−1（國際單位制）或 cm3·mol−1 （厘米 - 克 - 秒制）表示，定義為
: $\chi _{\text{mol}}=M\chi _{v}/\rho $
其中 ρ 是密度，單位 kg・m−3 （國際單位制）或 g・cm−3 （厘米 - 克 - 秒制）；_M_ 是摩爾質量，單位 kg・mol−1 （國際單位制）或 g・mol−1 （厘米 - 克 - 秒制）。
有很多方法可用於測量磁化率。
* 超導量子干涉儀是一種非常敏感的磁力計。
* 可用核磁共振測量溶液中物質的磁化率。
## 磁性的類型
由於原子中的每個電子行為都與磁鐵相似，當一個孤立原子被放在磁場中，即會與磁場產生相互作用，也就是說，電子具有磁矩。相互作用有兩種類型。
1. 抗磁性。當原子處於磁場中，原子會磁極化，即產生感應磁矩。這種相互作用力有將原子推出磁場的趨勢。通常用負符號標記抗磁性。抗磁的原子常沒有未配對電子，也就是說每個電子都與同一原子軌道上的另一個電子相配對。兩個電子的矩會互相抵消，於是原子便沒有淨磁矩。 然而，對於有六個未配對電子的 Eu3+ 離子而言，軌道角動量會抵消電子角動量，於是這個原子在 0K 溫度下具有抗磁性。
2. 順磁性。至少一個電子未與其他電子配對。這樣的原子具有永久的磁矩。當原子處於磁場中，就會被磁場所吸引。通常用正符號標記順磁性。
當原子處於化合物中，其磁性則會隨化學環境發生改變。測量磁矩可以獲取有用的化學信息。 | [
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磁化學 | 在特定晶體材料中，獨立的磁矩會相互排列起來（磁矩既有強度也有方向）。這就導致了鐵磁性、反鐵磁性和亞鐵磁性。這些晶體性質是整體而言的，和化學性質幾乎沒有關係。
## 科研
2020 年 10 月，中國科學院合肥物質科學研究院強磁場中心磁性功能材料與器件研究團隊利用洛倫茲透射電子顯微鏡研究了 Kagome 晶體 Fe3Sn2 中的磁結構，澄清了該類材料中的複雜多拓撲態的起源。2021 年 7 月，中國中科院合肥物質科學研究院強磁場中心盛志高研究團隊，利用穩態強磁場大科學裝置 SM1 超導磁體開展特殊功能材料研究，發現強磁場可以有效調控化學反應的速率、反應路徑以及反應產物。
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安时 | **安培**（英語：ampere，法語：ampère）簡稱**安**（amp），是電流強度的單位，國際單位制七個基本單位之一，符號 **A**。安培是以法國數學家和物理學家安德烈 - 馬里・安培命名的，為了紀念他在古典電磁學方面的貢獻。
實際情況中，安培是對單位時間內通過導體橫截面的電荷量的度量。1 秒內通過橫截面的電荷量為 1 庫侖（6.241×1018 個電子的電荷量）時，電流強度為 1 安培。1 安培也相當於 1 伏特電壓施加在 1 歐姆阻抗時，所通過的電流量（電流強度）。
比一安培小的電流可以用毫安、微安等單位表示。
* 1 安（A）= 1000 毫安（mA）
* 1 毫安（mA）= 1000 微安（μA）
## 定義
2018 年 11 月 16 日，第 26 屆國際計量大會決定，重新定義安培為：
其中 Δ_ν_Cs 是指銫 133 原子基態的兩個超精細能階之間的躍遷輻射頻率。新定義使用基本電荷 e 重新定義安培，這樣的條件可以在實驗室中重現，同時較簡單易明，於 2019 年 5 月 20 日生效（2019 年國際單位制基本單位重新定義）。
## 歷史
安培最初是被定義為厘米 - 克 - 秒制中電流單位絕對安培的十分之一。如此確定它的大小是為了保證從國際單位制中的其他單位推導安培得到的值比較合適。
「國際安培」是一個較早的電流單位，定義為使硝酸銀溶液中每秒析出 0.001118000 克銀的電流。之後，更精確地測量發現，這一電流是 0.99985 A。
1948 年第九屆國際計量大會決定定義安培為：
2005 年，國際計量委員會同意研究將元電荷電荷量用於安培定義的可能。新的定義在 2014 年的第 25 屆國際度量衡委員會上被討論，於 2019 年 5 月 20 日生效。
2018 年第二十六屆國際計量大會通過給予元電荷確定的電荷量，確定了安培的新定義。自 2019 年 5 月 20 日起，元電荷的電荷量被確定為 $1.602176634\times 10^{-19}C$ ，而 $C=A\cdot s$ 。由此，1 安培所代表的電流強度大小由元電荷電荷量和秒確定。
## 應用
安培這一單位與瓦特秤和歐姆定律中的單位伏特、歐姆相聯繫。而在約瑟夫森效應和量子霍爾效應中也都各自有所應用。
現在，建立一安培的技術已經能夠達到 10-7 的逼近誤差。
## 例子
* 計算器工作電流 1×10-4A | [
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] |
安时 | * 手機充電電流 0.5-2A （快充可到 5A 以上)
* 閃電約 1×104A
* 住宅配電箱約 60A
* 住宅斷路器，常見的有 10/20A
* 塑殼斷路器（MCCB），常見 200-800A
* 空氣斷路器（ACB），用於低壓電（380V），常見 1600/2500/4000A
* 用於配電的 1650kVA 三相變壓器，低壓線圈（380V）最大輸出 2500A
## 參見
* 安培計
* 電流
* 歐姆定律
* 庫侖
* 必歐 - 沙伐定律
## 注釋
## 參考資料
## 外部連結
* The NIST Reference on Constants, Units, and Uncertainty（頁面存檔備份，存於網際網路檔案館）
* NIST _Definition of ampere and μ0_（頁面存檔備份，存於網際網路檔案館） | [
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绝对黑体 | 在熱力學中，**黑體**（英語：Black body），是一個理想化的物體，它能夠吸收外來的全部電磁輻射，並且不會有任何的反射與透射。隨著溫度上升，黑體所輻射出來的電磁波與光線則稱做黑體輻射。這個名詞在 1862 年由古斯塔夫・克希荷夫所提出並引入熱力學內。
## 定義簡述
黑體對於任何波長的電磁波的吸收係數為 1，透射係數為 0。但黑體未必是黑色的，即使不反射任何的電磁波，它也可以放出電磁波，而這些電磁波的波長和能量則全取決於黑體的溫度，不因其他因素而改變。
對於人的視覺而言，黑體在 700K 以下時看起來是黑色的，這是由於在 700K 之下的黑體所放出來的輻射能量很小且輻射波長在可見光範圍之外。若黑體的溫度高過上述的溫度的話，黑體則不會再是黑色的了，它會開始變成紅色，並且隨著溫度的升高，而分別有橘色、黃色、白色等顏色出現，即黑體吸收和放出電磁波的過程遵循了光譜，其軌跡為普朗克軌跡（或稱為黑體軌跡）。黑體輻射實際上是黑體的熱輻射。在黑體的光譜中，由於高溫引起高頻率即短波長，因此較高溫度的黑體靠近光譜結尾的藍色區域而較低溫度的黑體靠近紅色區域。
在室溫下，黑體放出的基本為紅外線，但當溫度漲幅超過了百度之後，黑體開始放出可見光，根據溫度的升高過程，分別變為紅色，橙色，黃色，白色和藍色。當黑體變為白色的時候，它同時會放出大量的紫外線。
黑體單位表面積的輻射通量 $P$ 與其溫度的四次方成正比，即：
: $P=\sigma T^{4}$
式中 $\sigma $ 稱為斯特凡 - 波耳茲曼常數，又稱為斯特凡常數。
黑體的放射過程引發物理學家對量子場內的熱平衡狀態的興趣。在古典物理中，所有熱平衡的傅里葉模型都遵循能量均分定理。當物理學家使用古典物理解釋黑體時，不可避免的發生了紫外災變，即用於計算黑體輻射強度的瑞立 - 金斯定律在輻射頻率趨向於無窮大時計算結果也趨向於無窮大。由於黑體可以用於檢驗熱平衡的性質，因為它放出的輻射遵循熱力學散射，歷史上對黑體的研究成為了量子物理開始的契機。
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绝对黑体 | 在實驗室內，研究者們可以模擬最靠近黑體的設備是大型空腔表面所開的一個小洞。只要有光線射向這個小洞，光線便會在空腔內反射或者被空腔內的牆壁所吸收，而只剩下微乎極微的光線可以再由洞口射出，亦即入射的光線幾乎都被吸收了，而沒有反射。如此，這個小洞就有如一個黑體一般，而且當空腔開始加熱以後，小洞發出來的輻射所形成的光譜將會以量子化計算且和空腔材質無關。依據克希荷夫熱輻射定律，光譜的圖形只和空腔的溫度有關，而和其他因素沒有關係。
## 解釋
### 黑體模擬裝置
通常，可以用一個開有小孔的腔體模擬黑體。射入小孔的電磁波經過多次反射後也難以射出，可以認爲電磁波被完全吸收。因此這個小孔就成爲一個黑體。但對於波長大於小孔直徑的電磁波，將有部分被反射，因此對於這部分電磁波而言，小孔並不是黑體。
## 黑體的實現
_黑體的實現是指現實世界的物理體現。_
### Cavity with a hole
1898 年，奧托・魯默（Otto Lummer）和費迪南德・庫爾鮑姆（Ferdinand Kurlbaum）發表了他們的腔輻射源的論文。至今為止，它們的設計在輻射測量方面幾乎沒有改變。 它是鉑金盒壁上的一個孔，被隔膜隔開，其內部被氧化鐵塗黑。 它是逐步改進測量結果的重要組成部分，從而導致了普朗克定律的發現。1901 年描的版本，其內部被鉻，鎳和氧化鈷的混合物塗黑。
### Near-black materials
人們感興趣於用於偽裝的類黑體材料和用於雷達隱身的雷達吸收材料。它們還可以用作太陽能收集器和紅外熱探測器。作為一種理想的輻射發射器，具有黑體行為的熱物質會產生高效的紅外加熱器，尤其是在無法使用對流加熱的空間或真空中。它們還可以用作望遠鏡和照相機的減反射表面，以減少雜散光，並收集有關高對比度區域中物體的資訊（例如，觀察恆星周圍的行星），由於黑體的物質會吸收光來自非目標的來源。
人們早就知道，漆黑的塗層會使物體接近黑色。在製造的碳奈米管中發現了對碳煙 (lamp-black) 的改進。奈米多孔材料可以達到接近真空的折射率，在一種情況下，其平均反射率為 0.045％。 2009 年，一支日本科學家小組基於垂直排列的單壁碳奈米管，創建了一種接近於理想黑體的奈米黑材料。在從紫外線到遠紅外區域的光譜範圍內，它吸收了 98％至 99％的入射光。 | [
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] |
绝对黑体 | 幾乎完美的黑色材料的其他示例是通過化學刻蝕鎳磷合金 (nickel–phosphorus alloy)，垂直排列的碳奈米管陣列 (vertically aligned carbon nanotube arrays) 和花朵碳奈米結構 (flower carbon nanostructures) ； 都吸收 99.9％或更多的光。（奈米碳管黑體）
## 參考文獻
## 參見
* 黑體輻射
* 普朗克公式
* 白體 (物理學)
* 克希荷夫熱輻射定律
* 斯特凡－波茲曼定律 | [
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黑体辐射 | **黑體輻射**指處於熱力學平衡態的黑體發出的電磁輻射。黑體輻射的電磁波譜只取決於黑體的溫度。另一方面，所謂黑體輻射是光和物質達到平衡所表現出的現象。物質達到平衡，所以可以用一個溫度來描述物質的狀態，而光和物質的交互作用很強，如此光和光之間也可以用一個溫度來描述（光和光之間本身不會有交互作用，但光和物質的交互作用很強），而描述這關係的便是普朗克分佈（Planck distribution）。黑體輻射能量按波長的分布僅與溫度有關。
黑體不僅僅能全部吸收外來的電磁輻射，且散射電磁輻射的能力比同溫度下的任何其它物體強。對於黑體的研究，使自然現象中的量子效應被發現。而黑體作為一個理想化的物體，在現實中是不存在的，因此現實中物體的輻射也與理論上的黑體輻射有所出入。但是，可以觀察一些非常類似黑體的物質發出的輻射，例如一顆恆星或一個只有單一開口的空腔所發出的輻射。舉個例來說，人們觀測到宇宙背景輻射，對應到一個約 3K 的黑體輻射，這暗示宇宙早期光是和物質達到平衡的。而隨著時間演化，溫度慢慢降了下來，但方程式依然存在。（頻率和溫度的效應抵銷）
## 黑體輻射方程
### 黑體輻射本領
古斯塔夫・基爾霍夫證明，對於任意一個物體，輻射本領 $E(\nu ,T)$ 與吸收率 $A(\nu ,T)$ 之比是一個與組成物體的物質無關的普適函數 (以 $f(\nu ,T)$ 表示)
: ${\frac {E(\nu ,T)}{A(\nu ,T)}}=f(\nu ,T)$
其中，輻射本領 $E(\nu ,T)$ 為單位時間內從輻射體表面的單位面積上發射出的輻射能量的頻率分布，所以，在 $\Delta t$ 的時間，從 $\Delta S$ 面積上發射出頻率在 $\nu --\nu +\Delta \nu $ 範圍內的能量為 $E(\nu ,T)\Delta t\Delta S\Delta \nu $ 。因此 $E(\nu ,T)$ 的單位為 $J/m^{2}$ ，可以證明，黑體輻射本領與輻射體的能量密度分布 $u(\nu ,T)$ 的關係為
: $E(\nu ,T)={\frac {c}{4}}u(\nu ,T)$
$u(\nu ,T)$ 的單位為 $J\cdot s/m^{3}$ 吸收率 $A(\nu ,T)$ 則為照到物體上的輻射能量分布被吸收的份額，由於黑體的吸收率為 1，所以它的輻射本領
: $E(\nu ,T)=f(\nu ,T)$ | [
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黑体辐射 | 這意味著黑體輻射本領等價於普適函數（與物質無關）
同時也可以以用 $E(\lambda ,T)$ 來表達輻射本領
: $E(\lambda ,T)={\frac {\nu ^{2}}{c}}E(\nu ,T)$
$$
E(\lambda ,T)$ 的單位為 $J/m^{3}\cdot s
$$
### 黑體輻射的普朗克公式
用於描述在任意溫度 $T\,$ 下，從一個黑體中發射的電磁輻射的輻射率與電磁輻射的頻率的關係公式。這裡輻射率是頻率 $\nu $ 的函數：
: $I(\nu ,T)d\nu =\left({\frac {2h\nu ^{3}}{c^{2}}}\right){\frac {1}{e^{\frac {h\nu }{kT}}-1}}\,d\nu $
各個物理量的意義
: $I\,$ ：輻射率，在單位時間內從單位表面積和單位立體角內以單位頻率間隔或單位波長間隔輻射出的能量，[_J_·_s_ -1·_m_-2·_sr_ -1·_Hz_ -1];
: $h\,$ : 普朗克常數，[_J_·_s_ ];
: $c\,$ : 光速，[_m_/_s_ ];
: $k\,$ : 玻爾茲曼常數，[_J_ /_K_ ];
: $\nu \,$ : 電磁波的頻率，[_Hz_ ];
: $T\,$ : 黑體的溫度，[_K_ ].
