[{"data":1,"prerenderedAt":3978},["ShallowReactive",2],{"evidence-index":3,"evidence-doc-entropy":384},[4,19,35,48,62,75,94,106,117,129,141,153,164,175,192,206,216,227,240,259,270,280,290,300,312,321,332,344,358,369],{"path":5,"title":6,"description":7,"keywords":8,"field":17,"order":18},"\u002Fevidence\u002Fmaxwells-equations","Maxwell's Equations","Four equations that joined electricity, magnetism and light, and set physics on the road to relativity.",[9,10,11,12,13,14,15,16],"electromagnetism","electric fields","magnetic fields","electromagnetic waves","light","Gauss law","Faraday law","Ampere law","Physical",1,{"path":20,"title":21,"description":22,"keywords":23,"field":34,"order":18},"\u002Fevidence\u002Fnumerology-origin","Numerology Origin","How ancient number symbolism, built to describe cosmic order and interpret sacred texts, became modern personality numerology.",[24,25,26,27,28,29,30,31,32,33],"number symbolism","Pythagoras","gematria","sacred numbers","divination","history of mathematics","tetractys","monad","decad","perfect number","Historical",{"path":36,"title":37,"description":38,"keywords":39,"field":47,"order":18},"\u002Fevidence\u002Forigin-of-life","Origin of Life","What chemistry, geology and genomics currently tell us about how non-living matter became the first living cells.",[40,41,42,43,44,45,46],"abiogenesis","RNA world","Miller-Urey experiment","prebiotic chemistry","hydrothermal vents","protocells","LUCA","Biological",{"path":49,"title":50,"description":51,"keywords":52,"field":61,"order":18},"\u002Fevidence\u002Fsuffering","Suffering","How Siddhartha Gautama tested both luxury and severe self-denial, and what his diagnosis of suffering became.",[53,54,55,56,57,58,59,60],"Buddhism","Buddha","Siddhartha Gautama","dukkha","Four Noble Truths","Middle Way","Eightfold Path","craving","Cultural",{"path":63,"title":64,"description":65,"keywords":66,"field":74,"order":18},"\u002Fevidence\u002Fthe-evolution-of-trust","The Evolution of Trust","How the prisoner's dilemma, Axelrod's tournaments, and later models explain when cooperation beats selfishness, and when it does not.",[67,68,69,70,71,72,73],"game theory","prisoners dilemma","cooperation","Robert Axelrod","tit for tat","reciprocity","evolutionary dynamics","Social",{"path":76,"title":77,"description":78,"keywords":79,"field":34,"order":93},"\u002Fevidence\u002Fcivil-disobedience","Civil Disobedience","How Gandhi turned a refusal to obey into a political force, why open, nonviolent law-breaking works more often than violence, and where it fails.",[80,81,82,83,84,85,86,87,88,89,90,91,92],"civil disobedience","Gandhi","satyagraha","Salt March","nonviolence","ahimsa","Thoreau","Martin Luther King","non-cooperation","nonviolent resistance","Chenoweth","Orwell","wu wei",2,{"path":95,"title":96,"description":97,"keywords":98,"field":47,"order":93},"\u002Fevidence\u002Fplacebo","Placebo","How an inert treatment can change pain, symptoms and brain chemistry, why it works, and where its power ends.",[99,100,101,102,103,104,105],"placebo effect","nocebo","expectation","conditioning","pain relief","clinical trials","endogenous opioids",{"path":107,"title":108,"description":109,"keywords":110,"field":74,"order":93},"\u002Fevidence\u002Fpuer-aeternus","Puer Aeternus","The Jungian \"eternal youth\" archetype, what it describes, what empirical psychology can and cannot say about it, and where such people thrive or struggle.",[111,112,113,114,115,116],"Carl Jung","Jungian psychology","eternal youth","archetype","Peter Pan","adulthood",{"path":118,"title":119,"description":120,"keywords":121,"field":61,"order":93},"\u002Fevidence\u002Ftaoism","Taoism","What the Dao, wu wei and naturalness mean, and why Daoist texts separate caring for something from clinging to it.",[122,123,124,125,126,92,127,128],"Daoism","Laozi","Zhuangzi","Daodejing","Tao Te Ching","yin and yang","attachment",{"path":130,"title":131,"description":132,"keywords":133,"field":17,"order":93},"\u002Fevidence\u002Ftime-dilation","Time Dilation","Why moving clocks and clocks deeper in gravity tick slower, and how the twin paradox resolves.",[134,135,136,137,138,139,140],"special relativity","general relativity","twin paradox","Lorentz factor","proper time","atomic clocks","GPS",{"path":142,"title":143,"description":144,"keywords":145,"field":17,"order":152},"\u002Fevidence\u002Fhiggs-field","Higgs Field","How an invisible field filling all of space gives mass to fundamental particles, and how it was found in 2012.",[146,147,148,149,150,151],"Higgs boson","particle physics","Standard Model","CERN","Large Hadron Collider","symmetry breaking",3,{"path":154,"title":155,"description":156,"keywords":157,"field":47,"order":152},"\u002Fevidence\u002Fmeditation","Meditation","What meditation trains, how it seems to work in the brain, and how strong the evidence really is.",[158,159,160,161,162,163],"mindfulness","attention","neuroscience","brain","stress","contemplative practice",{"path":165,"title":166,"description":167,"keywords":168,"field":61,"order":152},"\u002Fevidence\u002Fthe-god-of-the-gaps","The God of the Gaps","Why explaining unknowns by appeal to God keeps losing ground, and why many theologians reject the move themselves.",[169,170,171,172,173,174],"theology","religion and science","argument from ignorance","naturalism","cosmological argument","fine-tuning",{"path":176,"title":177,"description":178,"keywords":179,"field":74,"order":152},"\u002Fevidence\u002Fthe-illusion-of-free-will","The Illusion of Free Will","A machine that knows what you will choose, what physics and psychology say about the feeling of choosing, and why philosophers still disagree about whether free will exists.",[180,181,182,183,184,185,186,187,188,189,190,191],"free will","Newcomb's problem","Newcomb's paradox","predictor","Laplace's demon","compatibilism","illusionism","decision theory","one box","two boxes","Wegner","conscious will",{"path":193,"title":194,"description":195,"keywords":196,"field":47,"order":205},"\u002Fevidence\u002Fif-the-world-was-vegan","If the World Was Vegan","A scenario analysis of whether global agriculture could feed everyone without animals, and what it would mean for land, soil and ecosystems.",[197,198,199,200,201,202,203,204],"veganism","plant-based diet","agriculture","livestock","food security","soil fertility","biodiversity","nutrition",4,{"path":207,"title":208,"description":209,"keywords":210,"field":17,"order":205},"\u002Fevidence\u002Fkaluza-klein-reduction","Kaluza–Klein Reduction","How one extra, curled-up dimension turns Einstein's gravity into gravity plus electromagnetism, and why we have not seen it.",[211,212,213,9,214,215],"extra dimensions","compactification","gravity","unification","charge quantization",{"path":217,"title":218,"description":219,"keywords":220,"field":17,"order":226},"\u002Fevidence\u002F13-dimensional-models","13-Dimensional Models","Where the number 13 comes from in theoretical physics, why a universe with two time dimensions is mathematically possible, and why none of it is confirmed.",[221,222,223,224,211,225,212],"string theory","M-theory","F-theory","S-theory","two-time physics",5,{"path":228,"title":229,"description":230,"keywords":231,"field":61,"order":226},"\u002Fevidence\u002F1984","1984","What Orwell's Nineteen Eighty-Four is about, why he wrote it, and why rival governments have banned or challenged it as propaganda for the other side.",[232,233,234,235,236,237,238,239],"George Orwell","Nineteen Eighty-Four","dystopia","totalitarianism","surveillance","Big Brother","Newspeak","censorship",{"path":241,"title":242,"description":243,"keywords":244,"field":47,"order":226},"\u002Fevidence\u002Fsentience","Sentience","What it means for a being to feel, how science tests it in animals from crabs to oysters, and why a simpler mind does not mean a smaller capacity to suffer.",[245,246,247,248,249,250,251,252,253,254,255,256,257,258,197],"sentience","consciousness","animal consciousness","animal pain","oyster","bivalves","hermit crabs","octopus","insects","hard problem","soul","moral status","Cambridge Declaration","New York Declaration",{"path":260,"title":261,"description":262,"keywords":263,"field":17,"order":269},"\u002Fevidence\u002Ftachyons","Tachyons","Hypothetical particles that always move faster than light, why they would carry messages into the past, and why modern physics reads them as a sign of instability instead.",[264,265,266,267,268,134],"faster than light","superluminal","causality","imaginary