### 黑體輻射的維恩位移定律
維恩位移定律表述了不同溫度的黑體波譜之間的聯繫。一旦某一個溫度下的黑體波譜形狀已知，則可通過維恩位移定律推導出同一黑體在其它溫度下的波譜形狀。
維恩位移定律計算出黑體輻射強度達到最大時的波長， $\lambda _{\mathrm {max} }$ ，這個物理量只和黑體的溫度相關：
: $\lambda _{\mathrm {max} }={\frac {b}{T}}$
其中 _b_ 為比例常數，稱為**維恩位移常數**，數值等於 2.897 7721 (26)×10–3 m K（2010 年國際科技數據委員會推薦值，括號中為 68.27% 置信度下的不確定尾數）。
注意到強度的峰值可以表達為單位波長強度或是單位頻率強度，在維恩位移定律中使用的是單位波長強度的表達式，而在上面的普朗克黑體輻射定律中則使用的是單位頻率強度。單位頻率能量達到最大時的波長為 | [
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黑体辐射 | : $\lambda '_{\mathrm {max} }={\frac {5.1\times 10^{-3}{\mbox{ m·K}}}{T}}$
### 黑體輻射的斯特藩 - 玻爾茲曼定律
這條定理指出，一個黑體表面單位面積放出的能量正比於其絕對溫度的 4 次方：
: $j^{\star }=\sigma T^{4},$
其中 $j^{\star }$ ：單位面積單位時間所放出的總能量，[ $J\cdot {s^{-1}}\cdot {m^{-2}}$ ]
$T$ : 黑體的絕對溫度，[_K_ ]
$\sigma $ ：斯特藩 - 玻爾茲曼常數，
: $\sigma ={\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}=5.670400(40)\times 10^{-8}\,J\cdot s^{-1}\cdot m^{-2}\cdot K^{-4}.$
## 人體的輻射
黑體輻射同樣適用於人體，因為人體的一部分能量以電磁波的形式散射出體表，其中大部分為紅外線。
淨放射功率是吸收功率和放出功率的差值：
: $P_{net}=P_{emit}-P_{absorb}.$
代入黑體輻射的斯特藩 - 玻爾茲曼定律：
: $P_{net}=A\sigma \epsilon \left(T^{4}-T_{0}^{4}\right)\,$
人體的表面積約為 2 平方米，皮膚和大多數衣物（非金屬材質）中，遠紅外線熱發射率基本相同。皮膚的溫度大概為 33℃，但在約為 20℃的環境溫度影響下，衣物會使體表溫度降為大約 28℃。因此，人體的淨放射功率約為
: $P_{net}=100\ \mathrm {W} \,$
人體一天以上述放射方式放出的總能量大約為 9000 千焦（100 W 乘以 86400 秒），即 2000 千卡。一個 40 歲的成年人的基礎代謝率約為 35 千卡 /（米 2・小時），即為 1700 千卡每天（以 2 平方米為基準）。實際上即便是靜坐的成年人每日的平均代謝率也比基礎代謝率高出約 50% 到 70%。 | [
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黑体辐射 | 熱對流和體液蒸發也是人體散失能量的重要因素。因為努塞爾特數遠遠大於單位一，所以熱傳導可以忽略。蒸發（汗液）這個要素只有在熱輻射和熱對流在某個恆溫環境內不起主導因素時才給予考慮。自由熱傳導率儘管比輻射率小，但是也是可以進行比較的。因此，人體在靜止涼爽的環境中散失的總熱能的三分之二是由於熱輻射導致的。由於使用了很多假設情況下的近似值，所以這個結果只能算是粗略的估計。人體周圍的空氣運動所引起的對流或是體液蒸發同樣和熱輻射一樣是導致體能流失的重要因素。
如果在人體這個黑體上使用維恩位移定律，可以得到人體放出輻射的波長峰值為：
: $\lambda _{peak}={\frac {2.898\times 10^{6}\ \mathrm {K} \cdot \mathrm {nm} }{305\ \mathrm {K} }}=9500\ \mathrm {nm} \,$ .
這個結果可以解釋為什麼人體熱像儀的波長一般設為非常靈敏的 7000 到 14000 奈米。
## 行星和其衛星之間的熱力學關係
黑體輻射定理的應用之一是用於概略的估計一個行星的溫度。其表面可能由於溫室效應而比估計溫度高。
### 因素
行星的溫度主要和以下幾個因素相關：
* 外來的輻射（比如來自某個恆星）
* 本身放出的輻射（比如地球自身紅外輻射）
* 反照率（行星反射的輻射）
* 溫室效應（只針對有大氣的行星）
* 行星自身產生的能量（由於放射性衰變，潮汐加熱和克赫歷程）
所有的輻射，無論是行星內部產生的，其他恆星還是其本身放出的，對行星的溫度都有很重要的影響。以下的推導即著重討論輻射。
### 推導
首先使用斯特藩 - 玻爾茲曼定律得到太陽放射出的總功率（能量 / 秒）：
: $P_{Semt}=\left(\sigma T_{S}^{4}\right)\left(4\pi R_{S}^{2}\right)\qquad \qquad (1)$
其中
: $\sigma \,$ ：斯特藩 - 玻爾茲曼常數
: $T_{S}\,$ ： 太陽的表面溫度
: $R_{S}\,$ ： 太陽的半徑
太陽平均的向各個方向放出能量，因此，地球實際上只是接受到其中很小的一部分。這部分能量為（指接觸到大氣層外部）： | [
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] |
黑体辐射 | : $P_{SE}=P_{Semt}\left({\frac {\pi R_{E}^{2}}{4\pi D^{2}}}\right)\qquad \qquad (2)$
其中
: $R_{E}\,$ : 地球的半徑
: $D\,$ ：天文單位， 太陽與地球的平均距離
由於本身的高溫，太陽發出的射線大多數屬於紫外線和可見光（UV-Vis）頻率範圍。在這個頻率範圍內，地球會反射一部分能量，其數量為 $\alpha $ ，即地球對 UV-Vis 範圍射線的反照率。反過來，即地球吸收了 $1-\alpha $ 的太陽光，並反射了剩下的。地球和其大氣層所吸收的能量為：
: $P_{abs}=(1-\alpha )\,P_{SE}\qquad \qquad (3)$
雖然地球僅僅以一個面積為 $\pi R^{2}$ 的圓形區域進行吸收，但是它同時以一個球體的形態向各個方向放出能量。假設地球是一個完全黑體，它將遵循斯特藩 - 玻爾茲定理：
: $P_{emt\,bb}=\left(\sigma T_{E}^{4}\right)\left(4\pi R_{E}^{2}\right)\qquad \qquad (4)$
其中 $T_{E}$ 是地球的溫度。由於地球的溫度明顯低於太陽，其放射的多為光系的紅外線（IR）部分。在這個頻率範圍內，地球會放出黑體總放射波的一部分，大約為 ${\overline {\epsilon }}$ ， ${\overline {\epsilon }}$ 是紅外線頻率的平均放射率。因此地球和其大氣層實際放出的能量為：
: $P_{emt}={\overline {\epsilon }}\,P_{emt\,bb}\qquad \qquad (5)$
假設地球處於熱平衡，則吸收的能量等於放射的能量：
: $P_{abs}=P_{emt}\qquad \qquad (6)$
代入所有關於太陽和地球能量的表達式（1-5）可以得到：
: $T_{E}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\epsilon }}}}}{2D}}}$
換句話說，考慮到所有的估計值，地球的溫度與下列因素有關：太陽的表面溫度，太陽的半徑，日地間距，以及地球的反照率和紅外發射率。
### 地球的溫度
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] |
黑体辐射 | : $T_{S}=5778\ \mathrm {K} ,$
: $R_{S}=6.96\times 10^{8}\ \mathrm {m} ,$
: $D=1.496\times 10^{11}\ \mathrm {m} ,$
: $\alpha =0.306\ $
並將平均放射率設為單位量，可以得到地球的「有效溫度」為：
: $T_{E}=$ 254.356 K or -18.8 ℃.
這個溫度值是基於地球是一個完全黑體的假設，忽略溫室效應並認為地球的反照率完全不變的基礎上得到的。而實際上地球僅是非常接近一個完美黑體，所以必須將估計溫度定為比有效溫度高出好幾度。如果想要估計地球在沒有大氣層的情況下的溫度，可以使用月球的反照率和發射率進行計算。月球的反照率和發射率大約為 0.1054 和 0.95， 因此，可以得到這種情況下的溫度約為 1.36 ℃.
地球的平均反照率的估計值在 0.3–0.4 之間，由此可以得到不同的估計溫度。進行計算時相較於太陽的溫度，尺寸和日地距離，人們更加常用日照常量（總日照量密度）。比如使用 0.4 為反照率並使用日照量密度 1400 W m−2，可以得到約為 245K 的地球溫度。同理，如果使用 0.3 的反照率以及 1372 W m−2 的日照常量，地球溫度為 255 K。
## 運動黑體的都卜勒效應
都卜勒效應是著名的物理現象，人們用它來描述當一個光源對觀察者作相對運動時，其光線頻率的變化。若一單色光源的發射頻率為 _f_，則當它相對觀察者做相對運動時，觀察者看起來其頻率為 _f'_：
: $f'=f{\frac {1}{\sqrt {1-v^{2}/c^{2}}}}(1-{\frac {v}{c}}\cos \theta )$
其中 _v_ 是這個光源對觀察者的相對速度，_θ_ 是速度向量和觀察者及光源間連線的夾角，_c_ 為光速。上面的公式為總公式，還可以從其得到一些更簡單的特殊例子，如光源直接向觀察者移動（_θ_ = π） 或是離開觀察者（_θ_ = 0），當然其移動速度遠遠小於 _c_。
## 參考文獻
## 參閱
* 盒中氣體
* 黑體
* 普朗克黑體輻射定律 | [
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代数几何 | **代數幾何**（英語：**algebraic geometry**）是數學的一個分支，經典代數幾何研究多項式方程的零點。現代代數幾何將抽象代數，尤其是交換代數，同幾何學的語言和問題結合起來。
代數幾何的基本研究對象為代數簇。代數簇是由空間坐標的若干代數方程的零點集。常見的例子有平面代數曲線，比如直線、圓、橢圓、拋物線、雙曲線、三次曲線（非奇異情形稱作橢圓曲線）、四次曲線（如雙紐線，以及卵形線）、以及一般 n 次曲線。代數幾何的基本問題涉及對代數簇的分類，比如考慮在雙有理等價意義下的分類，即雙有理幾何，以及模空間問題，等等。
代數幾何在現代數學占中心地位，與多複變函數論、微分幾何、拓撲學和數論等不同領域均有交叉。始於對代數方程組的研究，代數幾何延續解方程未竟之事；與其求出方程實在的解，代數幾何嘗試理解方程組的解的幾何性質。代數幾何的概念和技巧都催生了某些最深奧的數學的分支。
進入 20 世紀，代數幾何的研究又衍生出幾個分支：
* 研究代數簇中，坐標在有理數域或代數數域里的點；這一分支發展成算術幾何（更經典地，丟番圖幾何），屬於代數數論的分支。
* 研究代數簇的實點，即實代數幾何。
* 奇點理論的一大部分致力於研究代數簇中的奇異點，及關於奇異點的解消的存在性和方法。
* 代數簇的上同調理論，如晶體上同調、平展上同調、以及 Motive（頁面存檔備份，存於網際網路檔案館）上同調。
* 幾何不變量理論，起始於戴維・芒福德在二十世紀六十年代的研究，其思想起源於大衛・希爾伯特的古典不變量理論。
* 隨著計算機的興起，計算代數幾何作為代數幾何與符號運算兩支的交叉而嶄露頭角。這一分支本質上包含開發算法和軟體與尋找顯代數簇的性質這兩項工作。
20 世紀以來，代數幾何主流的許多進展都在抽象代數的框架內進行，越發強調代數簇「內蘊的」性質，即那些不取決於代數簇在射影空間的具體嵌入方式的性質，與拓撲學、微分幾何及復幾何等學科的發展相應。抽象代數幾何的一大關鍵成就是格羅滕迪克的概形論；概形論允許人們應用層論研究代數簇，某種意義上與應用層論研究微分流形與解析流形是否相似。概形論延伸了點的概念。在經典代數幾何中，根據希爾伯特零點定理，一個仿射代數簇的一點對應於坐標環上的一個極大理想，仿射概形上的子簇則對應於坐標環的素理想。而在概型論中，概型的點集包含了經典情況代數簇的點集，以及所有子簇的信息。這種方法使得經典代數幾何（主要涉及閉點）同時聯繫起了微分幾何、數論等主流分支的問題研究。 | [
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代数几何 | ## 基本概念
### 聯立多項式的零點
在古典代數幾何中，主要的研究對象是一組多項式的公共零點集，即同時滿足一個或多個多項式方程的所有點組成的集合。 例如，在三維歐幾里德空間 $\mathbb {R} ^{3}$ 中的單位球面被定義為滿足方程
: $x^{2}+y^{2}+z^{2}-1=0$
的所有點 $(x,y,z)$ 的集合。
一個 "傾斜的" 圓周在三維歐幾里德空間 $\mathbb {R} ^{3}$ 中可以被定義為同時滿足如下兩個方程
: $x^{2}+y^{2}+z^{2}-1=0$ ，
: $x+y+z=0$
的所有點 $(x,y,z)$ 的集合。
### 仿射簇
現在我們開始進入稍微抽象的領域。考慮一個數域 $k$ ，在古典代數幾何中這個域通常是複數域 $\mathbf {C} $ ，現在我們把它推廣為一個代數封閉的數域。我們定義數域 $k$ 上的 $n$ 維仿射空間 ${\mathbb {A} }_{k}^{n}$ ，簡單講來，它只是一些點的集合，以下為方便我們簡記為 ${\mathbb {A} }^{n}$ 。
如果函數
: $f:{\mathbb {A} }^{n}\to {\mathbb {A} }^{1}$
可以被寫為多項式，即如果有多項式 $p$ 在
: $k[x_{1},\cdots ,x_{n}]$ 上，
使得對 ${\mathbb {A} }^{n}$ 上的每個點
: $(t_{1},\cdots ,t_{n})$
都有
: $f(t_{1},\cdots ,t_{n})=p(t_{1},\cdots ,t_{n})$ ，定義這個函數是正則的。
$n$ 維仿射空間的正則函數正是數域 $k$ 上 $n$ 個變量的多項式。我們將 ${\mathbb {A} }^{n}$ 上的正則函數記為 $k[{\mathbb {A} }^{n}]$ 。
## 與拓撲場論的關係
拓撲場論是數學物理中對 sigma 模型的場做路徑積分量子化的理論。
sigma 模型是從一個實二維曲面到一個固定空間的映射，再加上此二維曲面上一些叢的平滑截面。其中映射部份被稱爲玻色場，截面部份被稱爲費米場。該理論的主要目的是通過路徑積分計算配分函數。 | [
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代数几何 | 在一些特殊情況下，可以用局部化方法把配分函數原在無限維空間的積分化簡爲在有限維空間的積分。對不同的作用量而言，這個過程給出了代數幾何的幾種計數理論，包括：
* Gromov Witten 不變量（即 IIA 型弦論）
* 辛流形裡的全純曲線計數
* Seiberg Witten 不變量
* Chern Simon 數規範場
IIB 型弦論則利用了 Hodge 結構的形變來計算。
## 主要研究者
* 亞歷山大・格羅滕迪克
* 讓 - 皮埃爾・塞爾
* 奧斯卡・扎里斯基
* 安德烈・韋伊
* 米高・弗朗西斯・阿蒂亞
* 愛德華・威滕
* 小平邦彥
* 森重文
* 廣中平祐
* 梅村浩
* 飯高茂
* 斎藤秀司
* 周煒良
## 註解
## 參見
* 微分幾何
* 幾何代數
* 代數拓樸
* 微分拓樸
## 參考書目
經典教科書，先於概形:
* W. V. D. Hodge; Daniel Pedoe. Methods of Algebraic Geometry: Volume 1. Cambridge University Press. 1994. ISBN 978-0-521-46900-5.
* Hodge, W. V. D.; Pedoe, Daniel. Methods of Algebraic Geometry: Volume 2. Cambridge University Press. 1994. ISBN 978-0-521-46901-2.
* Hodge, W. V. D.; Pedoe, Daniel. Methods of Algebraic Geometry: Volume 3. Cambridge University Press. 1994. ISBN 978-0-521-46775-9.
不使用概形的語言的現代教科書:
* Phillip Griffiths; Joe Harris. Principles of Algebraic Geometry. Wiley-Interscience. 1994. ISBN 978-0-471-05059-9.
* Joe Harris. Algebraic Geometry: A First Course. Springer-Verlag. 1995. ISBN 978-0-387-97716-4. | [
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代数几何 | * David Mumford. Algebraic Geometry I: Complex Projective Varieties 2nd ed. Springer-Verlag. 1995. ISBN 978-3-540-58657-9. 引文格式 1 維護：冗餘文本 (link)
* Miles Reid. Undergraduate Algebraic Geometry. Cambridge University Press. 1988. ISBN 978-0-521-35662-6.
* Igor Shafarevich. Basic Algebraic Geometry I: Varieties in Projective Space 2nd ed. Springer-Verlag. 1995. ISBN 978-0-387-54812-8. 引文格式 1 維護：冗餘文本 (link)
關於概形的教科書和參考書:
* David Eisenbud; Joe Harris. The Geometry of Schemes. Springer-Verlag. 1998. ISBN 978-0-387-98637-1.
* 亞歷山大・格羅滕迪克。代数几何基础. Publications mathématiques de l'IHÉS. 1960.
* 亞歷山大・格羅滕迪克。代数几何基础 **1** 2nd ed. Springer-Verlag. 1971. ISBN 978-3-540-05113-8. 引文格式 1 維護：冗餘文本 (link)
* Robin Hartshorne. Algebraic Geometry. Springer-Verlag. 1997. ISBN 978-0-387-90244-9.
* David Mumford. The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians 2nd ed. Springer-Verlag. 1999. ISBN 978-3-540-63293-1. 引文格式 1 維護：冗餘文本 (link) | [
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] |
代数几何 | * Igor Shafarevich. Basic Algebraic Geometry II: Schemes and Complex Manifolds. Springer-Verlag. 1995. ISBN 978-0-387-54812-8.
網際網路上的資料:
* Kevin R. Coombes: _Algebraic Geometry: A Total Hypertext Online System_
* _Algebraic geometry_entry on PlanetMath（頁面存檔備份，存於網際網路檔案館）
* _Algebraic Equations and Systems of Algebraic Equations_（頁面存檔備份，存於網際網路檔案館） at EqWorld: The World of Mathematical Equations | [
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数理模型方法 | **數學模型**（mathematical model）是使用數學來將一個系統簡化後予以描述。數學模型廣泛應用在自然科學（如物理學、化學、生物學、宇宙學）、工程學科（如電腦科學，人工智慧）、以及社會科學（如經濟學、心理學、社會學和政治科學）上。科學家和工程師用模型來解釋一個系統，研究不同組成部分的影響，以及對行為做出預測。常見的模型包括動態系統、機率模型、微分方程式或賽局模型等等。描述不同物件的模型可能有相同的形式，同一個模型也可能包含了不同的抽象結構。
## 分類
數學模型通常由關係與變數組成。關係可用算符描述，例如代數算符、函數、微分算符等。變數是關注的可量化的系統參數的抽象形式。算符可以與變數相結合發揮作用，也可以不與變數結合。 通常情況下，數學模型可被分為以下幾類：
* **線性與非線性：**在數學模型中，如果所有變數表現出線性關係，由此產生的數學模型為線性模型。否則，就為非線性模型。對線性與非線性的定義取決於具體資料，線性相依模型中也可能含有非線性表達式。例如，在一個線性統計模型中，假定參數之間的關係是線性的，但預測變數可能是非線性的。同理，如果一個微分方程式定義為線性微分方程式，指的是它可以寫成線性微分算子的形式，但其中仍可能有非線性的表達式。在數學規劃模型中，如果目標函數和約束條件都完全可以由線性方程式表示，那麼模型為線性模型。如果一個或多個目標函數或約束表示為非線性方程式，那麼模型是一個非線性模型。
即使在相對簡單的系統中，非線性也往往與混沌和不可逆性等現象有關。雖然也有例外，非線性系統和模型往往比線性研究起來更加困難。解決非線性問題的一個常見方法是線性化，但在嘗試用來研究對非線性依賴性很強的不可逆性等方面時就會出現問題。
* **靜態與動態：**動態模型對系統狀態隨時間變化情況起作用，而靜態（或穩態）模型是在系統保持平穩狀態下進行計算的，因而與時間無關。動態模型通常用微分方程式描述。 | [
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数理模型方法 | * **顯式與隱式：**如果整體模型的所有輸入參數都已知，且輸出參數可以由有限次計算求得（稱為線性規劃，不要與上面描述的線性模型相混淆），該模型稱作顯式模型。但有時輸出參數未知，相應的輸入必須通過迭代過程求解，如牛頓法（如果是線性模型）或布洛登法（如是非線性模型）。例如噴氣發動機物理特性如渦輪和噴管喉道面積，可以在給定特定飛行條件和功率設定的熱力學迴圈（空氣和燃油的流量、壓力、溫度）的情況下顯式計算出來，但不能用物理性質常數顯式計算出其他飛行條件和功率設定下發動機的工作週期。
* **離散與連續：**離散模型將物件視作離散的，例如分子模型中的微粒，又如機率模型中的狀態。而連續模型則由連續的物件所描述，例如管道中流體的速度場，固體中的溫度和壓力，電場中連續作用於整個模型的點電荷等。
* **確定性與機率性（隨機性）：**確定性模型是所有變數集合的狀態都能由模型參數和這些變數的先前狀態唯一確定的一種模型；因此，在一組給定的初始條件下確定性模型總會表現相同。相反，在隨機模型（通常成為「機率模型」）中存在隨機性，而且變數狀態並不能用唯一值來描述，而用機率分布來描述。
* **演繹，歸納與漂移：**演繹模型是建立在理論上的一種邏輯結構。歸納模型由實證研究及演繹模型推廣而得。漂移模型則既不依賴於理論，也不依賴於觀察，而僅僅是對預期結構的呼叫。 當數學應用在經濟學以外的社會科學時，此類模型一直被批評為毫無根據的模型。科學中在突變理論的應用已被定性為漂移模型。
## 建模的過程
### 分析問題
首先必須明白問題的本質，才能將之轉換成操作定義和數學符號。根據已知資訊的多寡，模型可以分為三類：
1. 白箱模型：指那些內部規律比較清楚的模型。如力學、熱學、電學以及相關的工程技術問題。
2. 灰箱模型：指那些內部規律尚不十分清楚，在建立和改善模型方面都還不同程度地有許多工作要做的問題。如氣象學、生態學、經濟學等領域的模型。
3. 黑箱模型：指一些其內部規律還很少為人們所知的現象。如生命科學、社會科學等方面的問題。但由於因素眾多、關係複雜，也可簡化為灰箱模型來研究。
### 簡化
模型描述的是理想化的情境，如 George E. P. Box 所言：「所有的模型都是錯的，但有些很有用」。判斷哪些核心部件必須保留、哪些可以簡化是建模的重要步驟。如果所有的細節都包含在內，模型和真實世界是一樣的，則沒有使用模型的意義。 | [
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数理模型方法 | 物理中常用的若干簡化模型包括無品質的繩子、點粒子、理想氣體以及無限深方形阱。用簡單方程式表示的物理定律有牛頓定律、馬克士威方程組和薛丁格方程式等。這些定律都是建立在實際情況的數學模型基礎上的。許多實際情況是非常複雜的，因此要用電腦進行類比，計算可行的模型是建立在基本定律或基本定律的近似模型上的。例如，分子可以用薛丁格方程式的近似解分子軌道模型進行類比。在工程中，物理模型通常運用的數學方法如有限元分析。不同數學模型使用不同的幾何學，但所使用的不一定是描述宇宙最準確的幾何學。歐幾里得幾何多用在古典物理學中，而狹義相對論和廣義相對論都是不使用歐幾里得幾何的理論。
在數理生物學中，哈溫定律描述一個無限大的族群、裡面隨機交配、沒有天擇或突變。族群遺傳學模型常假設固定的族群大小。計量遺傳學模型則假設連續性狀。
### 建立模型
決定實際問題中的各種因素，轉換為變數表示。接著應分析這些變數之間的關係，哪些是相互依存的，哪些是獨立的，他們具有什麼樣的關係，用合理的數學式表示這些關係。根據實際問題選用合適的數學框架（典型的有最佳化問題，組態問題等等），並具體的應用問題在這個數學框架下表出，並用合適的演算法求解數學框架下表出的問題。在這個過程中可能用到電腦類比和編程，常用的數學工具軟體套件括 MATLAB 和 Mathematica。
### 分析結果
最後使用計算結果解釋實際問題，並且分析結果的可靠性。這時常需用到各種訊息可視化的技巧。
## 科學應用
數學模型廣泛應用在各領域科學中，包括經濟學模型、族群生物學、計量生物學、流行病學、複雜系統、工程學等，物理理論幾乎無一例外會利用數學模型表示。
縱觀科學史，能夠量化和建立數學模型往往是一門學科發展成熟的指標，學科的進步常常源自新證據讓科學家可以建立更精確、更通用的模型。在生物學，現代生物綜論提出的各種數學模型讓演化生物學成為一門可以量化的學科。在物理學，牛頓運動定律準確地描述了許多日常現象，但在接近光速或是微觀層級時，這些定律就不適用，而需要用相對論或量子力學。
### 例子
* 電腦科學中的一個常見模型是各種自動機，如用抽象數學概念定義的確定有限狀態自動機（DFA），但是由於 DFA 的確定性性質，可以用硬體或軟體來實現以解決各種具體問題。
* 許多日常活動都暗含著數學模型的運用。把地球的一個區域投影在小的地圖平面上就是一個模型， 該模型可以用來規劃旅行。 | [
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] |
数理模型方法 | * **人口增長**：一個簡單（但粗略）的人口增長模型為馬爾薩斯增長模式；另一個較理想且被大量使用的人口增長模型為邏輯函數和其延伸。
* **位能場中的粒子模型**：在此模型中，粒子被視為一個品質為 _m_ 的點，其軌跡為一將時間對映至其空間座標的函數 _x_ : **R** → **R**3，位能場由一函數 _V_:**R**3 → **R** 給定，則其軌跡為如下微分方程式的解：
: : $-{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,$
: 也可以寫作
: $m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].$
: 需注意此模型假定粒子為一質點，但這在許多情形之下是錯誤的，如行星運動的模型之類。
* **鄰居遙感模型**解釋了從最初的混沌真菌網路形成蘑菇的過程。
* **電腦科學**：電腦網路模型，資料模型，表面模型等。
* **力學**：火箭模型運動等。
## 數學建模競賽
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] |
数理模型方法 | MCM/ICM 是 Mathematical Contest in Modeling 和 Interdisciplinary Contest in Modeling 的縮寫，即「數學建模競賽」和「交叉學科建模競賽」。MCM 始於 1985 年，ICM 始於 2000 年，由 COMAP（the Consortium for Mathematics and Its Application，美國數學及其應用聯合會）主辦，得到了 SIAM，NSA，INFORMS 等多個組織的贊助。MCM/ICM 與其他著名數學競賽（如 Putnam 數學競賽）的區別在於其著重強調研究問題、解決方案的原創性、團隊合作、交流以及結果的合理性。競賽以三人（本科生）為一組，在四天時間內，就指定的問題完成從建立模型、求解、驗證到論文撰寫的全部工作。競賽每年都吸引大量著名高校參賽。2008 年 MCM/ICM 有超過 2000 個隊伍參加，遍及五大洲。MCM/ICM 已經成為最著名的國際大學生競賽之一。
### 中國
全國大學生數學建模競賽創辦於 1992 年，每年一屆，是首批列入「高校學科競賽排行榜」的 19 項競賽之一。2020 年，來自全國及美國、英國、馬來西亞的 1470 所院校 / 校區、45680 隊 (本科 41826 隊、專科 3854 隊)、13 萬多人報名參賽。
## 參見
## 延伸閱讀
### 書籍
* Aris, Rutherford [ 1978 ] ( 1994 ). _Mathematical Modelling Techniques_, New York: Dover. ISBN 978-0-486-68131-3
* Bender, E.A. [ 1978 ] ( 2000 ). _An Introduction to Mathematical Modeling_, New York: Dover. ISBN 978-0-486-41180-4
* Gershenfeld, N. (1998) _The Nature of Mathematical Modeling_, Cambridge University Press ISBN 978-0-521-57095-4 . | [
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] |
数理模型方法 | * Lin, C.C. & Segel, L.A. ( 1988 ). _Mathematics Applied to Deterministic Problems in the Natural Sciences_, Philadelphia: SIAM. ISBN 978-0-89871-229-2
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紅位移 | 在物理學領域，**紅移**（Redshift）是指電磁輻射由於某種原因導致波長增加、頻率降低的現象，在可見光波段，表現為光譜的譜線朝紅端移動了一段距離。相反的，電磁輻射的波長變短、頻率升高的現象則被稱為藍移。紅移最初是在人們熟悉的可見光波段發現的，隨著對電磁波譜各個波段的了解逐步加深，任何電磁輻射的波長增加都可以稱為紅移。對於波長較短的 γ 射線、X - 射線和紫外線等波段，波長變長確實是波譜向紅光移動，「紅移」的命名並無問題；而對於波長較長的紅外線、微波和無線電波等波段，儘管波長增加實際上是遠離紅光波段，這種現象還是被稱為「紅移」。
當光源移動遠離觀測者時，觀測者觀察到的電磁波譜會發生紅移，這類似於聲波因為都卜勒效應造成的頻率變化。這樣的紅移現象在日常生活中有很多應用，例如都卜勒雷達、雷達槍，在天體光譜學裏，人們使用都卜勒紅移測量天體的物理行為。
另一種紅移稱為宇宙學紅移，其機制為空間的度規膨脹。這機制說明了在遙遠的星系、類星體，星系間的氣體雲的光譜中觀察到的紅移現象，其紅移增加的比例與距離成正比。這種關係為宇宙膨脹的觀點提供了有力的支持，比如大霹靂宇宙模型。
另一種形式的紅移是引力紅移，其為一種相對論性效應，當電磁輻射傳播遠離引力場時會觀測到這種效應；反過來說，當電磁輻射傳播接近引力場時會觀測到引力藍移，其波長變短、頻率升高。
紅移的大小由「紅移值」衡量，紅移值用 Z 表示，定義為：
: $z={\frac {\lambda -\lambda _{0}}{\lambda _{0}}}={\frac {f_{0}-f}{f}}$
這裡 $\lambda _{0}\,$ 是譜線原先的波長， $\lambda \,$ 是觀測到的波長， $f_{0}\,$ 是譜線原先的頻率， $f\,$ 是觀測到的頻率。
## 分類
* **都卜勒紅移**：物體和觀察者之間的相對運動可以導致紅移，與此相對應的紅移稱為都卜勒紅移，是由都卜勒效應引起的。
* **重力紅移**：根據廣義相對論，光從重力場中發射出來時也會發生紅移的現象。這種紅移稱為重力紅移。
* **宇宙學紅移**：20 世紀初，美國天文學家維斯托・斯里弗發現，觀測到的絕大多數星系的光譜線存在紅移現象。這是由於宇宙空間在膨脹，使天體發出的光波被拉長，譜線因此「變紅」，這稱為宇宙學紅移，並由此得到哈伯定律。20 世紀 60 年代發現了一類具有極高紅移值的天體：類星體，成為近代天文學中非常活躍的研究領域。 | [
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紅位移 | ## 簡史
這個主題的發展開始於 19 世紀對波動力學現象的探索，因而連結到了都卜勒效應。稍後，因為克里斯琴・都卜勒在 1842 年對這種現象提出了物理學上的解釋，而被稱為都卜勒效應。他的假說在 1845 年被荷蘭的科學家赫里斯托福魯斯・亨里克斯・迪德里克斯・白貝羅用聲波做實驗而獲得證實。都卜勒預言這種現象可以應用在所有的波上，並且指出恆星的顏色不同可能是由於它們相對於地球的運動速度不同而引起的。後來這個推論被否認。恆星呈現不同的顏色是因為溫度不同，而不是運動速度不同。
都卜勒紅移是法國物理學家阿曼德・斐索在 1848 年首先發現的，他指出恆星譜線位置的移動是由於都卜勒效應，因此也稱為「都卜勒 - 斐索效應」。1868 年，英國天文學家威廉・哈金斯首次測出了恆星相對於地球的運動速度。
在 1871 年，利用太陽的自轉測出在可見光太陽光譜的夫朗和斐譜線在紅光有 0.1 Å 的位移。在 1901 年，亞里史塔克・貝樓坡斯基在實驗室中利用轉動的鏡片證明了可見光的紅移。
在 1912 年開始的觀測，維斯托・斯里弗發現絕大多數的螺旋星雲都有不可忽視的紅移。然後，埃德溫・哈伯發現這些星雲（現在知道是星系）的紅移和距離有關聯性，也就是哈伯定律。這些觀察在今天被認為是造成宇宙膨脹大爆炸理論的強而有力證據。