mass","instability",6,{"path":271,"title":272,"description":273,"keywords":274,"field":17,"order":279},"\u002Fevidence\u002Fclosed-timelike-curves","Closed Timelike Curves","How Einstein's equations permit paths through spacetime that return to their own past, and why no one has ever found one.",[275,135,266,276,277,278],"time travel","wormholes","Godel","chronology protection",7,{"path":281,"title":282,"description":283,"keywords":284,"field":17,"order":289},"\u002Fevidence\u002Fnovikov-self-consistency-principle","Novikov Self-Consistency Principle","Why a journey into the past might never change it, and why that need not require any force preventing change.",[275,285,286,287,266,288],"grandfather paradox","causal loops","closed timelike curves","consistency",8,{"path":291,"title":292,"description":293,"keywords":294,"field":17,"order":299},"\u002Fevidence\u002Fbootstrap-paradox","Bootstrap Paradox","Causal loops in which an object or idea exists only because it was carried into its own past, and whether physics can tolerate them.",[275,295,296,266,297,298],"causal loop","information paradox","entropy","self-consistency",9,{"path":301,"title":302,"description":303,"keywords":304,"field":17,"order":311},"\u002Fevidence\u002Fblock-universe","Block Universe","The view, motivated by relativity, that past, present and future all exist and time does not flow.",[305,306,307,308,309,310],"eternalism","spacetime","relativity","simultaneity","presentism","arrow of time",10,{"path":313,"title":314,"description":315,"keywords":316,"field":17,"order":320},"\u002Fevidence\u002Fdeterminism","Determinism","Whether the present fixes the entire future, what physics says about it, and why free will may survive either way.",[180,266,317,185,318,319],"Laplace demon","chaos","quantum mechanics",11,{"path":322,"title":323,"description":324,"keywords":325,"field":17,"order":331},"\u002Fevidence\u002Fbig-bounce","Big Bounce","The idea that our universe began not from nothing but from the collapse of a previous one, and what physics can and cannot say about it.",[326,327,328,329,297,330],"cosmology","Big Bang","cyclic universe","loop quantum cosmology","eternal return",12,{"path":333,"title":334,"description":335,"keywords":336,"field":17,"order":343},"\u002Fevidence\u002Fgolden-ratio","Golden Ratio","Why one number, the golden ratio, keeps turning up in sunflowers, and why many famous claims about it are myths.",[337,338,339,340,341,342],"phi","Fibonacci","phyllotaxis","spirals","irrational numbers","geometry",13,{"path":345,"title":346,"description":347,"keywords":348,"field":17,"order":357},"\u002Fevidence\u002Fsha-256-encryption","SHA-256 Hashing","How the SHA-256 hash function works, why it cannot be run backwards, and what reversing it would actually require.",[349,350,351,352,353,354,355,356],"cryptography","hash function","SHA256","encryption","preimage resistance","collisions","Bitcoin","brute force",14,{"path":359,"title":360,"description":361,"keywords":362,"field":17,"order":368},"\u002Fevidence\u002Fgrovers-algorithm","Grover's Algorithm","How a quantum computer can search N possibilities in about √N steps, and why that is both remarkable and far less than popular accounts claim.",[363,364,365,366,367,349],"quantum computing","quantum search","qubits","amplitude amplification","oracle",15,{"path":370,"title":371,"description":372,"keywords":373,"field":17,"order":383},"\u002Fevidence\u002Fentropy","Entropy","Why heat flows one way, how a thought experiment about a tiny demon tied physics to information, and whether black holes destroy it.",[297,374,375,376,377,378,379,310,380,381,382],"second law of thermodynamics","Maxwell's demon","Szilard engine","Landauer's principle","information","Boltzmann","black hole information paradox","Hawking radiation","unitarity",16,{"id":385,"title":371,"body":386,"description":372,"extension":3971,"field":17,"keywords":3972,"meta":3973,"navigation":3974,"order":383,"path":370,"seo":3975,"stem":3976,"__hash__":3977},"evidence\u002Fevidence\u002Fentropy.md",{"type":387,"value":388,"toc":3946},"minimark",[389,392,396,399,409,414,417,429,519,746,839,843,848,859,870,947,950,954,957,1086,1331,1334,1338,1345,1359,1363,1367,1381,1384,1387,1393,1397,1405,1535,1606,1613,1617,1625,1748,1848,1856,1860,1872,1881,1885,1893,2180,2313,2324,2331,2335,2339,2346,2357,2360,2364,2376,2648,2773,2780,3049,3314,3318,3326,3336,3340,3343,3410,3417,3433,3454,3466,3473,3477,3517,3521,3547,3551,3567,3573,3576,3580,3599,3613,3625,3637,3654,3671,3687,3703,3720,3737,3750,3767,3784,3800,3816,3832,3849,3866,3883,3900,3916,3932],[390,391,371],"h1",{"id":297},[393,394,395],"p",{},"Pour milk into a cup of coffee. Within seconds it swirls, spreads, and settles into a uniform brown. Now wait for the reverse: for the milk to gather itself back into a white cloud and leave the coffee black. Nothing in the laws of motion forbids it. Every collision between molecules could, in principle, be run backwards. Yet no one has ever seen it happen.",[393,397,398],{},"The same asymmetry appears everywhere. Hot things cool down; cold things do not spontaneously heat up. A dropped glass shatters; the pieces never jump back into a glass. We remember the past and not the future.",[393,400,401,404,405,408],{},[402,403,371],"strong",{}," is the name physics gives to what increases in all these processes. It began as an engineering quantity for steam engines, became a way of counting molecular arrangements, and ended up as a measure of ",[406,407,378],"em",{},". That last step came from an unusual source: a thought experiment about a small, intelligent being that seemed able to break the laws of physics.",[410,411,413],"h2",{"id":412},"why-engineers-needed-it","Why Engineers Needed It",[393,415,416],{},"In the early nineteenth century, steam engines were transforming industry, and no one knew their limits. In 1824 the French engineer Sadi Carnot showed that an engine producing work from heat must take heat from a hot body and dump some into a cold one, and that there is a maximum efficiency set only by the two temperatures.",[393,418,419,420,422,423],{},"Rudolf Clausius turned this into a general principle. In 1865 he defined a new quantity, which he named ",[402,421,297],{}," from a Greek word for transformation, and stated the two laws of thermodynamics in their famous form: the energy of the universe is constant; the entropy of the universe tends to a maximum. ",[424,425,428],"sup",{"className":426},[427],"c-cite","[1]",[393,430,431,432,486,487,518],{},"In Clausius's definition, when a small amount of heat ",[433,434,437,463],"span",{"className":435},[436],"katex",[433,438,441],{"className":439},[440],"katex-mathml",[442,443,445],"math",{"xmlns":444},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[446,447,448,458],"semantics",{},[449,450,451,455],"mrow",{},[452,453,454],"mi",{},"δ",[452,456,457],{},"Q",[459,460,462],"annotation",{"encoding":461},"application\u002Fx-tex","\\delta Q",[433,464,468],{"className":465,"ariaHidden":467},[466],"katex-html","true",[433,469,472,477,483],{"className":470},[471],"base",[433,473],{"className":474,"style":476},[475],"strut","height:0.8889em;vertical-align:-0.1944em;",[433,478,454],{"className":479,"style":482},[480,481],"mord","mathnormal","margin-right:0.0379em;",[433,484,457],{"className":485},[480,481]," flows reversibly into a system at absolute temperature ",[433,488,490,504],{"className":489},[436],[433,491,493],{"className":492},[440],[442,494,495],{"xmlns":444},[446,496,497,502],{},[449,498,499],{},[452,500,501],{},"T",[459,503,501],{"encoding":461},[433,505,507],{"className":506,"ariaHidden":467},[466],[433,508,510,514],{"className":509},[471],[433,511],{"className":512,"style":513},[475],"height:0.6833em;",[433,515,501],{"className":516,"style":517},[480,481],"margin-right:0.1389em;",", the entropy changes by",[433,520,523],{"className":521},[522],"katex-display",[433,524,526,571],{"className":525},[436],[433,527,529],{"className":528},[440],[442,530,532],{"xmlns":444,"display":531},"block",[446,533,534,568],{},[449,535,536,539,542,546,564],{},[452,537,538],{},"d",[452,540,541],{},"S",[543,544,545],"mo",{},"=",[547,548,549,562],"mfrac",{},[449,550,551,553],{},[452,552,454],{},[554,555,556,558],"msub",{},[452,557,457],{},[559,560,561],"mtext",{},"rev",[452,563,501],{},[452,565,567],{"mathvariant":566},"normal",".",[459,569,570],{"encoding":461},"dS = \\frac{\\delta Q_{\\text{rev}}}{T}.",[433,572,574,600],{"className":573,"ariaHidden":467},[466],[433,575,577,581,584,588,593,597],{"className":576},[471],[433,578],{"className":579,"style":580},[475],"height:0.6944em;",[433,582,538],{"className":583},[480,481],[433,585,541],{"className":586,"style":587},[480,481],"margin-right:0.0576em;",[433,589],{"className":590,"style":592},[591],"mspace","margin-right:0.2778em;",[433,594,545],{"className":595},[596],"mrel",[433,598],{"className":599,"style":592},[591],[433,601,603,607,743],{"className":602},[471],[433,604],{"className":605,"style":606},[475],"height:2.0574em;vertical-align:-0.686em;",[433,608,610,615,739],{"className":609},[480],[433,611],{"className":612},[613,614],"mopen","nulldelimiter",[433,616,618],{"className":617},[547],[433,619,623,730],{"className":620},[621,622],"vlist-t","vlist-t2",[433,624,627,727],{"className":625},[626],"vlist-r",[433,628,632,646,657],{"className":629,"style":631},[630],"vlist","height:1.3714em;",[433,633,635,640],{"style":634},"top:-2.314em;",[433,636],{"className":637,"style":639},[638],"pstrut","height:3em;",[433,641,643],{"className":642},[480],[433,644,501],{"className":645,"style":517},[480,481],[433,647,649,652],{"style":648},"top:-3.23em;",[433,650],{"className":651,"style":639},[638],[433,653],{"className":654,"style":656},[655],"frac-line","border-bottom-width:0.04em;",[433,658,660,663],{"style":659},"top:-3.677em;",[433,661],{"className":662,"style":639},[638],[433,664,666,669],{"className":665},[480],[433,667,454],{"className":668,"style":482},[480,481],[433,670,672,675],{"className":671},[480],[433,673,457],{"className":674},[480,481],[433,676,679],{"className":677},[678],"msupsub",[433,680,682,718],{"className":681},[621,622],[433,683,685,713],{"className":684},[626],[433,686,689],{"className":687,"style":688},[630],"height:0.1514em;",[433,690,692,696],{"style":691},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[433,693],{"className":694,"style":695},[638],"height:2.7em;",[433,697,703],{"className":698},[699,700,701,702],"sizing","reset-size6","size3","mtight",[433,704,706],{"className":705},[480,702],[433,707,710],{"className":708},[480,709,702],"text",[433,711,561],{"className":712},[480,702],[433,714,717],{"className":715},[716],"vlist-s","​",[433,719,721],{"className":720},[626],[433,722,725],{"className":723,"style":724},[630],"height:0.15em;",[433,726],{},[433,728,717],{"className":729},[716],[433,731,733],{"className":732},[626],[433,734,737],{"className":735,"style":736},[630],"height:0.686em;",[433,738],{},[433,740],{"className":741},[742,614],"mclose",[433,744,567],{"className":745},[480],[393,747,748,749,777,778,806,807,835,836,838],{},"Here ",[433,750,752,765],{"className":751},[436],[433,753,755],{"className":754},[440],[442,756,757],{"xmlns":444},[446,758,759,763],{},[449,760,761],{},[452,762,541],{},[459,764,541],{"encoding":461},[433,766,768],{"className":767,"ariaHidden":467},[466],[433,769,771,774],{"className":770},[471],[433,772],{"className":773,"style":513},[475],[433,775,541],{"className":776,"style":587},[480,481]," is entropy, and ",[433,779,781,794],{"className":780},[436],[433,782,784],{"className":783},[440],[442,785,786],{"xmlns":444},[446,787,788,792],{},[449,789,790],{},[452,791,501],{},[459,793,501],{"encoding":461},[433,795,797],{"className":796,"ariaHidden":467},[466],[433,798,800,803],{"className":799},[471],[433,801],{"className":802,"style":513},[475],[433,804,501],{"className":805,"style":517},[480,481]," is measured in kelvin. The formula explains why heat flows from hot to cold. The same amount of heat raises the entropy of a cold body more than it lowers the entropy of a hot one, because ",[433,808,810,823],{"className":809},[436],[433,811,813],{"className":812},[440],[442,814,815],{"xmlns":444},[446,816,817,821],{},[449,818,819],{},[452,820,501],{},[459,822,501],{"encoding":461},[433,824,826],{"className":825,"ariaHidden":467},[466],[433,827,829,832],{"className":828},[471],[433,830],{"className":831,"style":513},[475],[433,833,501],{"className":834,"style":517},[480,481]," is in the denominator. So the transfer raises the total. The reverse would lower it, and the ",[402,837,374],{}," says that in an isolated system this does not happen.",[410,840,842],{"id":841},"what-entropy-counts","What Entropy Counts",[844,845,847],"h3",{"id":846},"a-box-of-coins","A box of coins",[393,849,850,851,854,855],{},"Clausius's entropy worked, but it did not say what entropy ",[406,852,853],{},"is",". Ludwig Boltzmann answered that in the 1870s by looking at the molecules. ",[424,856,858],{"className":857},[427],"[2]",[393,860,861,862,865,866,869],{},"Imagine 100 coins in a box. Shake it. The ",[402,863,864],{},"macrostate"," is what you can see at a glance: how many heads there are. The ",[402,867,868],{},"microstate"," is the full detail: which particular coins are heads.",[393,871,872,873,946],{},"There is exactly one microstate with all 100 coins showing heads. There are about ",[433,874,876,898],{"className":875},[436],[433,877,879],{"className":878},[440],[442,880,881],{"xmlns":444},[446,882,883,895],{},[449,884,885],{},[886,887,888,892],"msup",{},[889,890,891],"mn",{},"10",[889,893,894],{},"29",[459,896,897],{"encoding":461},"10^{29}",[433,899,901],{"className":900,"ariaHidden":467},[466],[433,902,904,908,912],{"className":903},[471],[433,905],{"className":906,"style":907},[475],"height:0.8141em;",[433,909,911],{"className":910},[480],"1",[433,913,915,919],{"className":914},[480],[433,916,918],{"className":917},[480],"0",[433,920,922],{"className":921},[678],[433,923,925],{"className":924},[621],[433,926,928],{"className":927},[626],[433,929,931],{"className":930,"style":907},[630],[433,932,934,937],{"style":933},"top:-3.063em;margin-right:0.05em;",[433,935],{"className":936,"style":695},[638],[433,938,940],{"className":939},[699,700,701,702],[433,941,943],{"className":942},[480,702],[433,944,894],{"className":945},[480,702]," microstates with exactly 50 heads. Shake the box and you will almost certainly land near 50–50, not because anything pushes the coins there, but because almost all the possibilities are there.",[393,948,949],{},"That is the second law in miniature. Systems drift towards macrostates that can be realised in the most ways, simply because there are overwhelmingly more of them.",[844,951,953],{"id":952},"boltzmanns-formula","Boltzmann's formula",[393,955,956],{},"Boltzmann expressed this as",[958,959,961],"equation",{"label":960},"Boltzmann Entropy",[433,962,964],{"className":963},[522],[433,965,967,1000],{"className":966},[436],[433,968,970],{"className":969},[440],[442,971,972],{"xmlns":444,"display":531},[446,973,974,997],{},[449,975,976,978,980,988,991,994],{},[452,977,541],{},[543,979,545],{},[554,981,982,985],{},[452,983,984],{},"k",[452,986,987],{},"B",[452,989,990],{},"ln",[543,992,993],{},"⁡",[452,995,996],{},"W",[459,998,999],{"encoding":461},"S = k_B \\ln W",[433,1001,1003,1021],{"className":1002,"ariaHidden":467},[466],[433,1004,1006,1009,1012,1015,1018],{"className":1005},[471],[433,1007],{"className":1008,"style":513},[475],[433,1010,541],{"className":1011,"style":587},[480,481],[433,1013],{"className":1014,"style":592},[591],[433,1016,545],{"className":1017},[596],[433,1019],{"className":1020,"style":592},[591],[433,1022,1024,1028,1072,1076,1080,1083],{"className":1023},[471],[433,1025],{"className":1026,"style":1027},[475],"height:0.8444em;vertical-align:-0.15em;",[433,1029,1031,1035],{"className":1030},[480],[433,1032,984],{"className":1033,"style":1034},[480,481],"margin-right:0.0315em;",[433,1036,1038],{"className":1037},[678],[433,1039,1041,1064],{"className":1040},[621,622],[433,1042,1044,1061],{"className":1043},[626],[433,1045,1048],{"className":1046,"style":1047},[630],"height:0.3283em;",[433,1049,1051,1054],{"style":1050},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[433,1052],{"className":1053,"style":695},[638],[433,1055,1057],{"className":1056},[699,700,701,702],[433,1058,987],{"className":1059,"style":1060},[480,481,702],"margin-right:0.0502em;",[433,1062,717],{"className":1063},[716],[433,1065,1067],{"className":1066},[626],[433,1068,1070],{"className":1069,"style":724},[630],[433,1071],{},[433,1073],{"className":1074,"style":1075},[591],"margin-right:0.1667em;",[433,1077,990],{"className":1078},[1079],"mop",[433,1081],{"className":1082,"style":1075},[591],[433,1084,996],{"className":1085,"style":517},[480,481],[393,1087,1088,1089,1117,1118,1149,1150,1330],{},"where ",[433,1090,1092,1105],{"className":1091},[436],[433,1093,1095],{"className":1094},[440],[442,1096,1097],{"xmlns":444},[446,1098,1099,1103],{},[449,1100,1101],{},[452,1102,996],{},[459,1104,996],{"encoding":461},[433,1106,1108],{"className":1107,"ariaHidden":467},[466],[433,1109,1111,1114],{"className":1110},[471],[433,1112],{"className":1113,"style":513},[475],[433,1115,996],{"className":1116,"style":517},[480,481]," is