## 測量、特性和解釋
紅移可以經由單一光源的光譜進行測量（參考上面理想化的光譜例証圖）。如果在光譜中有一些特徵，可以是吸收線、發射線、或是其他在光強上的變化，那麼原則上紅移就可以測量。這需要一個有相似特徵的光譜來做比較，例如當氫原子發出光線時，其光譜有明確的特徵譜線，而這一系列的特色譜線都有一定間隔的。如果有這種特性的譜線型態但在不同的波長上被比對出來，那麼這個物體的紅移就能測量了。因此，測量一個物體的紅移，只需要頻率或是波長的範圍。只觀察到一些孤立的特徵，或是沒有特徵的光譜，或是白噪音（一種相當無序雜亂的波），是無法計算紅移的。
紅移（和藍移）可能會在天體被觀測的和輻射的波長（或頻率）而帶有不同的變化特徵，天文學習慣使用無因次的數量 _z_ 來表示。如果 _λ_ 代表波長，_f_ 代表頻率（注意：_λf_ = _c_，此處的 _c_ 是光速），那麼 _z_ 可以由下面的公式來定義： | [
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紅位移 | 在 _z_ 被測量後，紅移和藍移的差別只是簡單的正負號的區別。依據下一章節的機制，無論被觀察到的是紅移或藍移，都有一些基本的說明。例如，都卜勒效應的藍移（_z_ < 0）會聯想到物體朝向觀測者接近並且能量增加，反過來說都卜勒紅移（_z_ > 0），就會聯想到物體遠離觀測者而去並且能量減少。同樣的，愛因斯坦效應的藍移可以聯想到光線進入強引力場，而愛因斯坦效應的紅移是離開引力場。
## 機制
一個光子在真空中傳播可以有幾種不同的紅移機制，每一種機制都能產生類似都卜勒紅移的現象，意謂著 _z_ 是與波長無關的。這些機制分別使用伽利略變換、勞侖茲變換、或相對論轉換在各個參考架構之間來比較。
### 都卜勒效應
如果一個光源是遠離觀測者而去，那麼會發生紅移（_z_ > 0），當然，如果光源是朝向觀測者移動，便會產生藍移（_z_ < 0）。這對所有的電磁波都適用，而且可以用都卜勒效應解釋。當然的結果是，這種形式的紅移被稱為都卜勒紅移。如果光源遠離觀測者的速度是 _v_，忽略掉相對論的效應，紅移可以表示為：
: $z\approx {\frac {v}{c}}$ (由於 $\gamma \approx 1$ , 參考下面的說明)
此處 _c_ 是光速，在古典的都卜勒效應，光源的的頻率是無需修正的，但是退行會造成低頻的錯覺。
對於聲波等機械波，也有這樣的效應。當波源向觀察者運動時，觀察者接受到的波的頻率變高；波源離開時則反之。
### 相對論性都卜勒效應
更完整的都卜勒紅移需要考慮相對論的效應，特別是在速度接近光速的情況下。完整的文章可以參考相對論性都卜勒效應。簡單的說，物體的運動接近光速時需要將狹義相對論介紹的時間膨脹因素勞侖茲變換因子 γ 引入古典的都卜勒公式中，改正後的形式如下：
: $1+z=\left(1+{\frac {v}{c}}\right)\gamma $
這種現象最早是在 1938 年赫伯特・艾凡斯和 G. R. 史迪威進行的實驗中被觀察到的，稱為艾凡斯 - 史迪威實驗。
由於羅倫茲因子只與速度的量值有關，這使得紅移與相對論的相關只獨立的與來源的運動取向有關。在對照時，古典這一部分的形式只與來源的運動投影在視線方向上的分量有關，因此在不同的方向上會得到不同的結果。同樣的，一個運動方向與觀測者之間有 _θ_ 的角度（正對著觀測者時角度為 0），完整的相對論性都卜勒效應形式為： | [
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紅位移 | : $1+z={\frac {1-v\cos(\theta )/c}{\sqrt {1-v^{2}/c^{2}}}}$
而正對著觀測者的運動物體（θ = 0°），公式可以簡化為：
: $1+z={\sqrt {\frac {1+{\frac {v}{c}}}{1-{\frac {v}{c}}}}}$
在特殊的狀況下，運動源與測器成直角（θ = 90°），相對性的紅移為橫向紅移，儘管物體並沒有移動遠離觀測者，仍舊會測量到紅移
: $1+z={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}$
即使來源是朝向觀測者運動，如果有橫向的分量，那麼在這個方向上的速度可以增加到抵消預期中的藍移，而且如果繼續增加速度，則會使朝向觀測者的來源呈現紅移。
### 膨脹的宇宙
在 20 世紀初期，史立佛、哈伯和其他人，首度測量到銀河系之外星系的紅移和藍移，他們起初很單純的解釋是都卜勒效應造成的紅移和藍移，但是稍後哈伯發現距離和紅移之間有著粗略的關聯性，距離越遠紅移的量也越大。理論學者幾乎立刻意識到這些觀察到的紅移可以用另一個不同的機制來解釋，哈伯定律就是紅移和距離之間交互作用的關聯性，需要使用廣義相對論空間的度規膨脹的宇宙論模型來解釋。結果是，光子在通過擴張的空間時被延展，產生了宇宙學紅移。這與都卜勒效應所描述的因速度增加所產生的紅移不同（這是勞侖茲變換），在光源和觀測者之間不是因為動量和能量的轉換，取代的是光子因為經過膨脹的空間使波長增加而紅移。這種效應在標準宇宙學模型中被解釋為可以觀測到與時間相關聯的宇宙標度因子（ $a$ ），如下的形式：
: $1+z={\frac {a_{\mathrm {now} }}{a_{\mathrm {then} }}}$ 。
這種型態的紅移稱為「宇宙學紅移」或「哈伯紅移」。如果宇宙是收縮而不是膨脹，我們將觀測到星系以相同比例的藍移取代紅移。
這些星系不是以實際的速度遠離觀測者而去，取代的是在其間的空間延展，這造成了大尺度下宇宙論原則所需要的各向同性的現象。在宇宙學紅移 z < 0.1 的情況下，時空擴展的作用對星系所造成的獨特效應與被觀察到的紅移，相對於都卜勒效應的紅移和藍移是極微小的。 | [
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紅位移 | 實際的速度和空間膨脹的之間的區別在膨脹的橡皮板宇宙有清楚的說明，一般的宇宙學也曾經描述過類似的空間擴展。如果以滾珠軸承來代表兩個物體，以有彈性的橡皮墊代表時空，都卜勒效應是軸承橫越過橡皮墊產生的獨特運動，宇宙學紅移則是橡皮墊向下沉陷的柱狀體的沉陷量。(很明顯的在模型上會有維度的問題，當軸承滾動時應該是在橡皮墊上，而如果兩個物體的距離夠遠時宇宙學紅移的速度會大於都卜勒效應的速度。)
儘管速度是由分別由都卜勒紅移和宇宙學紅移共同造成的，天文學家（特別是專業的）有時會以 "退行速度" 來取代在膨脹宇宙中遙遠的星系的紅移，即使很明顯的只是視覺上的退行。影響所及，在大眾化的講述中經常會以 "都卜勒紅移" 而不是 "宇宙學紅移" 來描述受到時空擴張影響下的星系運動，而不會注意到在使用相對論的場合下計算的 "宇宙學退行速度" 不會與都卜勒效應的速度相同。
明確的說，都卜勒紅移只適用於狹義相對論，因此 _v > c_ 是不可能的；而相對的，在宇宙學紅移中 _v > c_ 是可能的，因為空間會使物體（例如，從地球觀察類星體）遠離的速度超過光速。更精確的，"遙遠的星系退行" 的觀點和 "空間在星系之間擴展" 的觀點可以通過坐標系統的轉換來連繫。 要精確的表達必須要使用數學的羅伯遜 - 沃克度規。
### 重力紅移
在廣義相對論的理論中，引力會造成時間的膨脹，這就是所謂的引力紅移或是愛因斯坦位移。這個作用的理論推導從愛因斯坦場方程式的施瓦氏解，以一顆光子在不帶電荷、不轉動、球對稱質量的重力場運動，產生的紅移：
: $1+z={\frac {1}{\sqrt {1-\left({\frac {2GM}{rc^{2}}}\right)}}}$ ,
其中 $G$ 是萬有引力常數， $M$ 是產生引力場的質量， $r$ 是觀測者的徑向坐標，而 $c$ 則為光速。
引力紅移的結果可以從狹義相對論和等效原理導出，並不需要完整的廣義相對論。
在地球上這種效應非常小，但是經由梅斯堡效應依然可以測量出來，並且在龐德 - 雷布卡實驗中首次得到驗證。然而，在黑洞附近就很顯著，當一個物體接近事件視界時，紅移將變成無限大，他也是在宇宙微波背景輻射中造成大角度尺度溫度擾動的主要角色。
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紅位移 | 在天文觀測中可以測量到紅移，因為原子的發射光譜和吸收光譜，與在地球上的實驗室內的分光儀校準好的光譜比較時，是非常的明顯。當從同一個天體上測量到各種不同的吸收和發射譜線時，_z_ 被發現是一個常數。雖然來自遙遠天體的譜線可能會被污染，並且有輕微的變寬，但並不能夠用熱力學或機械的行為來解釋。基於這些和其他的理由，公眾的輿論已經將天文學上觀測到的紅移認定是三種類似的都卜勒紅移之一，而沒有任何一種假說能如此的振振有詞。
光譜學，用在測量上，比只要簡單的通過特定的濾光器來測定天體亮度的光度學要困難。當測光時，可以利用所有的數據（例如，哈伯深空和哈伯超深空），天文學家依靠的是紅移測光的技術，由於濾光器在某些波長的範圍內非常靈敏，依靠這樣的技術可以假定許多光譜的本質隱藏在光源之內，觀測誤差可以 δ_z_ = 0.5 為級距來排序，並且比分光鏡的更為可靠許多。然而，光度學無法考慮到紅移的定性描述。例如，一個與太陽相似的光譜，但紅移 _z_ = 1，最為明亮的是在紅外線的區域，而非以黃 - 綠為尖峰的黑體光譜，並且光的強度在經過濾光器時將減少二級（1+_z_，更多的細節請參考紅移測光的 K 校正。
### 在本地群的觀測
在附近的目標（在我們的銀河系內的天體）觀測到的紅移幾乎都與相對於視線方向上的速度有關。觀察這樣的紅移和藍移，讓天文學家可以測量速度和分光雙星的參考質量。這種方法是英國天文學家威廉・哈金斯在 1868 年最先採用的。相同的，從光譜儀中對單獨的一顆恆星所測得的微量的紅移和藍移是天文學家檢測是否有行星系環繞著恆星的診斷和測量的方法之一。對紅移更精確的測量被應用於日震學上，藉以精確的測量太陽光球的運動。紅移也被應用於第一次的行星自轉速率的測量、星際雲的速度、星系的自轉，還有吸積的動力學呈現在中子星和黑洞的都卜勒和重力紅移。
另外，還有各種不同輻射和吸收的溫度造成的都卜勒增寬 —— 對單一的吸收或輻射譜線造成的紅移和藍移的效應。測量來自不同方向的氫線 21 公分波的擴展和轉移，天文學家能測量出星際氣體的退行速度，揭露出我們銀河系的自轉曲線。相同的測量也被應用在其他的星系，例如仙女座星系。做為一種診斷的工具，紅移測量在天文學的分光學中是最重要的工具之一。
### 外星系的觀察 | [
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紅位移 | 宇宙中合於哈伯定律的天體距離越遠就有越大的紅移，因此被觀測到有最大紅移，對應於最遙遠的距離也有最長的回應時間的天體是宇宙微波背景輻射，紅移的數值高達 _z_ = 1089（_z_ = 0 相當於現在的時間），在宇宙年齡為 137 億年的狀態下，相當於大霹靂之後 379,000 年的時間。
核心像點光源的類星體是紅移（ $z>0.1$ ）最高的天體，是在望遠鏡改善之前，除了星系之外還能被發現的其他高紅移天體。目前，被發現紅移最高的類星體是 $z=6.4$ ，被證實紅移最高的星系是 $z=7.0$ 在尚未經確認的報告中顯示，透過重力透鏡觀測到的遙遠星系集團有紅移高達 $z=10$ 的星系。
對比本星系群遙遠，但仍在處女座星系團附近，距離為 10 億秒差距左右的星系，紅移與星系的距離是近似成比例的，這種關係最早是由哈伯發現的，也就是眾所皆知的哈伯定律。星系紅移最早是維斯托・斯萊弗大約在 1912 年發現的，而哈伯結合了斯萊弗的測量成為度量天體距離的另一種方法：哈伯定律。在建基於廣義相對論下被廣泛接受的宇宙模型中，紅移是空間擴展的主要結果：這意味著遙遠的星系都離我們而去，光離開星系越久，空間的擴展也越多，所以光也就被延伸越多，紅移的值也就越大，所以越遠的看起來就移動的越快。哈伯定律一樣適用哥白尼原則，由於我們通常不知道天體有多明亮，測量紅移會比直接測量距離容易，所以使用哈伯定律就可以得知天體大略的距離。
星系之間的和星系團的重力交互作用在正常的哈伯圖上導致值得注意的消散，星系的本動速度和在宇宙中的維理天體的迷蹤質量相疊加，這種作用導致在附近的星系（像仙女座星系）顯示出藍移的現象，並且向共同的重心接近，同時星系團的紅移圖像因紅移空間畸變作用使本動速度的消散大致成球型的分佈。這個增加的組合給了宇宙學家一個單獨測量質量的「質光比」（以太陽的質量和光為單位的星系的質量與光度比值），是尋找暗物質的重要工具。
對更遙遠的星系，目前的距離和紅移之間的關連性變得更為複雜。當你看見一個遙遠的星系，也就是看見相同久遠之前的星系，而那時的宇宙和現在是不同的。在那些早期的時刻，我們期待在俇展的速率上有所不同，原因至少有二個：
1. 星系之間相互的重力吸引會減緩宇宙的擴張行動
2. 可能存在的宇宙學常數或第五元素與可能會改變宇宙擴張的速率。 | [
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紅位移 | 最近的觀測卻建議宇宙的擴張不僅沒有如同第一點的預測減速，反而在加速中。這是廣泛的，雖然不是相當普遍的，相信這是因為有暗物質在控制著宇宙的發展。這樣的宇宙學常數暗示宇宙的最後命運不是大擠壓，反而可預見宇宙將長久存在。（可是在宇宙內多數的物理程序仍然朝向熱寂。）
擴張的宇宙是大霹靂理論的中心預言，如果往前追溯，理論預測「奇異點」的存在，而那時的宇宙有無限大的密度；廣義相對論的理論，大霹靂的理論依據，將不再能適用。最有可能取代的理論據信是尚未成熟的量子引力學，能在密度變得無窮大之前繼續適用。
### 紅移巡天
在先進的自動化望遠鏡和改良的光譜儀合作之下，以一定數量星空的紅移當成宇宙的投影，通過紅移與角度位置數據的結合，紅移巡天圖可以顯示天空中一定範圍內物質的立體分布狀態。這些觀測被用來研究宇宙的大尺度結構，長城、許多廣達 5 億光年的超星系團，紅移巡天的檢測提供了戲劇性的大尺度構造的例子。
第一次紅移巡天是 CfA 紅移巡天，開始於 1977 年，至 1982 年完成最初的資料蒐集。最近的有 2 度視場星系紅移巡天，測量宇宙在一個部份的大尺度結構，量測了 22 萬個星系的 _z_ 值，最後的結果已經在 2003 年 6 月釋出。（除了描繪星系在大尺度的模型，2 度視場也可以估計微中子質量的上限。）其他值得重視的研究還有史隆數位巡天（SDSS），在 2005 年仍在繼續進行中，目標瞄準在觀測一億個天體。SDSS 已經觀測到紅移高達 0.4 的星系和紅移超過 _z_ = 6 的類星體。深度 2 紅移巡天使用凱克望遠鏡和新的「DEIMOS」光譜儀，是深度 1 計劃的延續。深度 2 是設計來研究紅移 0.7 或更高的黯淡星系，因此可以填補 SDSS 和 2df 計劃的不足。
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紅位移 | 在輻射轉移和物理光學的主題中會總結電磁輻射中波長和頻率轉換可能發生的現象和交互作用導致位移的結果。在這些情況下位移和物理上對應的能量轉移到物質或其他的光子，而不是歸結於參考座標系的轉變之間。這些轉移可以歸結於沃爾夫效應或是來自於基本粒子、微粒物質、或來自電介質的折射率的漲落，其導致電磁輻射的散射。當這些現象對應於「紅移」或「藍移」的現象時，是物理的電磁輻射場本身的交互作用或是介入（干預）的物質來自參考座標系效應的現象。在天文物理，質 - 光交互作用的結果在輻射場的能量的遷移上通常是「紅化」（reddening）而不是「紅移」，而這個項目通常是保留在前面的效應中討論的。
在許多情況下散射會導致輻射的紅化，因為熵會使光子趨向最低能量而減少高能的光子（總能量守恆）。除了在小心控制的情況下，散射不會在同一個變化中橫跨整個光譜，換言之，任何一個波長上計算得到的 _z_ 只是一個對應於波長的函數，而且，來自介質的隨機散射通常可能發生在任何的角度上，而 _z_ 又是一個散射角的函數。如果多次的散射發生，或是散射的粒子在相對的運動中，那麼通常都會造成譜線的畸變。
在恆星際天文學，可見光譜會因為穿過星際物質的散射過程出現星際紅化：類似於在日出或日落時大氣層造成太陽光偏紅和天空是藍色的瑞立散射。這種明顯的轉移成紅色的現象，是因為譜線中的紅色部分沒有被轉移成其他的波長，以及額外的黯淡和畸變結合，這些現象使光子在視線中出現或消失。
## 注釋
## 參考資料
## 延伸閱讀
* Lineweaver, C. H.; Davis, T. M. Misconceptions about the big bang (PDF). Scientific American. 2005, **292** (3): 36–45 [2016-07-07]. （原始內容 (PDF) 存檔於 2016-08-22） （英語）.