the number of microstates consistent with the macrostate, ",[433,1119,1121,1137],{"className":1120},[436],[433,1122,1124],{"className":1123},[440],[442,1125,1126],{"xmlns":444},[446,1127,1128,1134],{},[449,1129,1130,1132],{},[452,1131,990],{},[543,1133,993],{},[459,1135,1136],{"encoding":461},"\\ln",[433,1138,1140],{"className":1139,"ariaHidden":467},[466],[433,1141,1143,1146],{"className":1142},[471],[433,1144],{"className":1145,"style":580},[475],[433,1147,990],{"className":1148},[1079]," is the natural logarithm, and ",[433,1151,1153,1195],{"className":1152},[436],[433,1154,1156],{"className":1155},[440],[442,1157,1158],{"xmlns":444},[446,1159,1160,1192],{},[449,1161,1162,1168,1171,1174,1177,1189],{},[554,1163,1164,1166],{},[452,1165,984],{},[452,1167,987],{},[543,1169,1170],{},"≈",[889,1172,1173],{},"1.38",[543,1175,1176],{},"×",[886,1178,1179,1181],{},[889,1180,891],{},[449,1182,1183,1186],{},[543,1184,1185],{},"−",[889,1187,1188],{},"23",[559,1190,1191],{}," J\u002FK",[459,1193,1194],{"encoding":461},"k_B \\approx 1.38 \\times 10^{-23}\\ \\text{J\u002FK}",[433,1196,1198,1253,1274],{"className":1197,"ariaHidden":467},[466],[433,1199,1201,1204,1244,1247,1250],{"className":1200},[471],[433,1202],{"className":1203,"style":1027},[475],[433,1205,1207,1210],{"className":1206},[480],[433,1208,984],{"className":1209,"style":1034},[480,481],[433,1211,1213],{"className":1212},[678],[433,1214,1216,1236],{"className":1215},[621,622],[433,1217,1219,1233],{"className":1218},[626],[433,1220,1222],{"className":1221,"style":1047},[630],[433,1223,1224,1227],{"style":1050},[433,1225],{"className":1226,"style":695},[638],[433,1228,1230],{"className":1229},[699,700,701,702],[433,1231,987],{"className":1232,"style":1060},[480,481,702],[433,1234,717],{"className":1235},[716],[433,1237,1239],{"className":1238},[626],[433,1240,1242],{"className":1241,"style":724},[630],[433,1243],{},[433,1245],{"className":1246,"style":592},[591],[433,1248,1170],{"className":1249},[596],[433,1251],{"className":1252,"style":592},[591],[433,1254,1256,1260,1263,1267,1271],{"className":1255},[471],[433,1257],{"className":1258,"style":1259},[475],"height:0.7278em;vertical-align:-0.0833em;",[433,1261,1173],{"className":1262},[480],[433,1264],{"className":1265,"style":1266},[591],"margin-right:0.2222em;",[433,1268,1176],{"className":1269},[1270],"mbin",[433,1272],{"className":1273,"style":1266},[591],[433,1275,1277,1281,1284,1319,1323],{"className":1276},[471],[433,1278],{"className":1279,"style":1280},[475],"height:1.0641em;vertical-align:-0.25em;",[433,1282,911],{"className":1283},[480],[433,1285,1287,1290],{"className":1286},[480],[433,1288,918],{"className":1289},[480],[433,1291,1293],{"className":1292},[678],[433,1294,1296],{"className":1295},[621],[433,1297,1299],{"className":1298},[626],[433,1300,1302],{"className":1301,"style":907},[630],[433,1303,1304,1307],{"style":933},[433,1305],{"className":1306,"style":695},[638],[433,1308,1310],{"className":1309},[699,700,701,702],[433,1311,1313,1316],{"className":1312},[480,702],[433,1314,1185],{"className":1315},[480,702],[433,1317,1188],{"className":1318},[480,702],[433,1320,1322],{"className":1321},[591]," ",[433,1324,1326],{"className":1325},[480,709],[433,1327,1329],{"className":1328},[480],"J\u002FK"," is Boltzmann's constant, which converts the count into thermodynamic units. The equation is carved on Boltzmann's tombstone in Vienna.",[393,1332,1333],{},"On this view, entropy measures how many ways the hidden details could be arranged without changing what we observe. Low entropy means the macrostate pins down the details; high entropy means it hides almost all of them.",[844,1335,1337],{"id":1336},"a-law-of-probability","A law of probability",[393,1339,1340,1341,1344],{},"This changes the status of the second law. It is not an absolute prohibition but a statistical one. The milk ",[406,1342,1343],{},"could"," unmix; the probability is just so small that it would not be expected to happen in many lifetimes of the universe. For small systems of a few molecules, fluctuations that briefly lower entropy do happen, and are measured.",[393,1346,1347,1348,1352,1353,1356,1357,567],{},"It also creates a puzzle. If the microscopic laws run equally well forwards and backwards, why does entropy increase towards the future and not towards the past? The standard answer is that the universe began in a state of extraordinarily low entropy, and everything since has been running downhill from it. ",[424,1349,1351],{"className":1350},[427],"[3]"," Why the beginning was like that is unknown. It is one of the central problems for any theory of the early universe, including the ",[1354,1355,323],"a",{"href":322},", and it is closely tied to the question of whether time \"flows\" at all, discussed in ",[1354,1358,302],{"href":301},[410,1360,1362],{"id":1361},"maxwells-demon","Maxwell's Demon",[844,1364,1366],{"id":1365},"the-thought-experiment","The thought experiment",[393,1368,1369,1370,1372,1373,1376,1377],{},"In 1867, James Clerk Maxwell, whose equations had already unified electricity, magnetism and light (see ",[1354,1371,6],{"href":5},"), described a challenge to the second law in a letter, and later published it in his book ",[406,1374,1375],{},"Theory of Heat"," (1871). ",[424,1378,1380],{"className":1379},[427],"[4]",[393,1382,1383],{},"Take a box of gas at uniform temperature, divided in two by a wall with a tiny door. The molecules are not all moving at the same speed: some are fast, some slow. A small being stands by the door. When a fast molecule approaches from the left, it opens the door and lets it through. When a slow molecule approaches from the right, it lets that one through the other way. It never pushes anything; it only opens and closes a frictionless door.",[393,1385,1386],{},"After a while, the right side is full of fast molecules and is hot. The left side is full of slow ones and is cold. A temperature difference has appeared from nothing, and it could be used to run an engine. Entropy has decreased, apparently with no cost.",[393,1388,1389,1390,1392],{},"William Thomson (Lord Kelvin) later named this being ",[402,1391,375],{},". Maxwell did not believe it could beat the second law. His point was that the law is statistical: it holds because we cannot see and sort individual molecules, not because sorting them is impossible in principle.",[844,1394,1396],{"id":1395},"szilard-the-demon-needs-to-know","Szilard: the demon needs to know",[393,1398,1399,1400,1404],{},"For more than half a century, the demon resisted a clean answer. In 1929 Leo Szilard stripped it down to a single molecule in a box. ",[424,1401,1403],{"className":1402},[427],"[5]"," The demon learns which half the molecule is in, inserts a partition, and lets the molecule push the partition like a piston. One bit of knowledge, \"left or right\", produces a definite amount of work:",[433,1406,1408],{"className":1407},[522],[433,1409,1411,1446],{"className":1410},[436],[433,1412,1414],{"className":1413},[440],[442,1415,1416],{"xmlns":444,"display":531},[446,1417,1418,1443],{},[449,1419,1420,1423,1425,1431,1433,1435,1437,1440],{},[559,1421,1422],{},"work",[543,1424,545],{},[554,1426,1427,1429],{},[452,1428,984],{},[452,1430,987],{},[452,1432,501],{},[452,1434,990],{},[543,1436,993],{},[889,1438,1439],{},"2",[543,1441,1442],{"separator":467},",",[459,1444,1445],{"encoding":461},"\\text{work} = k_B T \\ln 2 ,",[433,1447,1449,1470],{"className":1448,"ariaHidden":467},[466],[433,1450,1452,1455,1461,1464,1467],{"className":1451},[471],[433,1453],{"className":1454,"style":580},[475],[433,1456,1458],{"className":1457},[480,709],[433,1459,1422],{"className":1460},[480],[433,1462],{"className":1463,"style":592},[591],[433,1465,545],{"className":1466},[596],[433,1468],{"className":1469,"style":592},[591],[433,1471,1473,1476,1516,1519,1522,1525,1528,1531],{"className":1472},[471],[433,1474],{"className":1475,"style":476},[475],[433,1477,1479,1482],{"className":1478},[480],[433,1480,984],{"className":1481,"style":1034},[480,481],[433,1483,1485],{"className":1484},[678],[433,1486,1488,1508],{"className":1487},[621,622],[433,1489,1491,1505],{"className":1490},[626],[433,1492,1494],{"className":1493,"style":1047},[630],[433,1495,1496,1499],{"style":1050},[433,1497],{"className":1498,"style":695},[638],[433,1500,1502],{"className":1501},[699,700,701,702],[433,1503,987],{"className":1504,"style":1060},[480,481,702],[433,1506,717],{"className":1507},[716],[433,1509,1511],{"className":1510},[626],[433,1512,1514],{"className":1513,"style":724},[630],[433,1515],{},[433,1517,501],{"className":1518,"style":517},[480,481],[433,1520],{"className":1521,"style":1075},[591],[433,1523,990],{"className":1524},[1079],[433,1526],{"className":1527,"style":1075},[591],[433,1529,1439],{"className":1530},[480],[433,1532,1442],{"className":1533},[1534],"mpunct",[393,1536,1088,1537,1565,1566,1605],{},[433,1538,1540,1553],{"className":1539},[436],[433,1541,1543],{"className":1542},[440],[442,1544,1545],{"xmlns":444},[446,1546,1547,1551],{},[449,1548,1549],{},[452,1550,501],{},[459,1552,501],{"encoding":461},[433,1554,1556],{"className":1555,"ariaHidden":467},[466],[433,1557,1559,1562],{"className":1558},[471],[433,1560],{"className":1561,"style":513},[475],[433,1563,501],{"className":1564,"style":517},[480,481]," is