## 外部連結
* 內德・賴特對紅移所做的講解（頁面存檔備份，存於網際網路檔案館）（英文）
* 邁克・路西烏克對紅移所做的講解（英文）
* 由美國紅外處理和分析中心提供的對於紅移的講解（英文）
## 相關條目
* 藍移
* 相對論性都卜勒效應
* 都卜勒效應
* 重力紅移
* 類星體
* 哈伯定律
* 宇宙大爆炸理論 | [
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穩定流 | **流體動力學**（英語：Fluid dynamics）是流體力學的一門子學科。流體動力學研究的對象是運動中的流體（含液體和氣體）的狀態與規律。流體動力學底下的子學科包括有空氣動力學和液體動力學。
解決一個典型的流體動力學問題，需要計算流體的多項特性，主要包括速度、壓力、密度、溫度。
流體動力學有很大的應用，比如在預測天氣，計算飛機所受的力和力矩，輸油管線中石油的流率等方面上。其中的的一些原理甚至運用在交通工程，因交通運輸本身可被視為一連續流體運動。
## 流體動力學方程式
流體動力學的基本公理為守恆律，特別是質量守恆、動量守恆（也稱作牛頓第二與第三定律）以及能量守恆。這些守恆律以古典力學為基礎，並且在量子力學及廣義相對論中有所修改。它們可用雷諾傳輸定理（Reynolds transport theorem）來表示。
除了上面所述，流體還假設遵守「連續性假設」（continuum assumption）。流體由分子所組成，彼此互相碰撞，也與固體相碰撞。然而，連續性假設考慮了流體是連續的，而非離散的。因此，諸如密度、壓力、溫度以及速度等性質都被視作是在無限小的點上具有良好定義的，並且從一點到另一點是連續變動。流體是由離散的分子所構成的這項事實則被忽略。
若流體足夠緻密，可以成為一連續體，並且不含有離子化的組成，速度相對於光速是很慢的，則牛頓流體的動量方程式為「納維 - 斯托克斯方程式」。其為非線性微分方程式，描述流體的流所帶有的應力是與速度及壓力呈線性相依。未簡化的納維 - 斯托克斯方程式並沒有一般閉形式解，所以只能用在計算流體力學，要不然就需要進行簡化。方程式可以通過很多方法來簡化，以容易求解。其中一些方法允許適合的流體力學問題能得到閉形式解。
除了質量、動量與能量守恆方程式之外，另外還有熱力學的狀態方程式，使得壓力成為流體其他熱力學變數的函數，而使問題得以被限定。其中一個例子是所謂的理想氣體方程式：
: $p={\frac {\rho R_{u}T}{M}}$
其中 $p$ 為壓力， $\rho $ 為密度， $R_{u}$ 為氣體常數， $M$ 為分子量，以及 $T$ 為溫度。
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穩定流 | 所有流體某種程度上而言都是可壓縮的，換言之，壓力或溫度的改變會造成流體密度的改變。然而，許多情況下，壓力或溫度改變所造成的密度改變相當微小，是可以被忽略的。此種流體可以用不可壓縮流進行模擬，否則必須使用更普遍性的可壓縮流方程式進行描述。
數學上而言，「不可壓縮性」代表著流體流動時，其密度 $\rho \;$ 維持不變，換言之：
: ${\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0\,$ ，
其中， $\mathrm {D} /\mathrm {D} t$ 為隨質導數（substantial derivative）。此條件可以簡化許多描述流體的方程式，尤其是運用在均勻密度的流體。而隨質導數又可分解成局部導數與對流導數，前者代表位置不變時，性質隨時間之變化率，而後者代表質點運動時，該性質隨速度方向之變化率。若為不可壓縮流，則代表對密度做隨質導數與對流導數，都各別為 0 時，代表密度不隨位置跟時間改變，即不可壓縮流。
對於氣體要辨別是否具有可壓縮性，馬赫數是一個衡量的指標。概略來說，在馬赫數低於 0.3 左右時，可以用不可壓縮流的行為解釋。
至於液體，較符合可壓縮流還是不可壓縮流的性質，主要取決於液體本身的性質（特別是液體的臨界壓力與臨界溫度）和流體的條件（液體壓力是否接近和液體臨界壓力）。
聲學的問題往往需要引進壓縮性的考量，因為聲波算是可壓縮波，其性質會隨著傳播的介質以及壓力變化而改變。
### 黏性流與非黏性流
當流體內的阻力越大時，描述流體須考慮其黏性的影響。雷諾數可用來估算流體的黏性對描述問題的影響。
所謂史托克流指雷諾數相當小的流動。在此情況，流體的慣性相較於黏性可忽略。而流體的雷諾數大代表流體流動時慣性大於黏性。因此當流體有很大的雷諾數，假設它是非黏性流，忽略其黏性，可當成一個近似。
這樣的近似，當雷諾數大時，可得到很好的結果，即便是在某些不得不考慮黏性的問題上（例如邊界問題）。但在流體與管壁的邊界，有所謂的不滑移條件，局部會有很大的速率應變率，使得黏性的作用放大而有渦度，黏性因而不可被忽略。
因此，計算管壁對流體的淨力，需要使用黏性方程式。如同達朗白謬論的說明，物體在非黏性流裡，不會感受到力。歐拉方程式是描述非黏性流的標準方程式。在這種情況，一個常使用的模型，使用歐拉方程式描述遠離邊界的流體，在接觸的邊界，使用邊界層方程式。 | [
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] |
穩定流 | 在某一個流線上，將歐拉方程式積分，可得到白努利定律。如果流體每一處都是無旋轉渦動，白努利方程式可描述整個流動。
### 穩定流與非穩定流
穩定流即在流場中任一特定位置上，此位置上流體質點的任何物理性質不會隨時間改變。在流場中若有流線，線上任一位置上的切線方向與質點之速度向量相同。
非穩定流：水在滲流場內運動過程中各個運動要素隨時間改變的水流運動。運動要素包括水位、流速、流向等
### 層流與紊流
當流動由漩渦和表觀的隨機性所主導時，此種流動稱為紊流。當亂流效應不明顯時，則稱為層流。然而值得注意的是，流動之中存在於漩渦不一定表示此流動為亂流──這些現象可能也存在於層流之中。數學上，紊流通常以雷諾分離法來表示，也就是紊流可以表示成穩定流與擾動部分的和。
亂流遵守納維 - 斯托克斯方程式。數值直解法（Direct numerical simulation，DNS），基於納維 - 斯托克斯方程式可應用在不可壓縮流，可使用雷諾數對紊流進行模擬（必須在電腦性能與演算結果準確性均能負荷的條件下）。而此數值直解法的結果，可以解釋所得的實驗資料。
然而，大部分我們有興趣的流動都是雷諾數比 DNS 能夠模擬的範圍大上許多，即使電腦性能在接下來的數十年間持續發展，仍難以實行模擬。任何飛行交通工具，要足夠能承載一個人（L >3 m）以 72 km/h（20 m/s）的速度移動，此情況都遠遠在 DNS 能夠模擬的範圍之外（雷諾數為 4 百萬）。像是空中巴士 A300 或波音 747 這類的飛行工具，機翼上的雷諾數超過 4 千萬（以翼弦為標準）。為了能夠處理這些生活上實際的問題，需要建立紊流模型。雷諾平均納維 - 斯托克斯方程式（Reynolds-averaged Navier-Stokes equations）結合了紊流的效果，提供了一個紊流的模型，將額外的動量傳遞表示由雷諾應力所造成；然而，亂流也會增加熱傳與質傳速度。大渦數值模擬計算（Large eddy simulation，LES）也是一個模擬方法，外觀與分離渦流模型（detached eddy simulation, DES）甚相似，是一種紊流模擬與大渦數值模擬計算的結合。
### 牛頓式流體與非牛頓式流體
牛頓流體為在定溫及定壓之下，流體的動力黏制係數不會隨速度梯度變化，且保持定值，非牛頓流體的動力黏制係數則會隨速度梯度改變。
### 其他近似
## 參考文獻
## 相關條目 | [
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數學物理家 | **數學物理**是數學和物理學的交叉領域，指應用特定的數學方法來研究物理學的某些部分。對應的數學方法也叫**數學物理方法**。數學和物理學的發展在歷史上一直密不可分，許多數學理論是在物理問題的基礎上發展起來的；很多數學方法和工具通常也只在物理學中找到實際應用。不過，也只是互相參考而已，所有沒有所謂的一定。
## 主要內容
* 微分方程式的解算：很多物理問題，比如在古典力學和量子力學中求解運動方程式，都可以被歸結為在一定邊界條件下的對微分方程式的求解。因此求解微分方程式成為數學物理的最重要組成部分。相關的數學工具包括：
* 常微分方程式的求解
* 偏微分方程式的求解
* 特殊函數
* 積分變換
* 複變函數論
* 場的研究（場論）：場是現代物理的主要研究對象。電動力學研究電磁場；廣義相對論研究引力場；規範場論研究規範場。對不同的可使用不同的數學工具，包括：
* 矢量分析
* 張量分析
* 微分幾何
* 代數幾何
* 拓樸學
* 對稱性的研究：對稱性是物理中的重要概念。它是守恆律的基礎，在晶體學和量子場論中都有重要應用。對稱性由對稱群或相關的代數結構描述，研究它的數學工具是：
* 群論
* 表示論
* 作用量（action）理論：作用量理論被廣泛應用於物理學的各個領域，例如分析力學和路徑積分。相關的數學工具包括：
* 變分法
* 泛函分析
## 參見
* 希爾伯特第六問題
* 理論物理學
## 參考
## 文獻
* Abraham, Ralph; Marsden, Jerrold E., _Foundations of mechanics: a mathematical exposition of classical mechanics with an introduction to the qualitative theory of dynamical systems_ 2nd, Providence, [RI.]: AMS Chelsea Pub., 2008, ISBN 978-0-8218-4438-0 | [
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數學物理家 | * Arnold, Vladimir I.; Vogtmann, K.; Weinstein, A. (tr.), _Mathematical methods of classical mechanics / [Matematicheskie metody klassicheskoĭ mekhaniki]_ 2nd, New York, [NY.]: Springer-Verlag, 1997, ISBN 0-387-96890-3
* Courant, Richard; Hilbert, David, Methods of mathematical physics, New York, [NY.]: Interscience Publishers, 1989
* Glimm, James; Jaffe, Arthur, _Quantum physics: a functional integral point of view_ 2nd, New York, [NY.]: Springer-Verlag, 1987, ISBN 0-387-96477-0 (pbk.)
* Haag, Rudolf, _Local quantum physics: fields, particles, algebras_ 2nd rev. & enl., Berlin, [Germany] ; New York, [NY.]: Springer-Verlag, 1996, ISBN 3-540-61049-9 (softcover)
* Hawking, Stephen W.; Ellis, George F. R., _The large scale structure of space-time_, Cambridge, [England]: Cambridge University Press, 1973, ISBN 0-521-20016-4
* Kato, Tosio, _Perturbation theory for linear operators_ 2nd repr., Berlin, [Germany]: Springer-Verlag, 1995, ISBN 3-540-58661-X (This is a reprint of the second (1980) edition of this title.) | [
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數學物理家 | * Margenau, Henry; Murphy, George Moseley, _The mathematics of physics and chemistry_ 2nd repr., Huntington, [NY.]: R. E. Krieger Pub. Co., 1976, ISBN 0-88275-423-8 (This is a reprint of the 1956 second edition.)
* Morse, Philip McCord; Feshbach, Herman, _Methods of theoretical physics_ repr., Boston, [Mass.]: McGraw Hill, 1999, ISBN 0-07-043316-X (This is a reprint of the original (1953) edition of this title.)
* von Neumann, John; Beyer, Robert T. (tr.), _Mathematical foundations of quantum mechanics_, Princeton, [NJ.]: Princeton University Press, 1955
* Reed, Michael C.; Simon, Barry, _Methods of modern mathematical physics (4 vol.)_, New York. {NY.]: Academic Press, 1972–1977, ISBN 0-12-585001-8
* Titchmarsh, Edward Charles, _The theory of functions_ 2nd, London, [England]: Oxford University Press, 1939 (This tome was reprinted in 1985.)