the temperature of the gas and ",[433,1567,1569,1587],{"className":1568},[436],[433,1570,1572],{"className":1571},[440],[442,1573,1574],{"xmlns":444},[446,1575,1576,1584],{},[449,1577,1578,1580,1582],{},[452,1579,990],{},[543,1581,993],{},[889,1583,1439],{},[459,1585,1586],{"encoding":461},"\\ln 2",[433,1588,1590],{"className":1589,"ariaHidden":467},[466],[433,1591,1593,1596,1599,1602],{"className":1592},[471],[433,1594],{"className":1595,"style":580},[475],[433,1597,990],{"className":1598},[1079],[433,1600],{"className":1601,"style":1075},[591],[433,1603,1439],{"className":1604},[480]," reflects the two equally likely possibilities.",[393,1607,1608,1609,1612],{},"Szilard's key insight was that ",[402,1610,1611],{},"information has a thermodynamic value",". The demon converts knowledge into energy. So where does the entropy go?",[844,1614,1616],{"id":1615},"landauer-and-bennett-forgetting-costs-heat","Landauer and Bennett: forgetting costs heat",[393,1618,1619,1620,1624],{},"The final answer came from computing. In 1961 Rolf Landauer at IBM argued that erasing information is physically irreversible. ",[424,1621,1623],{"className":1622},[427],"[6]"," Resetting a memory bit to a fixed state, whatever it held before, must release at least",[958,1626,1628],{"label":1627},"Landauer Bound",[433,1629,1631],{"className":1630},[522],[433,1632,1634,1665],{"className":1633},[436],[433,1635,1637],{"className":1636},[440],[442,1638,1639],{"xmlns":444,"display":531},[446,1640,1641,1662],{},[449,1642,1643,1645,1648,1654,1656,1658,1660],{},[452,1644,457],{},[543,1646,1647],{},"≥",[554,1649,1650,1652],{},[452,1651,984],{},[452,1653,987],{},[452,1655,501],{},[452,1657,990],{},[543,1659,993],{},[889,1661,1439],{},[459,1663,1664],{"encoding":461},"Q \\ge k_B T \\ln 2",[433,1666,1668,1687],{"className":1667,"ariaHidden":467},[466],[433,1669,1671,1675,1678,1681,1684],{"className":1670},[471],[433,1672],{"className":1673,"style":1674},[475],"height:0.8778em;vertical-align:-0.1944em;",[433,1676,457],{"className":1677},[480,481],[433,1679],{"className":1680,"style":592},[591],[433,1682,1647],{"className":1683},[596],[433,1685],{"className":1686,"style":592},[591],[433,1688,1690,1693,1733,1736,1739,1742,1745],{"className":1689},[471],[433,1691],{"className":1692,"style":1027},[475],[433,1694,1696,1699],{"className":1695},[480],[433,1697,984],{"className":1698,"style":1034},[480,481],[433,1700,1702],{"className":1701},[678],[433,1703,1705,1725],{"className":1704},[621,622],[433,1706,1708,1722],{"className":1707},[626],[433,1709,1711],{"className":1710,"style":1047},[630],[433,1712,1713,1716],{"style":1050},[433,1714],{"className":1715,"style":695},[638],[433,1717,1719],{"className":1718},[699,700,701,702],[433,1720,987],{"className":1721,"style":1060},[480,481,702],[433,1723,717],{"className":1724},[716],[433,1726,1728],{"className":1727},[626],[433,1729,1731],{"className":1730,"style":724},[630],[433,1732],{},[433,1734,501],{"className":1735,"style":517},[480,481],[433,1737],{"className":1738,"style":1075},[591],[433,1740,990],{"className":1741},[1079],[433,1743],{"className":1744,"style":1075},[591],[433,1746,1439],{"className":1747},[480],[393,1749,1750,1751,1847],{},"of heat into the surroundings. At room temperature this is about ",[433,1752,1754,1782],{"className":1753},[436],[433,1755,1757],{"className":1756},[440],[442,1758,1759],{"xmlns":444},[446,1760,1761,1779],{},[449,1762,1763,1766,1768],{},[889,1764,1765],{},"3",[543,1767,1176],{},[886,1769,1770,1772],{},[889,1771,891],{},[449,1773,1774,1776],{},[543,1775,1185],{},[889,1777,1778],{},"21",[459,1780,1781],{"encoding":461},"3 \\times 10^{-21}",[433,1783,1785,1803],{"className":1784,"ariaHidden":467},[466],[433,1786,1788,1791,1794,1797,1800],{"className":1787},[471],[433,1789],{"className":1790,"style":1259},[475],[433,1792,1765],{"className":1793},[480],[433,1795],{"className":1796,"style":1266},[591],[433,1798,1176],{"className":1799},[1270],[433,1801],{"className":1802,"style":1266},[591],[433,1804,1806,1809,1812],{"className":1805},[471],[433,1807],{"className":1808,"style":907},[475],[433,1810,911],{"className":1811},[480],[433,1813,1815,1818],{"className":1814},[480],[433,1816,918],{"className":1817},[480],[433,1819,1821],{"className":1820},[678],[433,1822,1824],{"className":1823},[621],[433,1825,1827],{"className":1826},[626],[433,1828,1830],{"className":1829,"style":907},[630],[433,1831,1832,1835],{"style":933},[433,1833],{"className":1834,"style":695},[638],[433,1836,1838],{"className":1837},[699,700,701,702],[433,1839,1841,1844],{"className":1840},[480,702],[433,1842,1185],{"className":1843},[480,702],[433,1845,1778],{"className":1846},[480,702]," joules per bit: tiny, but not zero.",[393,1849,1850,1851,1855],{},"In 1982 Charles Bennett applied this to the demon. ",[424,1852,1854],{"className":1853},[427],"[7]"," Measuring a molecule can, in principle, be done without cost. But the demon has a finite memory. To keep working, it must eventually erase what it learned about earlier molecules. That erasure produces at least as much entropy as the sorting removed. The demon does not break the second law; it pays with forgetting.",[844,1857,1859],{"id":1858},"experimental-confirmation","Experimental confirmation",[393,1861,1862,1863,1867,1868],{},"Both halves of the story have now been tested. In 2010 Shoichi Toyabe and colleagues used feedback control on a microscopic bead to convert information about its position into free energy, a working version of Szilard's engine. ",[424,1864,1866],{"className":1865},[427],"[8]"," In 2012 Antoine Bérut and colleagues erased single bits stored in a colloidal particle held by lasers, and measured the heat released approaching the Landauer limit. ",[424,1869,1871],{"className":1870},[427],"[9]",[393,1873,1874,1877,1878],{},[402,1875,1876],{},"Status:"," the link between information and thermodynamic entropy is established experimentally, not merely theoretically. Philosophers still debate how general Landauer's argument is and whether it has been derived without circularity. ",[424,1879,1380],{"className":1880},[427],[410,1882,1884],{"id":1883},"information-entropy","Information Entropy",[393,1886,1887,1888,1892],{},"In 1948 Claude Shannon, working on telephone signals at Bell Labs, needed a measure of how much information a message carries. ",[424,1889,1891],{"className":1890},[427],"[10]"," He arrived at a formula of the same shape as Boltzmann's:",[433,1894,1896],{"className":1895},[522],[433,1897,1899,1952],{"className":1898},[436],[433,1900,1902],{"className":1901},[440],[442,1903,1904],{"xmlns":444,"display":531},[446,1905,1906,1949],{},[449,1907,1908,1911,1913,1915,1924,1930,1941,1947],{},[452,1909,1910],{},"H",[543,1912,545],{},[543,1914,1185],{},[1916,1917,1918,1921],"munder",{},[543,1919,1920],{},"∑",[452,1922,1923],{},"i",[554,1925,1926,1928],{},[452,1927,393],{},[452,1929,1923],{},[554,1931,1932,1939],{},[449,1933,1934,1937],{},[452,1935,1936],{},"log",[543,1938,993],{},[889,1940,1439],{},[554,1942,1943,1945],{},[452,1944,393],{},[452,1946,1923],{},[543,1948,1442],{"separator":467},[459,1950,1951],{"encoding":461},"H = -\\sum_i p_i \\log_2 p_i ,",[433,1953,1955,1974],{"className":1954,"ariaHidden":467},[466],[433,1956,1958,1961,1965,1968,1971],{"className":1957},[471],[433,1959],{"className":1960,"style":513},[475],[433,1962,1910],{"className":1963,"style":1964},[480,481],"margin-right:0.0813em;",[433,1966],{"className":1967,"style":592},[591],[433,1969,545],{"className":1970},[596],[433,1972],{"className":1973,"style":592},[591],[433,1975,1977,1981,1984,1987,2039,2042,2083,2086,2134,2137,2177],{"className":1976},[471],[433,1978],{"className":1979,"style":1980},[475],"height:2.3277em;vertical-align:-1.2777em;",[433,1982,1185],{"className":1983},[480],[433,1985],{"className":1986,"style":1075},[591],[433,1988,1991],{"className":1989},[1079,1990],"op-limits",[433,1992,1994,2030],{"className":1993},[621,622],[433,1995,1997,2027],{"className":1996},[626],[433,1998,2001,2014],{"className":1999,"style":2000},[630],"height:1.05em;",[433,2002,2004,2008],{"style":2003},"top:-1.8723em;margin-left:0em;",[433,2005],{"className":2006,"style":2007},[638],"height:3.05em;",[433,2009,2011],{"className":2010},[699,700,701,702],[433,2012,1923],{"className":2013},[480,481,702],[433,2015,2017,2020],{"style":2016},"top:-3.05em;",[433,2018],{"className":2019,"style":2007},[638],[433,2021,2022],{},[433,2023,1920],{"className":2024},[1079,2025,2026],"op-symbol","large-op",[433,2028,717],{"className":2029},[716],[433,2031,2033],{"className":2032},[626],[433,2034,2037],{"className":2035,"style":2036},[630],"height:1.2777em;",[433,2038],{},[433,2040],{"className":2041,"style":1075},[591],[433,2043,2045,2048],{"className":2044},[480],[433,2046,393],{"className":2047},[480,481],[433,2049,2051],{"className":2050},[678],[433,2052,2054,2075],{"className":2053},[621,622],[433,2055,2057,2072],{"className":2056},[626],[433,2058,2061],{"className":2059,"style":2060},[630],"height:0.3117em;",[433,2062,2063,2066],{"style":691},[433,2064],{"className":2065,"style":695},[638],[433,2067,2069],{"className":2068},[699,700,701,702],[433,2070,1923],{"className":2071},[480,481,702],[433,2073,717],{"className":2074},[716],[433,2076,2078],{"className":2077},[626],[433,2079,2081],{"className":2080,"style":724},[630],[433,2082],{},[433,2084],{"className":2085,"style":1075},[591],[433,2087,2089,2097],{"className":2088},[1079],[433,2090,2092,2093],{"className":2091},[1079],"lo",[433,2094,2096],{"style":2095},"margin-right:0.0139em;","g",[433,2098,2100],{"className":2099},[678],[433,2101,2103,2125],{"className":2102},[621,622],[433,2104,2106,2122],{"className":2105},[626],[433,2107,2110],{"className":2108,"style":2109},[630],"height:0.207em;",[433,2111,2113,2116],{"style":2112},"top:-2.4559em;margin-right:0.05em;",[433,2114],{"className":2115,"style":695},[638],[433,2117,2119],{"className":2118},[699,700,701,702],[433,2120,1439],{"className":2121},[480,702],[433,2123,717],{"className":2124},[716],[433,2126,2128],{"className":2127},[626],[433,2129,2132],{"className":2130,"style":2131},[630],"height:0.2441em;",[433,2133],{},[433,2135],{"className":2136,"style":1075},[591],[433,2138,2140,2143],{"className":2139},[480],[433,2141,393],{"className":2142},[480,481],[433,2144,2146],{"className":2145},[678],[433,2147,2149,2169],{"className":2148},[621,622],[433,2150,2152,2166],{"className":2151},[626],[433,2153,2155],{"className":2154,"style":2060},[630],[433,2156,2157,2160],{"style":691},[433,2158],{"className":2159,"style":695},[638],[433,2161,2163],{"className":2162},[699,700,701,702],[433,2164,1923],{"className":2165},[480,481,702],[433,2167,717],{"className":2168},[716],[433,2170,2172],{"className":2171},[626],[433,2173,2175],{"className":2174,"style":724},[630],[433,2176],{},[433,2178,1442],{"className":2179},[1534],[393,2181,1088,2182,2253,2254,2283,2284,2312],{},[433,2183,2185,2203],{"className":2184},[436],[433,2186,2188],{"className":2187},[440],[442,2189,2190],{"xmlns":444},[446,2191,2192,2200],{},[449,2193,2194],{},[554,2195,2196,2198],{},[452,2197,393],{},[452,2199,1923],{},[459,2201,2202],{"encoding":461},"p_i",[433,2204,2206],{"className":2205,"ariaHidden":467},[466],[433,2207,2209,2213],{"className":2208},[471],[433,2210],{"className":2211,"style":2212},[475],"height:0.625em;vertical-align:-0.1944em;",[433,2214,2216,2219],{"className":2215},[480],[433,2217,393],{"className":2218},[480,481],[433,2220,2222],{"className":2221},[678],[433,2223,2225,2245],{"className":2224},[621,622],[433,2226,2228,2242],{"className":2227},[626],[433,2229,2231],{"className":2230,"style":2060},[630],[433,2232,2233,2236],{"style":691},[433,2234],{"className":2235,"style":695},[638],[433,2237,2239],{"className":2238},[699,700,701,702],[433,2240,1923],{"className":2241},[480,481,702],[433,2243,717],{"className":2244},[716],[433,2246,2248],{"className":2247},[626],[433,2249,2251],{"className":2250,"style":724},[630],[433,2252],{}," is the probability of the ",[433,2255,2257,2270],{"className":2256},[436],[433,2258,2260],{"className":2259},[440],[442,2261,2262],{"xmlns":444},[446,2263,2264,2268],{},[449,2265,2266],{},[452,2267,1923],{},[459,2269,1923],{"encoding":461},[433,2271,2273],{"className":2272,"ariaHidden":467},[466],[433,2274,2276,2280],{"className":2275},[471],[433,2277],{"className":2278,"style":2279},[475],"height:0.6595em;",[433,2281,1923],{"className":2282},[480,481],"-th possible message and ",[433,2285,2287,2300],{"className":2286},[436],[433,2288,2290],{"className":2289},[440],[442,2291,2292],{"xmlns":444},[446,2293,2294,2298],{},[449,2295,2296],{},[452,2297,1910],{},[459,2299,1910],{"encoding":461},[433,2301,2303],{"className":2302,"ariaHidden":467},[466],[433,2304,2306,2309],{"className":2305},[471],[433,2307],{"className":2308,"style":513},[475],[433,2310,1910],{"className":2311,"style":1964},[480,481]," is measured in bits. A fair coin toss carries 1 bit. A coin that always lands heads carries 0 bits, because you learn nothing from seeing it.",[393,2314,2315,2316,2319,2320],{},"Shannon's entropy measures ",[406,2317,2318],{},"missing information",": how uncertain you are before the message arrives. Boltzmann's entropy measures the missing information about which microstate a system is in. The demon story shows that the resemblance is not a coincidence. Thermodynamic entropy is, in a precise sense, information we do not have about the molecules. ",[424,2321,2323],{"className":2322},[427],"[4,11]",[393,2325,2326,2327,2330],{},"This also connects to modern cryptography. A hash function such as ",[1354,2328,2329],{"href":345},"SHA-256"," maps many inputs to one output, so it discards information and cannot be run backwards. In a physical computer, Landauer's principle says that the discarded information does not vanish: it leaves as heat.",[410,2332,2334],{"id":2333},"is-information-ever-lost","Is Information Ever Lost?",[844,2336,2338],{"id":2337},"what-the-laws-say","What the laws say",[393,2340,2341,2342,2345],{},"Throw a book into a fire. The words seem gone. But the fundamental laws of physics, as currently understood, suggest they are not. Classical mechanics preserves the volume of possible states over time (Liouville's theorem), and quantum mechanics evolves states by ",[402,2343,2344],{},"unitary"," transformations, which can always be undone. In both cases, the exact present fixes the exact past as well as the future.",[393,2347,2348,2349,2352,2353,2356],{},"In principle, then, the smoke, ash, heat and light from the burning book still encode every word. Reading them back is hopelessly impractical, but the information is scrambled, not destroyed. This principle, sometimes called ",[402,2350,2351],{},"conservation of information",", is really a statement about the reversibility of the microscopic laws. It is closely related to ",[1354,2354,2355],{"href":313},"determinism",", which asks whether the present fixes the future. Conservation of information asks, in addition, whether the present fixes the past.",[393,2358,2359],{},"There is one caveat. In interpretations of quantum mechanics where measurement causes a genuine, random collapse, that collapse is not reversible. Whether it happens is an open question of interpretation.",[844,2361,2363],{"id":2362},"black-holes-have-entropy","Black holes have entropy",[393,2365,2366,2367,2371,2372],{},"Black holes seemed to offer a way around the second law: drop a box of hot gas into one, and its entropy disappears from the universe. In 1973 Jacob Bekenstein argued that this could not be the whole story, and that a black hole must itself have entropy, proportional to the area of its event horizon. ",[424,2368,2370],{"className":2369},[427],"[12]"," Stephen Hawking made this precise and, in 1974–75, showed that quantum effects make black holes emit thermal radiation. ",[424,2373,2375],{"className":2374},[427],"[13]",[958,2377,2379],{"label":2378},"Bekenstein–Hawking Entropy",[433,2380,2382],{"className":2381},[522],[433,2383,2385,2437],{"className":2384},[436],[433,2386,2388],{"className":2387},[440],[442,2389,2390],{"xmlns":444,"display":531},[446,2391,2392,2434],{},[449,2393,2394,2401,2403],{},[554,2395,2396,2398],{},[452,2397,541],{},[559,2399,2400],{},"BH",[543,2402,545],{},[547,2404,2405,2423],{},[449,2406,2407,2413,2420],{},[554,2408,2409,2411],{},[452,2410,984],{},[452,2412,987],{},[886,2414,2415,2418],{},[452,2416,2417],{},"c",[889,2419,1765],{},[452,2421,2422],{},"A",[449,2424,2425,2428,2431],{},[889,2426,2427],{},"4",[452,2429,2430],{},"G",[452,2432,2433],{"mathvariant":566},"ℏ",[459,2435,2436],{"encoding":461},"S_{\\text{BH}} = \\frac{k_B c^3 A}{4 G \\hbar}",[433,2438,2440,2503],{"className":2439,"ariaHidden":467},[466],[433,2441,2443,2447,2494,2497,2500],{"className":2442},[471],[433,2444],{"className":2445,"style":2446},[475],"height:0.8333em;vertical-align:-0.15em;",[433,2448,2450,2453],{"className":2449},[480],[433,2451,541],{"className":2452,"style":587},[480,481],[433,2454,2456],{"className":2455},[678],[433,2457,2459,2486],{"className":2458},[621,622],[433,2460,2462,2483],{"className":2461},[626],[433,2463,2465],{"className":2464,"style":1047},[630],[433,2466,2468,2471],{"style":2467},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[433,2469],{"className":2470,"style":695},[638],[433,2472,2474],{"className":2473},[699,700,701,702],[433,2475,2477],{"className":2476},[480,702],[433,2478,2480],{"className":2479},[480,709,702],[433,2481,2400],{"className":2482},[480,702],[433,2484,717],{"className":2485},[716],[433,2487,2489],{"className":2488},[626],[433,2490,2492],{"className":2491,"style":724},[630],[433,2493],{},[433,2495],{"className":2496,"style":592},[591],[433,2498,545],{"className":2499},[596],[433,2501],{"className":2502,"style":592},[591],[433,2504,2506,2510],{"className":2505},[471],[433,2507],{"className":2508,"style":2509},[475],"height:2.1771em;vertical-align:-0.686em;",[433,2511,2513,2516,2645],{"className":2512},[480],[433,2514],{"className":2515},[613,614],[433,2517,2519],{"className":2518},[547],[433,2520,2522,2637],{"className":2521},[621,622],[433,2523,2525,2634],{"className":2524},[626],[433,2526,2529,2546,2554],{"className":2527,"style":2528},[630],"height:1.4911em;",[433,2530,2531,2534],{"style":634},[433,2532],{"className":2533,"style":639},[638],[433,2535,2537,2540,2543],{"className":2536},[480],[433,2538,2427],{"className":2539},[480],[433,2541,2430],{"className":2542},[480,481],[433,2544,2433],{"className":2545},[480],[433,2547,2548,2551],{"style":648},[433,2549],{"className":2550,"style":639},[638],[433,2552],{"className":2553,"style":656},[655],[433,2555,2556,2559],{"style":659},[433,2557],{"className":2558,"style":639},[638],[433,2560,2562,2602,2631],{"className":2561},[480],[433,2563,2565,2568],{"className":2564},[480],[433,2566,984],{"className":2567,"style":1034},[480,481],[433,2569,2571],{"className":2570},[678],[433,2572,2574,2594],{"className":2573},[621,622],[433,2575,2577,2591],{"className":2576},[626],[433,2578,2580],{"className":2579,"style":1047},[630],[433,2581,2582,2585],{"style":1050},[433,2583],{"className":2584,"style":695},[638],[433,2586,2588],{"className":2587},[699,700,701,702],[433,2589,987],{"className":2590,"style":1060},[480,481,702],[433,2592,717],{"className":2593},[716],[433,2595,2597],{"className":2596},[626],[433,2598,2600],{"className":2599,"style":724},[630],[433,2601],{},[433,2603,2605,2608],{"className":2604},[480],[433,2606,2417],{"className":2607},[480,481],[433,2609,2611],{"className":2610},[678],[433,2612,2614],{"className":2613},[621],[433,2615,2617],{"className":2616},[626],[433,2618,2620],{"className":2619,"style":907},[630],[433,2621,2622,2625],{"style":933},[433,2623],{"className":2624,"style":695},[638],[433,2626,2628],{"className":2627},[699,700,701,702],[433,2629,1765],{"className":2630},[480,702],[433,2632,2422],{"className":2633},[480,481],[433,2635,717],{"className":2636},[716],[433,2638,2640],{"className":2639},[626],[433,2641,2643],{"className":2642,"style":736},[630],[433,2644],{},[433,2646],{"className":2647},[742,614],[393,2649,748,2650,2678,2679,2708,2709,2737,2738,2768,2769,2772],{},[433,2651,2653,2666],{"className":2652},[436],[433,2654,2656],{"className":2655},[440],[442,2657,2658],{"xmlns":444},[446,2659,2660,2664],{},[449,2661,2662],{},[452,2663,2422],{},[459,2665,2422],{"encoding":461},[433,2667,2669],{"className":2668,"ariaHidden":467},[466],[433,2670,2672,2675],{"className":2671},[471],[433,2673],{"className":2674,"style":513},[475],[433,2676,2422],{"className":2677},[480,481]," is the area of the horizon, ",[433,2680,2682,2695],{"className":2681},[436],[433,2683,2685],{"className":2684},[440],[442,2686,2687],{"xmlns":444},[446,2688,2689,2693],{},[449,2690,2691],{},[452,2692,2417],{},[459,2694,2417],{"encoding":461},[433,2696,2698],{"className":2697,"ariaHidden":467},[466],[433,2699,2701,2705],{"className":2700},[471],[433,2702],{"className":2703,"style":2704},[475],"height:0.4306em;",[433,2706,2417],{"className":2707},[480,481]," is the speed of light, ",[433,2710,2712,2725],{"className":2711},[436],[433,2713,2715],{"className":2714},[440],[442,2716,2717],{"xmlns":444},[446,2718,2719,2723],{},[449,2720,2721],{},[452,2722,2430],{},[459,2724,2430],{"encoding":461},[433,2726,2728],{"className":2727,"ariaHidden":467},[466],[433,2729,2731,2734],{"className":2730},[471],[433,2732],{"className":2733,"style":513},[475],[433,2735,2430],{"className":2736},[480,481]," is Newton's gravitational constant, and ",[433,2739,2741,2755],{"className":2740},[436],[433,2742,2744],{"className":2743},[440],[442,2745,2746],{"xmlns":444},[446,2747,2748,2752],{},[449,2749,2750],{},[452,2751,2433],{"mathvariant":566},[459,2753,2754],{"encoding":461},"\\hbar",[433,2756,2758],{"className":2757,"ariaHidden":467},[466],[433,2759,2761,2765],{"className":2760},[471],[433,2762],{"className":2763,"style":2764},[475],"height:0.6889em;",[433,2766,2433],{"className":2767},[480]," is the reduced Planck constant. The formula is unusual: it contains the constants of thermodynamics, relativity, gravity and quantum mechanics at once. The entropy grows with the ",[406,2770,2771],{},"area"," of the horizon, not the volume inside, a clue that has shaped much of modern theoretical physics.",[393,2774,2775,2776,2779],{},"The ",[402,2777,2778],{},"temperature"," of this Hawking radiation is inversely proportional to the mass:",[433,2781,2783],{"className":2782},[522],[433,2784,2786,2839],{"className":2785},[436],[433,2787,2789],{"className":2788},[440],[442,2790,2791],{"xmlns":444,"display":531},[446,2792,2793,2836],{},[449,2794,2795,2801,2803,2834],{},[554,2796,2797,2799],{},[452,2798,501],{},[452,2800,1910],{},[543,2802,545],{},[547,2804,2805,2815],{},[449,2806,2807,2809],{},[452,2808,2433],{"mathvariant":566},[886,2810,2811,2813],{},[452,2812,2417],{},[889,2814,1765],{},[449,2816,2817,2820,2823,2825,2828],{},[889,2818,2819],{},"8",[452,2821,2822],{},"π",[452,2824,2430],{},[452,2826,2827],{},"M",[554,2829,2830,2832],{},[452,2831,984],{},[452,2833,987],{},[543,2835,1442],{"separator":467},[459,2837,2838],{"encoding":461},"T_H = \\frac{\\hbar c^3}{8 \\pi G M k_B},",[433,2840,2842,2898],{"className":2841,"ariaHidden":467},[466],[433,2843,2845,2848,2889,2892,2895],{"className":2844},[471],[433,2846],{"className":2847,"style":2446},[475],[433,2849,2851,2854],{"className":2850},[480],[433,2852,501],{"className":2853,"style":517},[480,481],[433,2855,2857],{"className":2856},[678],[433,2858,2860,2881],{"className":2859},[621,622],[433,2861,2863,2878],{"className":2862},[626],[433,2864,2866],{"className":2865,"style":1047},[630],[433,2867,2869,2872],{"style":2868},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[433,2870],{"className":2871,"style":695},[638],[433,2873,2875],{"className":2874},[699,700,701,702],[433,2876,1910],{"className":2877,"style":1964},[480,481,702],[433,2879,717],{"className":2880},[716],[433,2882,2884],{"className":2883},[626],[433,2885,2887],{"className":2886,"style":724},[630],[433,2888],{},[433,2890],{"className":2891,"style":592},[591],[433,2893,545],{"className":2894},[596],[433,2896],{"className":2897,"style":592},[591],[433,2899,2901,2905,3046],{"className":2900},[471],[433,2902],{"className":2903,"style":2904},[475],"height:2.3271em;vertical-align:-0.836em;",[433,2906,2908,2911,3043],{"className":2907},[480],[433,2909],{"className":2910},[613,614],[433,2912,2914],{"className":2913},[547],[433,2915,2917,3034],{"className":2916},[621,622],[433,2918,2920,3031],{"className":2919},[626],[433,2921,2923,2983,2991],{"className":2922,"style":2528},[630],[433,2924,2925,2928],{"style":634},[433,2926],{"className":2927,"style":639},[638],[433,2929,2931,2934,2938,2943],{"className":2930},[480],[433,2932,2819],{"className":2933},[480],[433