* Thirring, Walter E.; Harrell, Evans M. (tr.), _A course in mathematical physics / [Lehrbuch der mathematischen Physik] (4 vol.)_, New York, [NY.]: Springer-Verlag, 1978–1983 | [
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數學物理家 | * Weyl, Hermann; Robertson, H. P. (tr.), _The theory of groups and quantum mechanics / [Gruppentheorie und Quantenmechanik]_, London, [England]: Methuen & Co., 1931
* Whittaker, Edmund Taylor; Watson, George Neville, A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions 4th, Cambridge: Cambridge University Press, 1927, ISBN 978-0-521-58807-2 | [
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量子化学 | **量子化學**（英語：Quantum chemistry），或稱**分子量子力學**，是應用量子力學的規律和方法來研究化學問題的一門學科。將量子理論應用於原子體系還是分子體系是區分量子物理學與量子化學的標準之一。
目前認為最早的量子化學計算是 1927 年布勞（Ø.Burrau）對 H+
2 離子以及同年瓦爾特・海特勒和弗里茨・倫敦對 H2 分子的計算，開創量子化學這一個交叉學科。經過近八十年發展之後，量子化學已經成為化學家們廣泛應用的一種理論方法。
## 量子化學發展史
1925 年和 1926 年，物理學家維爾納・海森伯和埃爾溫・薛丁格各自建立了矩陣力學和波動力學，標誌著量子力學的誕生，同時也為化學家提供了認識物質化學結構的新理論工具。1927 年物理學家瓦爾特・海特勒和弗里茨・倫敦將量子力學處理原子結構的方法應用於氫氣分子，成功地定量闡釋了兩個中性原子形成化學鍵的過程，他們的成功標誌著量子力學與化學的交叉學科 —— 量子化學的誕生。
在海特勒和倫敦之後，化學家們也開始應用量子力學理論，並且在兩位物理學家對氫氣分子研究的基礎上建立了三套闡釋分子結構的理論。萊納斯・鮑林在最早的氫分子模型基礎上發展了價鍵理論，並且因為這一理論獲得了 1954 年度的諾貝爾化學獎；1928 年，物理化學家羅伯特・S・馬利肯提出了最早的分子軌道理論，1931 年，埃里希・休克爾（E. Hückel）發展了馬利肯的分子軌道理論，並將其應用於對苯分子等共軛體系的處理；漢斯・貝特於 1931 年提出了配位場理論並將其應用於過渡金屬元素在配位場中能級裂分狀況的理論研究，後來，配位場理論與分子軌道理論相結合發展出了現代配位場理論。價鍵理論、分子軌道理論以及配位場理論是量子化學描述分子結構的三大基礎理論。早期，由於計算手段非常有限，計算量相對較小，且較為直觀的價鍵理論在量子化學研究領域占據著主導地位，1950 年代之後，隨著計算機的出現和飛速發展，以及高斯函數的引進，海量計算已經是可以輕鬆完成的任務，分子軌道理論的優勢在這樣的背景下凸現出來，逐漸取代了價鍵理論的位置，目前在化學鍵理論中占主導地位。 | [
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] |
量子化学 | 1928 年道格拉斯・哈特里提出了哈特里方程，方程將每一個電子都看作是在其餘的電子所提供的平均勢場中運動的，通過迭代法給出每一個電子的運動方程。1930 年，弗拉基米爾・福克對哈特里方程補充了泡利原理，提出**哈特里 - 福克方程**，進一步完善了由哈特里發展的哈特里方程。為了求解哈特里 - 福克方程，1951 年克萊門斯・C・J・羅特漢進一步提出將方程中的分子軌道用組成分子的原子軌道線性展開，發展出了著名的 RHF 方程，這個方程以及在這個方程基礎上進一步發展的方法是現代量子化學處理問題的主要方法。
雖然量子力學以及量子化學的基本理論早在 1930 年代就已經基本成型，但是所涉及的多體薛丁格方程形式非常複雜，至今仍然沒有精確解法，而即便是近似解，所需要的計算量也是驚人的，例如：一個擁有 100 個電子的小分子體系，在求解 RHF 方程的過程中僅僅雙電子積分一項就有 1 億個之巨。這樣的計算顯然是人力所不能完成的，因而在此後的數十年中，量子化學進展緩慢，甚至為從事實驗的化學家所排斥。1953 年美國的帕里瑟（Parise）、帕爾（R. Parr）和英國的約翰・波普使用手搖計算器分別獨立地實現了對氮氣分子的 RHF 自洽場計算，雖然整個計算過程耗時整整兩年，但是這一成功向試驗化學家證明了量子化學理論確實可以準確地描述分子的結構和性質，並且為量子化學打開了計算機時代的大門，因而這一計算結果有著劃時代的意義。
1952 年日本化學家福井謙一提出了前線軌道理論，1965 年美國有機化學家羅伯特・伯恩斯・伍德沃德和量子化學家羅阿爾德・霍夫曼聯手提出了有機反應中的分子軌道對稱守恆原理。福井、伍德瓦爾德和霍夫曼的理論使用簡單的模型，以簡單分子軌道理論為基礎，迴避那些高深的數學運算而以一種直觀的形式將量子化學理論應用於對化學反應的定性處理，通過他們的理論，實驗化學家得以直觀地窺探分子軌道波函數等抽象概念。福井和霍夫曼憑藉他們這一貢獻獲得了 1981 年度的諾貝爾化學獎。 | [
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量子化学 | 在計算方法方面，隨著計算機的發展，量子化學計算方法也飛速發展，在 1920 年代至今的數十年內，湧現出了組態相互作用方法（**CI**）、多體微擾理論（**MP**）、密度泛函理論（**DFT**）以及數量眾多形式不一的旨在減少計算量的半經驗計算方法，由於量子化學家們的工作，現在已經有大量商用量子化學計算軟體出現，其中很多都能夠在普通 PC 機上實現化學精度的量化計算，昔日神秘的量子化學理論，已經成為化學家常用的理論工具。約翰・波普與瓦爾特・科恩分別因為發展首個普及的量子化學軟體（Gaussian）和提出密度泛函理論 （Density Functional Theory）而獲得 1998 年諾貝爾化學獎。2009 年日本山口大學堀憲次教授和行政法人科技振興機構聯合成立過渡狀態科技公司，開拓量子化學於製藥業的實用商用化。
## 量子化學理論方法
* **分子軌道理論**：分子體系中的電子用單電子波函數滿足 Pauli 不相容原理的直積（如 Slater 行列式）來描述，其中每個單電子波函數通常由原子軌道線性組合得到（類似於原子體系中的原子軌道），被稱作分子軌道，分子軌道理論是目前應用最為廣泛的量子化學理論方法。
* **HF 方法**：Hartree-Fock 方法，用單個 Slater 行列式構造近似的波函數，通過對該行列式形式的波函數變分極小來確定據確定具體的波函數形式，是其他高級分子軌道理論方法的基礎。
* **CI 方法**：組態相互作用（Configuration Interaction）方法。用 HF 自洽場方法計算獲得的波函數和各級激發的波函數為基展開體系波函數。完全的組態相互作用（Full-CI）是指定基組下最精確的方法，但其計算量約以基函數的階乘規模增加，目前僅限於對小分子作為 Benchmark 以檢測其他方法的可靠性，在實際應用中常採用截斷 CI 方法，如 DCI、SDCI 等。由於截斷 CI 不滿足體積加合性（size-extensive），雖然有一些經驗方法校正（如 Davidson correction），仍然限制了其應用。 | [
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量子化学 | * **MP 方法**：多體微擾方法，將多電子體系的總哈密頓算符與 Fock 算符的差作為體系的微擾項，應用 Rayleigh-Schroedinger 微擾方法計算。一級微擾等價於 HF，二級微擾可以達到甚至超過 DCI 方法的精度水平，但計算量（N^5）小於 DCI（N^6）。一般不適用於能級接近簡併的體系。任意階的 MP 都是 size-extensive 的。
* **多組態自洽場方法**：同時對組態和展開係數進行優化，恰當地應用可以克服截斷 CI 不滿足體積加合性的缺點，並比較地準確處理能級近簡併的體系。
* **半經驗計算方法**：Extended Huckel method, AM1, PM3 等方法。在計算過程中根據實驗數據，將一些波函數積分用經驗常數替代，可以上千倍地減少計算量，採用的經驗常數不同，半經驗算法的應用範圍也不同，應用時需要根據研究體系的具體情況進行選擇。
* **密度泛函理論**：當分子體系各原子核空間位置確定後，電子密度在空間中的分布也確定，可以將體系的能量表示為電子密度的泛函，密度泛函分析變分法求出能量最低時的電子密度分布和體系能量。
* **價鍵理論**
## 量子化學研究內容
### 分子結構
通過計算不同分子結構的體系能量，量子化學方法可以找到分子勢能面上的能量最低點，從而確定分子在某一電子態的穩定構型。
### 化學反應
化學反應的過程可以看做分子體系在勢能面上滑動的過程，通過量子化學的計算，可以找到勢能面上的「駐點」：處於最低點的反應物和產物以及處於鞍點的過渡態，對比所有可能的反應途徑及其相對應的反應活化能，可以找到最有可能的反應途徑。
### 分子性質
量子化學計算可以獲得分子體系的電子波函數，通過這些電子波函數可以求算偶極矩、極化率等分子性質的計算，但是由於數學方法的局限，量子化學計算方法只能從上方逼近真實的分子體系能量，是一種近似計算，雖然能量的計算可以獲得較好的結果，但是獲得的電子波函數質量卻很差，因而分子性質計算的精度遠遠不及分子體系能量的計算。另一方面改進量子化學計算方法以獲得質量更好的電子波函數也是量子化學家目前面臨的挑戰之一。
## 量子化學計算軟體 | [
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量子化学 | * **Gaussian**[1] （頁面存檔備份，存於網際網路檔案館） ：量子化學領域最著名和應用最廣泛的軟體之一，由量子化學家約翰・波普的實驗室開發，可以應用從頭算方法、半經驗計算方法等進行分子能量和結構；過渡態能量和結構；化學鍵及反應能量；分子軌道；偶極矩；多極矩；紅外光譜和拉曼光譜，核磁共振，極化率和超極化率，熱力學性質，反應路徑等分子相關計算。可以運行在 Windows、Linux、Mac 作業系統中運行，目前最新版本為 Gaussian 16。但由於 Gaussian Inc.（Gaussian 的發展者）排斥其他軟體發展者的行為，而引來不少批評；其行為包括逐使原開發者約翰・波普離開 Gaussian Inc.（因而成為學術界其中一件為人齒冷的事件）；禁止其他開發者（包括約翰・波普在內）使用 Gaussian（引發起 Banned By Gaussian 運動）和禁止任何使用者發表比較 Gaussian 與其他量子化學軟體效能的報告等。
* **Gamess-US**[2] ：由於免費與開放源碼，成為除 Gaussian 以外，最廣泛應用的量子化學軟體，目前由 Iowa State Uinversity 的 Mark Gorden 教授的研究組主理。
* **ORCA**[3] ：一款從頭算量子化學計算軟體，包含了密度泛函理論、多體微擾、耦合簇方法、多參考方法以及半經驗計算方法。程序包致力於解決較大分子體系、過渡金屬及其光譜的計算。程序包由馬克斯普朗克煤炭研究所的 Frank Neese 教授課題組開發與維護。該程序包為閉源，學術用戶可免費申請。
* **Q-Chem**[4] （頁面存檔備份，存於網際網路檔案館） ：由一群隨約翰・波普離開 Gaussian Inc. 的學者創立的一個商業量子化學軟體。
* **Spartan**[5] （頁面存檔備份，存於網際網路檔案館） ：提供一個十分優良的圖形介面作量子化學計算，尤其適合非從事量子化學研究的學者使用。此軟體中有很大部的電子相關方法的計算程序實源自 Q-Chem，所以當使用到該些計算方法而作出報告時，應同時於文獻參考中引述 Spartan 與 Q-Chem。 | [
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量子化学 | * **CASTEP**[6] （頁面存檔備份，存於網際網路檔案館） ：為一量子力學為基礎的週期性固態材料化學計算的套裝軟體，此程式由英國劍橋大學卡文迪許實驗室（Cavendish Laboratory）的凝態物理理論組所共同研究開發。CASTEP 是由密度泛函理論為基礎的計算程式所組成，同時採用平面波（plane wave）為基底處理波函數，可針對具有週期性的固態材料表面進行化學模擬計算，而此軟體更具有高精準度以及高效能計算能力的表現。
* **MOLPRO**[7] （頁面存檔備份，存於網際網路檔案館） ：應用廣泛的量子化學計算軟體，軟體長於高精度計算，用多參考 CI，耦合簇等方法處理電子相關問題能夠對較大的分子體系進行準確的從頭計算。
* **MOPAC**[8] （頁面存檔備份，存於網際網路檔案館） ：應用最廣泛的半經驗量子化學計算軟體，能夠以 AM1，AM1-_d_，PM3，MNDO，MNDO-_d_，MINDO/3 等半經驗計算方法計算分子體系的自由能，活化能，反應路徑，偶極矩，非線性光學特性以及紅外光譜等性質。MOPAC 7 以前的版本為免費軟體，自 MOPAC 7 開始轉為商業軟體，目前最新版本為 MOPAC2007。
* **MOLCAS**[9] ：是一套包含各種量子化學計算方法的軟體，可以進行分子結構計算，鍵能，化學反應的能壘，激發能（包括自旋 - 軌道耦合），振動分辨吸收光譜，以及各種分子特性的計算，並且可以產生分子間的作用力，用於分子動力學的計算。目前最新版本為 MOLCAS 7。
* **CP2K**[10]: 應用廣泛的量子化學和凝聚態物理計算軟體，它可以模擬固體材料，溶液，分子，晶體，生物體系。CP2K 提供了多元化的理論方法包括 DFTB, LDA, GGA, MP2, RPA, 半經驗方法包括 (AM1, PM3, PM6, RM1, MNDO, xTB), 和精典力場 (AMBER, CHARMM, …). CP2K 可以模擬分子動力學，Metadynamics，模特卡羅，Ehrenfest 動力學，頻率分析，核激發光譜，能量優化，過渡態尋找（NEB, Dimer）。 | [
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] |
量子化学 | * **其他量子化學計算軟體**目前，除了上面提到的幾版著名量子化學計算軟體之外，還有大量商業和免費的量子化學計算軟體，其中絕大部分是從事量子化學或計算化學研究的實驗室自行開發的，此外，一些著名的大型化學軟體如 HyperChem、Chem3D、Sybyl 等，也包含有量子化學計算包。
## 參考文獻
## 外部連結
* Early Ideas in the History of Quantum Chemistry （頁面存檔備份，存於網際網路檔案館）
## 參見
* 計算化學
* 波恩 - 奧本海默近似
* 密度泛函理論
* 多體微擾理論
* Hartree-Fock 方程 | [
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光化学 | **光化學**（英語：**photochemistry**），是化學的一個分支，是一門研究物質因受光的影響而產生化學效應的學科。這裡的光通常指紫外光或可見光。光化學與其他化學的本質區別在於光化學涉及激發態。
## 歷史
最早進行光化學研究的學者是義大利化學家賈科莫・恰米奇安，從 1886 年開始，他與義大利化學家 Paolo Silber 共同完成了「苯醌向對苯二酚的轉化」以及「硝基苯在醇溶液中的光化學作用」等研究，他也可被認為是太陽能電池板之父。在 1912 年的第 8 屆國際應用化學大會上，他以「光化學的未來」為題發表了一篇演講，展望了光化學在未來可能起到的重要作用。
## 定義
光化學的定義有不同的表述。C. H. Wells 認為，光化學研究的是「吸收了紫外光或可見光的分子所經歷的化學行為和物理過程」。N. J. Turro 則認為「光化學研究的是電子激發態分子的化學行為和物理過程」。由於電子激發態通常由分子吸收紫外光或可見光形成，所以上述兩種定義的實質是一樣的。
## 激發態
**激發態**，即**電子激發態**。當分子中的電子排布不完全遵從構造原理時，則稱此分子處於激發態。物質的基態分子吸收了可見光或紫外光後，其電子會被激發，躍遷到更高的能級，形成處於激發態的分子。物質和光的能量都是量子化的，也就是說其能量只能取特定的值。因此，物質可以吸收具有特定能量的光子到達具有特定能量的激發態。相應地，激發態也可以通過各種方式失去多餘的能量回到基態。
### 激發態的形成
當一個光子的能量大於或等於兩個分子軌道間的能量差時，分子可以吸收這個光子，產生處於激發態的分子。這一過程需要遵從光化學第一定律。普通光引起的吸收過程還要遵循光化學第二定律，但雷射引發的吸收過程有可能引發雙光子效應，亦即光化學第二定律不再嚴格適用。
物質吸收光的能力可以用摩爾消光係數表徵。這一能力與化合物發生躍遷的概率有關。而躍遷發生的概率由躍遷選擇規則決定，但由於分子運動、旋軌耦合和旋旋耦合等因素，躍遷選擇規則並不完全適用。
### 單重態和三重態
**單重態**，也稱**單線態**，指的是自旋多重性為 1 的分子。單重態分子的能級在磁場中不裂分，在光譜中只能看到一條能級線。絕大多數分子的基態是單重態。 | [
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光化学 | **三重態**，也稱**三線態**，指的是自旋多重性為 3 的分子。三重態分子的能級在磁場中裂分，在光譜中原來的一條能級線裂分為三條線。當分子中含有兩個未配對的、自旋方向相同的電子時，該分子處於三重態。
### 激發態的失活
這些激發態的壽命較短，會在非常短的時間內失去多餘的能量回到基態構型，這個過程可以通過不同途徑完成。
#### 激發態的物理失活
如果失活過程中分子回到了其原始的基態，那麼就是一個物理失活過程。物理失活過程可以分為：
1. **分子內過程**
* **輻射躍遷**：在分子從激發態變為基態的過程中發射電磁輻射的躍遷。即發光過程，包括螢光、磷光和延時螢光。
* **非輻射躍遷**：在失活過程中不發無線電磁輻射的躍遷。包括內轉換和系間竄越。
2. **分子間過程**
* **振動弛豫**：當分子具有多餘的振動能時，將會快速地相互碰撞或與溶劑分子碰撞，從而生成某一電子能級的最低振動能級的分子，這一過程成為振動弛豫。
* **能量轉移**：此過程中，一個電子激發態分子（給體）將能量傳遞給另外一個分子（受體）從而變成更低電子能級的狀態，相應地，受體分子變成更高的電子能態。受體又稱猝滅劑，給體又稱敏化劑。
* **電子轉移**：此過程中一個光激發的給體分子與基態的受體分子相互作用，從而形成一個離子對。這個離子對有可能發生反向電子轉移，從而使激發態給體猝滅。
#### 激發態的化學失活
如果失活過程中形成了新的分子，則是化學失活過程。
### 雅布隆斯基圖
**雅布隆斯基圖**（英文：**Jablonski Diagram**）是分子激發和失活過程的圖示表示。
## 研究內容
### d - 區元素光化學
### 有機光化學
* 烯烴光化學
* 芳烴光化學
* 羰基化合物光化學
* 共軛烯酮光化學
* 偶氮化合物光化學
* 重氮化合物光化學
* 疊氮化合物光化學
* 有機硫化物光化學
* 光敏氧化反應
### 光催化
研究物質因受光的影響而產生催化效應的一個學科。如 TiO2 是最常用的光催化劑，它無毒、價廉、抗腐蝕並具有較高的催化活性。
### 超分子光化學
### 光電化學
### 生物光化學
## 參考文獻
### 引用
### 書籍
1. 張建成；王奪元。现代光化学。北京：化學工業出版社. [2006-09]. ISBN 7-5025-8947-3 （中文（簡體））. | [
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光化学 | 2. Wardle, Brian. Principles and Applications of Photochemistry. John Wiley and Sons. 2010. ISBN 9780470014936 （英語）. | [
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輻射化學 | **放射化學**（英語：Radiochemistry）是化學的一個分支，旨在研究那些參與化學反應的物質屬於或帶有放射性同位素的化學反應的一門學科。放射化學是關於放射性材料的化學；其中，放射性同位素用於研究非放射性同位素（在放射化學領域，由於缺乏放射性的物質在放射性衰變上是_穩定的_，因而往往又將這些非放射性物質稱為_冷物質_或_惰性物質_）的性質和化學反應。放射化學在很大程度上是利用放射性來研究普通化學反應。
## 歷史
此學科起源在 19 世紀末 20 世紀初，亨利・貝克勒於 1896 年發現鈾化合物所發射的輻射，能使底片感光變黑。其後皮埃爾・居里及瑪麗・居禮則在鐳的相關研究中，進一步發展了輻射化學的研究。**輻射化學**是研究普通的化學變化，所有元素不穩定的同位素經歷放射性衰變及放射性物質的產生（稱為放射性同位素）。
## 放射性同位素
放射化學研究所採用的放射性同位素包括天然的和人造的兩類。
## 主要衰變類型
放射化學之中所涉及的放射性衰變主要分為三種類型：
1. α 衰變 —— 主要產生 α 粒子
2. β 衰變 —— 主要產生 β 粒子
3. γ 衰變 —— 產生伽瑪射線
### 放射物
而放射性物質可產生三種放射物（每次放射性衰變大多出現一種），分別是：
1. （alpha）粒子
2. （beta）粒子
3. （gamma）伽瑪射線
它們有不同的特質。
#### α 粒子
由 α 衰變產生，只帶有兩個中子及兩個質子，不帶電子，是氦的同位素**氦 - 4**。
#### β 粒子
由 β 衰變產生，帶有一個電子，但不帶任何中子及質子。
#### 伽瑪射線
可由 α 衰變、β 衰變或 γ 衰變產生，不帶任何中子、質子及電子。在量子物理學上，只有光子，是電磁輻射的一種。
## 應用示例
### 核醫學
1. 採用碘的放射性同位素 **125I** 標記各種蛋白質或激素，以便利用放射免疫分析技術，檢測血清標本之中相應物質的濃度。
2. 採用氟的放射性同位素 **18F**，藉助於親核取代反應，合成 2 - 氟 - 2 - 脫氧 - D - 葡萄糖，作為一種常用的代謝顯像劑，用於 PET 成像檢查，診斷腫瘤等疾病。
## 參見
* 檢驗醫學
* 放射免疫分析
* 放射藥理學
* 核醫學放射性藥物列表 | [
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共价 | **共價鍵**（英語：covalent bond）是種化學鍵。兩粒或以上非金屬原子共同使用它們的外層電子（除砷化鎵），在理想情況下達到電子飽和的狀態，由此組成比較穩定和堅固的化學結構。與離子鍵不同，它們並沒有獲得或損失電子，進入共價鍵的原子向外不顯示電荷。共價鍵的強度比氫鍵強，比離子鍵小。同一種元素的原子或不同元素的原子都可以通過共價鍵結合，一般共價鍵結合的產物是分子，在少數情況下也可以形成晶體。又稱為共價鏈。
吉爾伯特・路易斯於 1916 年最早提出「共價鍵」這一概念。他認為 —— 在簡單原子軌域模型中，要形成共價鍵的原子必須互相提供電子（一個或多個）從而組成電子對，電子對則扮演了共價鍵的角色，而形成共價鍵後的原子共同擁有那對電子對。
在量子力學中，最早的共價鍵形成的解釋是由電子的複合而構成完整的軌域來解釋的。第一件量子力學共價鍵模型是 1927 年提出，當時人們還只能計算最簡單的共價鍵：氫氣分子的共價鍵。今天的計算表明，原子間的距離非常近時，它們的電子軌域會相互作用而形成整粒分子共用的電子軌域。
## 吸引和排斥
共價鍵中，以共價鍵連接的原子吸引共用的電子，使這些原子結合在一起。雖然其原子核之間和電子之間會因電荷相同而互相排斥，但原子核間的電子會減弱這些排斥作用，而電子與原子核間的相互作用更強。
## 電子飽和
按照簡單的電子殼模型，原子的外層電子在達到飽和狀態下最穩定。對大多數原子來說，外層電子數為 8 時它們達到飽和，即「八隅體」。這時它們的外層電子數與同週期的惰性氣體元素的外層電子數相同。
以氯化氫為例，在氯化氫分子中氫原子並沒有將它的外層電子交給氯原子。而是兩粒原子共用一對外層電子而達到飽和狀態。
## 鍵參數
**鍵參數**，是可以用於描述共價鍵性質的物理量的統稱，參數詳情可見於下列條目：
## 例子
### 氧分子
氧原子的外層電子數為 6，這六粒電子中的四粒組成兩對，其它兩粒單獨存在。
這兩粒單獨的電子與另一粒原子中相應的單獨電子結合組成兩對新的共用電子對，由此達到電子飽和狀態。
注意：此處的氧分子模型是簡化模型，實際上的氧分子要比這裡描述的要複雜得多，因為這 6 粒外層電子分布在不同的軌域上，所以它們不能形成這樣簡單的電子對。在實際之中，氧分子有 3 對共用的電子對和 2 粒單獨的電子。
## 分類
共價鍵是電子雲的重疊，所以共價鍵最本質的分類方式就是它們的重疊方式。現在已知有 3 種重疊方式，分別稱作： | [
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共价 | * σ 鍵
* π 鍵
* δ 鍵
在有機化合物中，通常把共價鍵以其共用的電子對數分為**單鍵**、**雙鍵**以及**叄鍵**。單鍵是一根 σ 鍵；雙鍵和叄鍵都含一根 σ 鍵，其餘 1 根或 2 根是 π 鍵。
但無機化合物不用此法。原因是，無機化合物中經常出現的共軛體系（離域 π 鍵）使得某兩粒原子之間共用的電子對數很難確定，因此無機物中常取**平均鍵級**，作為鍵能的粗略標準。
共價鍵還可以分為極性共價鍵和非極性共價鍵
## 極性共價鍵
假如組成共價鍵的原子的電負度不同，那麼其中一粒原子核可能吸引共用的電子對，電子對在分子中的分布也不相等，電子較集中的地方顯負性，電子較稀疏的地方顯正性。這樣整粒分子就會顯示出一定的極性。分子的電極的分布除其原子的電負度外還與其分子的組成有關。
## 網狀共價鍵
網狀共價鍵（Network covalent bonding）是原子晶體的主要化學鍵，是晶體結構的主要連接鍵。
網狀共價鍵是共價鍵中的一種，它所形成的物質不是一粒粒的分子，而是大的原子晶體。網狀共價鍵的連接之力很強，若想使鍵斷開則需要有很大的能量，所以以網狀共價鍵作為化學鍵的物質的熔點和沸點都較高、硬度也很大，最典型的例子之一就是金剛石。
## 配位鍵
配位鍵（又稱配位共價鍵）是特殊的共價鍵，它的特點在於共用的一對電子出自同一原子。形成配位鍵的條件是，一粒原子有孤電子對，而另一粒原子有空軌域。配位化合物，尤其是過渡金屬錯合物，種類繁多，用途廣泛，現已形成配位化學。
## 參考文獻
## 外部連結
* Covalent Bonds and Molecular Structure （頁面存檔備份，存於網際網路檔案館）
* Structure and Bonding in Chemistry—Covalent Bonds | [
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电学 | **電學**（electricity，electrical science）是涵蓋一切以電為研究基礎的學科，屬物理學的重要分支學科。19 世紀末隨著電報、電力系統的應用逐漸奠定了此工程的學科基礎，並廣泛地應用在各個領域。在技職教育上，以基本電學作為起始基礎教育學科，電機工程包括許多「次領域」如：電路學、電子學、電力學、電磁學等等，並且與其他物理科學領域有相互關係。
## 發展歷史
在 20 世紀初期發明的真空管，可以放大和處理電子訊號，因此產生了廣播、電視、和電話等遠程通訊事業，使人類的通訊和娛樂方式為之改觀。但是以真空管裝配的電子產品，不但體積大而且耗電量多，因此早期的電子工業對人類生活的影響，主要是在通訊和娛樂方面。在 20 世紀中期所發明的電晶體和其後發展的積體電路（_Inte-grated Circuit_，簡稱 IC）技術，幾乎完全取代了真空管，電子工業產生了革命性的變化，電子電路的設計和製造縮小到微米的尺寸。利用新的電子技術，機器生產得以自動化，通訊更為便捷且多樣化，對人類生活的各個層面產生了深遠的影響。在 20 世紀的末期，個人電腦和網際網路的普及，縮短了人與人之間的距離，使得天涯若比鄰。
## 電學基本概念
### 基本電學
* 電子（electron）：在原子中，圍繞在原子核外面帶負電荷的稱為電子。
* 電路（electrics circuit）: 由電源、用電器、導線等連接組成的電流通道，分為閉合電路和開合電路。不經負載的閉合電路被稱之為短路。電子元裝置在電路中的連接方法有串聯和並聯兩種基本形式。
* 電壓（voltage）或稱電位差，是趨使電子流經導線的一種潛能，若把電荷從一點移到另一點必須對電場作功就稱兩點之間存在**電壓**（電位差）。
* 電子流 在電路中正電荷其實不移動，實際移動為電子，電子流的方向為電流的反向
* 電荷（electric charge）是電子負荷的量，電場之源。當正電荷發生淨移動時，在其移動方向上即構成電流。
* 電阻（electric resistance）：限制電路中電流的量，亦稱為電流的阻力。
* 阻抗（impedance）：交流電路中對電流限制能力 (以同電阻用於直流電路非常相似的方式) 的一種度量。定義為電壓除以電流
* 電功率（electric Power）：定義為單位時間內所作之功。因導線不積存電荷，故在一閉合電路中有多少電荷通過電池必有相同量之電荷通過電阻。 | [
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电学 | * 電場（electric field）：正或負電荷周圍產生電作用的區域，電場方向由高電位指向低電位。
* 電容（capacitance）：加電壓至金屬平行板上，電荷會分布於其上，而其所表現的比例常數值，也是存儲電荷能力的度量。
* 電感（inductance）：線圈由變化磁場對另一個線圈（互感，M）或自身（自感，L）產生電壓能力的度量
### 應用電學
* 電源（power supply）：乾電池與家用的 110V/220V 交流電源是常見的電壓源。
* 電壓源：可以維持定值大小的電壓且不受負載變動的影響的來源。
* 電流源：可以維持定值大小的電流且不受負載變動的影響的來源。
* 充電（electrify）
* 變壓 / 整流（rectification/commutation）：把交流電 (不斷改變方向的電流) 變為直流電，只允許電流朝一個方向流動。電燈和電機使用交流電，但大多數電子設備需用直流電。
* 導體 （conductor）
* 接地（ground connection; grounding; earthing）
* 電擊（electric shock）：經由導體接觸到某程度的電壓源，人體只要 1mA 就會有觸電之感覺，5mA 以上就會有肌肉痙攣現象，在嚴格控制下可作為醫療使用，但未受控制下將會造成生命危險。
## 研究領域
主要的研究領域是：
* 靜電學是研究「靜止電荷」的特性及規律的一門學科。
* 電子學（Electronics）以電子電洞 (電洞) 為基礎，探討的電流量通常較小 (弱電), 藉由控制帶電粒子，以達到儲存資料或是控制開關等目的。相關概念如：半導體、積體電路（IC）、印刷電路板（PCB）、固態元件等。
* 微電子學（Microelectronics）是電子學中的子領域，是一項專門研究與學習如何將電子組件（或稱：電子零件）以極小型的方式研製生產的學問。
* 電力學是以探討大電流 (強電), 高功率的電路為主的科學，常應用於發電、供電系統。相關概念如：發電機 / 馬達、變壓器 / 整流器、功率元件等。
* 電路學（circuitry）以克希荷夫定律（Kirchhoff's rules）為基礎，探討元裝置的「電壓」與「電流」關係；或是探討放大，雜音的關係。工程師利用電子元件來設計「電路」以實現所需的功能。 | [
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] |
电学 | 次領域有機電整合、壓電力學、數位控制、自動控制、電工學 (electrotechnics)、電磁學、控制系統、機器人學（robotics）、電機（electric machinery）等。
交叉領域有光學（光電學 / 光電子學）、電化學、電子物理、量子力學、核子工程、電子材料、生物電子學、醫學電子學等。
其他相關的基礎學科有微積分、工程數學、離散數學、工程英文等。
## 代表科學家及其貢獻
## 參見
* 電
* 電現象
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千禧年大奖难题 | **千禧年大獎難題**（英語：**Millennium Prize Problems**）是七條由美國的克雷數學研究所（Clay Mathematics Institute，CMI）於 2000 年 5 月 24 日公佈的數學難題，解題總獎金 700 萬美元。根據克雷數學研究所制定的規則，這系列挑戰不限時間，題解必須發表在知名的國際期刊，並經過各方驗證，只要通過兩年驗證期和專家小組審核，每解破一題可獲獎金 100 萬美元。
這些難題旨在呼應 1900 年德國數學家大衛・希爾伯特在巴黎提出的 23 個歷史性數學難題，經過一百年，約 17 條難題至少已局部解答。而千禧年大獎難題的破解，極有可能為密碼學、航天、通訊等領域帶來突破進展。
迄今為止，在七條問題中，龐加萊猜想是唯一已解決的，2003 年，俄羅斯數學家格里戈里・佩雷爾曼證明了它的正確性。而其它六道難題仍有待研究者探索。
## 緣起與公佈
2000 年 5 月 24 日，克雷數學研究所在巴黎的法蘭西公學院召開了巴黎千年會議（Paris Millennium Event）。百多年前的 1900 年，德國數學家大衛・希爾伯特宣佈了著名的希爾伯特的 23 個問題，地點正是在巴黎舉行的第二屆國際數學家大會。在二十世紀，對此系列問題的研究極大地推動了數學的發展。出此考慮，克雷數學研究所決定邀請世界上有影響力的數學家參會，並在會上宣佈二十一世紀須解決的七大數學難題。宣佈這些問題前，當天會議首先播放了 1930 年希爾伯特退休時演講的錄音，包括他的名言：「我們必須知道，我們必將知道。」隨後，美國數學家約翰・泰特登台，依如下順序宣佈了七條問題中的三條：黎曼猜想、貝赫和斯維訥通－戴爾猜想和 P/NP 問題，並逐一簡單介紹。之後是英國數學家麥可・阿蒂亞演講，介紹剩下四題，分別是龐加萊猜想、霍奇猜想、楊－米爾斯存在性與質量間隙和納維－斯托克斯存在性與光滑性。 | [
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千禧年大奖难题 | 這七大難題是由克雷數學研究所的科學顧問委員會（Scientific Advisory Board）在諮詢其他頂尖數學家後共同選出。小組有五位國際數學專家，領導者是美國數學家、哈佛大學教授亞瑟・賈菲，此外還包括解決了費馬大定理的安德魯・懷爾斯、法國數學家阿蘭・科納、數學物理學家愛德華・威滕，以及上述提及過的約翰・泰特和麥可・阿蒂亞。他們旨在記錄當今數學家面對最難的問題，引起大眾對數學研究的注意，強調為難題尋找答案的重要性。問題甄選完成後，克雷數學研究所董事會撥款七百萬美元，為每條問題設立一百萬美元獎金，並寫出授獎規則。
## 獎勵與規則
總計 700 萬美元的獎金來源於從事投資基金美國商人蘭登・克雷，作為業餘數學愛好者，他在 1999 年創立了克雷數學研究所，隨即捐出了這筆款項。克雷數學研究所董事會（Board of Directors）將這筆錢設立為千禧年難題的獎金，每題價值一百萬美元。理論上只有該董事會有權授獎。董事會接受研究所科學顧問委員會（Scientific Advisory Board）對得獎人的推薦。
克雷數學研究所規定，任何解題並意在獲獎的研究者，不應把答案直接呈交至研究所，而須先在有國際聲譽的同行評審數學出版物發表完整解答，否則研究所的科學顧問委員會將評定發表方式是否合格。解答須在兩年內獲數學界廣泛接受。若滿足以上條件，科學顧問委員會將成立特別小組以評定獲獎資格，該小組至少有兩位非委員會成員，其中至少一人會完整校驗解答。特別的，對於 P/NP 問題和納維－斯托克斯問題，證明或證否皆有獲獎資格。至於其他幾條問題，提出反例亦可獲獎，但若原問題在重構後可以剔除特殊情況而不傷本質，提出者可能只獲得一小筆獎金。此外，若多位數學家對題解做出關鍵貢獻，可由多人分享獎金。
## 已解問題
### 龐加萊猜想
在拓撲學，二維球面是緊緻且單連通。通俗地說，意味球面不會無限延伸，並且其上任何閉合的圈都可收緊至一點。龐加萊猜想考慮的是更高維的情況：若閉合三維空間中每條閉曲線都可連續收縮到一點，那麼拓撲地看，這空間是否就是球面。它的數學陳述為：一個單連通三維閉流形同胚於三維球面。這猜想是三維流形的分類問題的核心。1962 年，史蒂芬・斯梅爾證明了龐加萊猜想在五維以上的等價結論，四維的情況則在二十年後由麥可・弗里德曼證明，但數學界始終對三維流形束手無策，而人類所處的宇宙是三維流形，更顯出問題重要。
龐加萊猜想的官方陳述由約翰・米爾諾寫出。 | [
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千禧年大奖难题 | 2003 年，俄羅斯數學家格里戈里・佩雷爾曼在 arXiv 貼出了完整證明，先後有兩篇文章，但文字簡略且原創度極強，數學界經過近三年才完成校驗。2006 年，多組研究者先後發表論文闡釋了佩雷爾曼的成果，並認定其無誤。由於這一貢獻，國際數學家大會決定授予佩雷爾曼菲爾茲獎，但他本人卻拒絕領獎。2010 年 3 月 18 日，千禧年大獎正式頒發給佩雷爾曼，但他又一次拒絕領獎，也包括克雷數學研究所的百萬獎金。根據俄羅斯國際文傳電訊社的消息，佩雷爾曼認為此獎不公，他相信哥倫比亞大學數學家理察・哈密頓對這問題的貢獻絲毫不遜於自己。
## 未解問題
### P/NP 問題
在理論計算機科學，複雜度類 P 指所有可由確定型圖靈機在多項式時間內解決的問題，類 NP 是所有可在多項式時間內驗證解的正確性的問題。這裡所謂「多項式時間」指的是求解算法運行時間至多是輸入規模的多項式函數。粗略說，P 類問題是可以在計算機上快速求解的問題，而對 NP 問題則可快速確定某個可能的解是否正確。可以看出 P 類問題也是 NP 類問題，而兩者是否完全相等便是 P/NP 問題，即是否所有 NP 類問題都是 P 類問題，擁有多項式時間的求解算法。P/NP 不單是抽象的數學難題；若得以解決，它在運籌學和密碼學等應用領域也將有重大影響，此外還被認為有特別的哲學意義。
2001 年一項針對 100 名數學和計算機科學家的調查發現其中 61 人相信 P≠NP，2012 年調查者重複同一問卷發現 84% 受訪者相信 P≠NP，在可能的解決方法上，他們給出了組合數論、邏輯學和代數幾何等答案。在研究方面，對 P/NP 問題的重大進展來自 1970 年代史蒂芬・庫克和列奧尼德・列文的成果，他們證明存在這樣一類問題，若能對任意一條 NP 問題找到多項式時間的求解算法，那麼所有 NP 問題都是多項式時間可解的。他們將此類命題命名為 NP 完全問題。而對 P/NP 難題最近一次引起大量討論的嘗試來自惠普實驗室的印度科學家維奈・地奧萊里卡（Vinay Deolalikar）在 2010 年 8 月網上發表長達 100 頁的論文，宣稱證明了 P≠NP，在計算機科學和數學界的一番討論和校閱，尼爾・伊莫爾曼等人發現論文有致命錯誤。
P/NP 問題的官方陳述由史蒂芬・庫克寫出。
### 霍奇猜想 | [
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千禧年大奖难题 | 數學的一大分支代數幾何的中心研究對象是代數簇，簡言之它是由代數方程產生的代數對象，是幾何對象的推廣，人們所熟知的任何幾何對象（如圓）都是一個代數簇，但並非所有代數簇都是幾何的、可以直觀描繪的。在此猜想中，代數幾何學家關心的是非奇異射影代數簇，粗略而言它是一面光滑的多維曲面，由代數方程解定義產生。霍奇猜想所說的是在這種「形狀完美」的代數簇上，本可能不是幾何對象的霍奇閉鏈（Hodge cycle）卻是由名為代數閉鏈的幾何對象組成的。其嚴謹的數學表述為：在非奇異復射影代數簇上，任何一條霍奇閉鏈都可以表示為代數閉鏈類的有理線性組合。誠然，霍奇猜想中的數學名詞可說是令人生畏，在七大千年難題中，它也被認為是對非專業人士而言最難理解的一個。然而霍奇猜想的證明將為代數幾何、拓撲學和數學分析三個領域建立一種基本的聯繫，因此具有重大意義。
蘇格蘭數學家威廉・霍奇在 1950 年公佈猜想後不久，唐納德・斯賓塞和小平邦彥就證明了其中一種簡單情況。近年來的研究方向分為兩支：美國數學家菲利普・格里菲斯等人嘗試將這一猜想化約為霍奇類導出的多元可容正規函數（admissible normal function）的奇點（singularities）存在性問題，克萊爾・瓦贊則力圖在算術簇上證明霍奇猜想。大體而言，霍奇猜想的證明仍然難見突破，它甚至被稱為是漫無邊際的猜測，暫時沒有有力證據證據表明霍奇的直覺正確。
霍奇猜想的官方陳述由皮埃爾・德利涅寫出。
### 黎曼猜想
數論分支解析數論的一大研究主題是素數分佈。1740 年瑞士數學家歐拉研究了如下用希臘字母 $\zeta $ 命名的函數：
: $\zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \;\;\;\;\;\;\;\!$
用一種與埃拉托斯特尼篩法頗有相通之處的證明法，他證明了對於任意 $s>1$ ， | [
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千禧年大奖难题 | : $\zeta (s)={\frac {1}{\left(1-{\frac {1}{2^{s}}}\right)\left(1-{\frac {1}{3^{s}}}\right)\left(1-{\frac {1}{5^{s}}}\right)\left(1-{\frac {1}{7^{s}}}\right)\left(1-{\frac {1}{11^{s}}}\right)\ldots }}=\prod _{p}{\frac {1}{1-p^{-s}}}$
此處 $p$ 為全體素數。這稱歐拉乘積公式的等式標誌解析數論的肇始，它表明 ζ 函數與素數有著隱約而緊密的關係。19 世紀的德國數學家黎曼對這一函數的性質做出了更深入的研究，他證實了通過解析開拓， $\zeta $ 函數可以被定義在複數域。由於他是認識到這一點第一人，此函數通常稱黎曼 ζ 函數。在 1859 年的論文中，黎曼首先觀察到 ζ 函數在負偶數上有零點，它們稱「平凡零點」，在注意到一些規律後，他猜測 ζ 函數所有非平凡零點的實數部分均為 1/2，也即 ζ 函數非平凡解都位於直線 ${\frac {1}{2}}+t\mathbf {i} $ （「臨界線」）上，這便是黎曼猜想。為數不少的數學命題以黎曼猜想成立為前提，其在數學上的影響力已遠超素數分佈模式一點。它還對密碼學和物理學意義重大。
黎曼猜想是千禧年七大難題和希爾伯特的 23 個問題唯一一條共同問題，是 150 年來一直吸引數學家的難題，對它的研究極大推動了解析數論發展。有分析和數值方面證據支持黎曼猜想正確。例如，1914 年哈代證明了 ζ 函數有無限多個零點的實部等於 1/2，1989 年布萊恩・康瑞（Brian Conrey）證明了 ζ 函數全部零點中有 2/5 位於臨界線上，2012 年這結果提升到 41.28%。2004 年，數學家通過計算機驗證了 ζ 函數前 1013 個零點，沒有找到黎曼猜想反例，但這些離真正證明黎曼猜想仍相去甚遠。
黎曼猜想的官方陳述由恩里科・邦別里寫出。
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千禧年大奖难题 | 在物理學，楊－米爾斯理論是基於非阿貝爾群的量子規範理論。20 世紀初，物理學家期待量子理論和經典場論兩種思想可以融合。在這一方向上，最早出現的理論是英國物理學家保羅・狄拉克 1927 年創立的量子電動力學，簡稱 QED，它提供了對電磁現象的量子描述，成為麥克斯韋理論的一個量子版本，能極為精確地解釋電磁場和電磁力。自然而然，物理學家期待後續的理論能將電磁現象與弱力和強力一道統一起來。1954 年楊振寧和羅伯特・米爾斯提出了楊－米爾斯理論，它是對 QED 的進一步推廣。在此基礎上統一電磁力和強弱相互作用時，物理學家發現這一理論的「無質量性」成為癥結所在。經典楊－米爾斯理論的核心是一組非線性偏微分方程，楊－米爾斯存在性與質量間隙難題旨在證明楊－米爾斯方程組有唯一解，並且該解滿足「質量間隙」這特徵，其官方表述為：對任意緊緻、單的規範群，四維歐幾里得空間中的量子楊－米爾斯理論存在一個正的質量間隙。質量間隙問題是量子色動力學理解強相互作用的理論關鍵，關乎理論物理學的數學基礎，其解決將意味著一個數學上完整的量子規範場論的產生。
這問題的解決前景不甚樂觀，愛德華・威滕也直言「（它）對現在而言實在太難。」物理學家普遍相信質量間隙存在，但至今未能找到確鑿的數學和物理學證明。
楊－米爾斯存在性與質量間隙問題的官方陳述由亞瑟・賈菲和愛德華・威滕寫出。
### 納維－斯托克斯存在性與光滑性
在流體力學，納維－斯托克斯方程描述了包括空氣和水在內的流體的運動。該方程組早在 1821 年便由法國工程師克勞德－路易・納維發現了，他通過引入黏度的概念而推廣了 17 世紀建立的歐拉方程，隨後愛爾蘭數學家喬治・斯托克斯又對其多次完善。長久以來，數學家和物理學家相信對此方程的解能解釋和預測流體行爲，但至今對它的理解也頗爲有限。具體說來，對 $i=1,2,3$ ，都有如下納維－斯托克斯方程，
: ${\frac {\partial v_{i}}{\partial t}}+\sum _{j=1}^{3}v_{j}{\frac {\partial v_{i}}{\partial x_{j}}}=-{\frac {\partial p}{\partial x_{i}}}+\nu \sum _{j=1}^{3}{\frac {\partial ^{2}v_{i}}{\partial x_{j}^{2}}}+f_{i}({\boldsymbol {x}},t)$ 。 | [
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千禧年大奖难题 | 其中 $\nabla \cdot \mathbf {v} =0$ 。而欲解的未知數是速度向量 $\mathbf {v} ({\boldsymbol {x}},t)$ 和流體壓力 $p({\boldsymbol {x}},t)$ 。納維－斯托克斯存在性與光滑性問題要求解答者證明該方程存在光滑的解。
在此問題上數學家已取得了部分成果。1934 年，讓・勒雷證明了方程弱解的存在性，該解滿足方程均值，在每一點上則不一定。另外，給定初始條件，總能找到一個正數 $T$ ，使方程在 $[0,T)$ 的時間段上可解，這正數也稱「爆裂時間」（blowup time）；只是一般而言由於 $T$ 實在太小，這些解未必有用。陶哲軒在 2014 年 2 月發表了一項關於三維納維－斯托克斯方程均值版本的爆裂時間的新成果。
納維－斯托克斯存在性與光滑性問題的官方陳述由查爾斯・費夫曼寫出。
### 貝赫和斯維訥通－戴爾猜想
在數論和代數幾何，由形如 $y^{2}=x^{3}+ax+b$ 的等式定義、且沒有奇點的曲線稱橢圓曲線。橢圓曲線是數論研究的重要領域，例如安德魯・懷爾斯對費馬大定理的證明的關鍵便是橢圓曲線。它在密碼學和數據傳輸上也均有應用。在此一問題上，數學家關心給定一橢圓曲線，其有多少個有理解，即有多少組有理數對 $(x,y)$ 滿足橢圓曲線方程。這一問題與該曲線對應的哈瑟 - 韋伊 L 函數 $L(C,s)$ 密切相關。基於計算機的數值依據，數學家布萊恩・貝赫和彼得・斯維訥通 - 戴爾猜想，一橢圓曲線 $C$ 有無窮多有理解，當且僅當 $L(C,s)$ 在 $s=1$ 時取 0。
有大量數值依據表明猜想正確。而在 1994 年前該猜想是否有意義都不甚明確，當時的數學家並不知是否存在一個合適的 $L(C,s)$ 函數，使得對所有的 $s$ 都有一個答數，這猜想直到 1994 年才作爲谷山－志村定理一個特殊形式被解決。近年來此問題進展不多，特別是對於橢圓曲線秩大於 1 的情況，數學家所知甚少。
貝赫和斯維訥通－戴爾猜想的官方陳述由安德魯・懷爾斯寫出。
## 參見
* 希爾伯特的 23 個問題
* 斯梅爾問題
## 注釋
## 參考來源
## 外部連結
* The Millennium Prize Problems（英文） | [
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] |
马克斯·普朗克科学促进会 | **馬克斯・普朗克學會**（德語：Max-Planck-Gesellschaft，縮寫 **MPG**），全稱**馬克斯・普朗克科學促進協會**（Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V.），台灣常譯為**普朗克研究院**，中國大陸常譯為**馬普所**。其為德國一流科學研究機構的聯合機構。至 2021 年共有 37 名研究員獲得諾貝爾獎。協會標誌為羅馬神話中的智慧女神米娜瓦。
## 歷史
馬克斯・普朗克的前身是威廉皇帝學會，成立於 1911 年。在 1947 年為了紀念前一年過世的前會長，量子物理學家馬克斯・普朗克而更名。
## 運作
協會為一非營利性法人機構，登記在柏林，但中央行政單位是在慕尼黑。主要運作財源由德國聯邦政府科研部及各邦政府各出資一半。2004 年協會總預算約為 12 億 5 千萬歐元。2018 年協會總預算為 18 億歐元。除了各研究所外，尚有國際馬克斯・普朗克研究學院等附屬機構。截止 2011 年 1 月 1 日，馬克斯・普朗克學會下設了 80 個研究所和研究實驗室，大多數分佈在德國各地，其中，4 個研究所和 1 個研究實驗室建立在德國本土之外。共雇用約 12,000 名人員，另經常約 9,000 名訪問學者於此工作。
馬克斯・普朗克學會所的任務是研究基礎科學，分為以下三個領域：生物學和醫學、物理化學技術、人文科學，涵蓋所有基礎科學研究領域。各馬克斯・普朗克研究所經常緊密與所在地大學合作，但仍為大學外之獨立機構，機構人員不負責大學內部教學工作，並擁用較大學更良好的設備和更充裕的資金，維持其世界級的學術地位。由於不受大學分科的影響，在跨領域研究上特別擅長。
各研究所底下有數個大致上獨立運作的學系和研究組。各由系主任領導，為終身職，在自己的系上有近乎絕對的權力，可自行決定雇用和經費使用，研究任何他感興趣的科學問題。在系主任退休時，該系結束運作。
## 影響
中央研究院的制度設計參考馬克斯・普朗克學會，將研究領域區分為數理科學組、生命科學組和人文社會科學組三個領域。
## 馬克斯・普朗克研究所列表
馬克斯・普朗克學會下轄超過 80 個研究所。
## 參考文獻
## 外部連結
* MPG 官方網站 （頁面存檔備份，存於網際網路檔案館）
## 參見
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銫元素 | **銫**（拼音：sè，注音：ㄙㄜˋ；英語：**Caesium** 或 **Cesium**；舊譯**鏭**），是一種化學元素，其化學符號為 **Cs**，原子序數為 55，原子量為 132.9054519 u。銫屬於鹼金屬，帶銀金色。
銫色白質軟，熔點低，28.44 ℃時即會熔化。它是在室溫或者接近室溫的條件下為液體的五種金屬元素之一。銫的物理性質和化學性質與同為鹼金屬的銣和鉀相似。該金屬極度活潑，並且能夠自燃。它是具有穩定同位素的元素中電負性最低的，其穩定同位素為銫 - 133。銫通常是從銫榴石中提取出來的，而其放射性同位素，尤其是銫 - 137，是更重元素的衰變產物，可從核反應爐產生的廢料中提取。
1860 年，兩位德國化學家羅伯特・威廉・本生和古斯塔夫・基爾霍夫通過剛剛研究出來的焰色反應發現銫，並以拉丁文「caesius」（意為天藍色）作為新元素的名稱。銫最早的小規模應用是作為真空管以及光電池的吸收劑。1967 年，國際單位制中的秒開始以銫 - 133 的發射光譜中一個特殊的頻率作為定義。自此之後，銫廣泛地用於原子鐘。二十世紀九十年代以來，用於鑽井液的甲酸銫成為銫元素的最大應用。該元素在化學工業以及電子產業等有重要用途。其放射性同位素銫 - 137 的半衰期大約為 30 年，可以用於醫學、工業測量儀器以及水文學。雖然銫僅有輕微的毒性，但其金屬卻是一種有害的材料；若其放射性同位素釋放到了環境中，將對健康造成較大的威脅。
## 性質
### 物理性質
銫是一種非常柔軟、延展性很強的的黃白色金屬，其莫氏硬度在所有的元素中最低，僅 0.2。在存在痕量的氧氣的情況下會變暗。銫的熔點為 28.4℃，為少數幾種在接近室溫的條件下為液態的金屬元素。汞是唯一的熔點低於銫的金屬元素。此外，該金屬具有較低的沸點，僅有 641℃，為除了汞以外沸點最低的金屬元素。銫的化合物燃燒時具有藍色或者紫色。
銫可以和其它鹼金屬以及金形成合金，並且與汞形成汞齊。在溫度低於 650℃的條件下，銫無法和鈷、鐵、鉬、鎳、鉑、鉭或者鎢形成合金。銫可以和銻、鎵、銦以及釷形成具有感光性的金屬互化物。銫可以和除鋰之外的鹼金屬混合形成合金，並且摩爾比例為 41% 銫，47% 鉀以及 12% 鈉的合金的熔點為 - 78℃，在所有已知的金屬合金中熔點最低。
已經研究過若干種汞齊：CsHg
2 為黑色並具有紫色金屬光澤，而 CsHg 具有金色，同樣具有金屬光澤。
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