,2935,2822],{"className":2936,"style":2937},[480,481],"margin-right:0.0359em;",[433,2939,2942],{"className":2940,"style":2941},[480,481],"margin-right:0.109em;","GM",[433,2944,2946,2949],{"className":2945},[480],[433,2947,984],{"className":2948,"style":1034},[480,481],[433,2950,2952],{"className":2951},[678],[433,2953,2955,2975],{"className":2954},[621,622],[433,2956,2958,2972],{"className":2957},[626],[433,2959,2961],{"className":2960,"style":1047},[630],[433,2962,2963,2966],{"style":1050},[433,2964],{"className":2965,"style":695},[638],[433,2967,2969],{"className":2968},[699,700,701,702],[433,2970,987],{"className":2971,"style":1060},[480,481,702],[433,2973,717],{"className":2974},[716],[433,2976,2978],{"className":2977},[626],[433,2979,2981],{"className":2980,"style":724},[630],[433,2982],{},[433,2984,2985,2988],{"style":648},[433,2986],{"className":2987,"style":639},[638],[433,2989],{"className":2990,"style":656},[655],[433,2992,2993,2996],{"style":659},[433,2994],{"className":2995,"style":639},[638],[433,2997,2999,3002],{"className":2998},[480],[433,3000,2433],{"className":3001},[480],[433,3003,3005,3008],{"className":3004},[480],[433,3006,2417],{"className":3007},[480,481],[433,3009,3011],{"className":3010},[678],[433,3012,3014],{"className":3013},[621],[433,3015,3017],{"className":3016},[626],[433,3018,3020],{"className":3019,"style":907},[630],[433,3021,3022,3025],{"style":933},[433,3023],{"className":3024,"style":695},[638],[433,3026,3028],{"className":3027},[699,700,701,702],[433,3029,1765],{"className":3030},[480,702],[433,3032,717],{"className":3033},[716],[433,3035,3037],{"className":3036},[626],[433,3038,3041],{"className":3039,"style":3040},[630],"height:0.836em;",[433,3042],{},[433,3044],{"className":3045},[742,614],[433,3047,1442],{"className":3048},[1534],[393,3050,1088,3051,3079,3080,3150,3151,3246,3247,3313],{},[433,3052,3054,3067],{"className":3053},[436],[433,3055,3057],{"className":3056},[440],[442,3058,3059],{"xmlns":444},[446,3060,3061,3065],{},[449,3062,3063],{},[452,3064,2827],{},[459,3066,2827],{"encoding":461},[433,3068,3070],{"className":3069,"ariaHidden":467},[466],[433,3071,3073,3076],{"className":3072},[471],[433,3074],{"className":3075,"style":513},[475],[433,3077,2827],{"className":3078,"style":2941},[480,481]," is the black hole's mass. For a black hole the mass of the Sun, ",[433,3081,3083,3101],{"className":3082},[436],[433,3084,3086],{"className":3085},[440],[442,3087,3088],{"xmlns":444},[446,3089,3090,3098],{},[449,3091,3092],{},[554,3093,3094,3096],{},[452,3095,501],{},[452,3097,1910],{},[459,3099,3100],{"encoding":461},"T_H",[433,3102,3104],{"className":3103,"ariaHidden":467},[466],[433,3105,3107,3110],{"className":3106},[471],[433,3108],{"className":3109,"style":2446},[475],[433,3111,3113,3116],{"className":3112},[480],[433,3114,501],{"className":3115,"style":517},[480,481],[433,3117,3119],{"className":3118},[678],[433,3120,3122,3142],{"className":3121},[621,622],[433,3123,3125,3139],{"className":3124},[626],[433,3126,3128],{"className":3127,"style":1047},[630],[433,3129,3130,3133],{"style":2868},[433,3131],{"className":3132,"style":695},[638],[433,3134,3136],{"className":3135},[699,700,701,702],[433,3137,1910],{"className":3138,"style":1964},[480,481,702],[433,3140,717],{"className":3141},[716],[433,3143,3145],{"className":3144},[626],[433,3146,3148],{"className":3147,"style":724},[630],[433,3149],{}," is about ",[433,3152,3154,3181],{"className":3153},[436],[433,3155,3157],{"className":3156},[440],[442,3158,3159],{"xmlns":444},[446,3160,3161,3178],{},[449,3162,3163,3166,3168],{},[889,3164,3165],{},"6",[543,3167,1176],{},[886,3169,3170,3172],{},[889,3171,891],{},[449,3173,3174,3176],{},[543,3175,1185],{},[889,3177,2819],{},[459,3179,3180],{"encoding":461},"6 \\times 10^{-8}",[433,3182,3184,3202],{"className":3183,"ariaHidden":467},[466],[433,3185,3187,3190,3193,3196,3199],{"className":3186},[471],[433,3188],{"className":3189,"style":1259},[475],[433,3191,3165],{"className":3192},[480],[433,3194],{"className":3195,"style":1266},[591],[433,3197,1176],{"className":3198},[1270],[433,3200],{"className":3201,"style":1266},[591],[433,3203,3205,3208,3211],{"className":3204},[471],[433,3206],{"className":3207,"style":907},[475],[433,3209,911],{"className":3210},[480],[433,3212,3214,3217],{"className":3213},[480],[433,3215,918],{"className":3216},[480],[433,3218,3220],{"className":3219},[678],[433,3221,3223],{"className":3222},[621],[433,3224,3226],{"className":3225},[626],[433,3227,3229],{"className":3228,"style":907},[630],[433,3230,3231,3234],{"style":933},[433,3232],{"className":3233,"style":695},[638],[433,3235,3237],{"className":3236},[699,700,701,702],[433,3238,3240,3243],{"className":3239},[480,702],[433,3241,1185],{"className":3242},[480,702],[433,3244,2819],{"className":3245},[480,702]," kelvin, far colder than the cosmic microwave background. Such a black hole would take roughly ",[433,3248,3250,3269],{"className":3249},[436],[433,3251,3253],{"className":3252},[440],[442,3254,3255],{"xmlns":444},[446,3256,3257,3266],{},[449,3258,3259],{},[886,3260,3261,3263],{},[889,3262,891],{},[889,3264,3265],{},"67",[459,3267,3268],{"encoding":461},"10^{67}",[433,3270,3272],{"className":3271,"ariaHidden":467},[466],[433,3273,3275,3278,3281],{"className":3274},[471],[433,3276],{"className":3277,"style":907},[475],[433,3279,911],{"className":3280},[480],[433,3282,3284,3287],{"className":3283},[480],[433,3285,918],{"className":3286},[480],[433,3288,3290],{"className":3289},[678],[433,3291,3293],{"className":3292},[621],[433,3294,3296],{"className":3295},[626],[433,3297,3299],{"className":3298,"style":907},[630],[433,3300,3301,3304],{"style":933},[433,3302],{"className":3303,"style":695},[638],[433,3305,3307],{"className":3306},[699,700,701,702],[433,3308,3310],{"className":3309},[480,702],[433,3311,3265],{"className":3312},[480,702]," years to evaporate.",[844,3315,3317],{"id":3316},"the-paradox","The paradox",[393,3319,3320,3321,3325],{},"In 1976 Hawking drew the uncomfortable conclusion. ",[424,3322,3324],{"className":3323},[427],"[14]"," If a black hole radiates away its mass, it eventually disappears. The radiation, in his calculation, is exactly thermal: it depends only on the black hole's mass, charge and spin, not on what fell in. A black hole made from an encyclopaedia and one made from the same mass of rock would leave behind identical radiation.",[393,3327,3328,3329,3332,3333,3335],{},"If that is right, the information about what fell in is ",[402,3330,3331],{},"destroyed",". A pure quantum state would evolve into a mixed one, something unitary quantum mechanics forbids. Hawking called this a \"breakdown of predictability\". That is the ",[402,3334,380],{},": two of our best theories, applied together, disagree about whether information can be lost.",[844,3337,3339],{"id":3338},"the-main-proposals","The main proposals",[393,3341,3342],{},"The positions differ on where the information ends up. Here, the physics column describes the proposal and the status column describes how it is regarded; none has been tested by observation.",[3344,3345,3346,3362],"table",{},[3347,3348,3349],"thead",{},[3350,3351,3352,3356,3359],"tr",{},[3353,3354,3355],"th",{},"Proposal",[3353,3357,3358],{},"Where the information goes",[3353,3360,3361],{},"Status",[3363,3364,3365,3377,3388,3399],"tbody",{},[3350,3366,3367,3371,3374],{},[3368,3369,3370],"td",{},"Information loss",[3368,3372,3373],{},"Destroyed when the hole evaporates",[3368,3375,3376],{},"Minority view; defended as consistent by some",[3350,3378,3379,3382,3385],{},[3368,3380,3381],{},"Escape in radiation",[3368,3383,3384],{},"Encoded subtly in the Hawking radiation",[3368,3386,3387],{},"Majority view among theorists",[3350,3389,3390,3393,3396],{},[3368,3391,3392],{},"Remnants",[3368,3394,3395],{},"Kept in a tiny, stable leftover",[3368,3397,3398],{},"Widely disfavoured",[3350,3400,3401,3404,3407],{},[3368,3402,3403],{},"Baby universes",[3368,3405,3406],{},"Carried into a disconnected region",[3368,3408,3409],{},"Speculative",[393,3411,3412,3413,3416],{},"A common intuition is that the information is simply ",[402,3414,3415],{},"kept at the singularity",", the point at the centre where, in general relativity, infalling matter ends up. The difficulty is that this only postpones the problem. In classical general relativity the singularity is where the theory stops giving answers, and once the black hole has evaporated there is no interior left to keep anything in. Information stored inside must either come out, remain in a leftover object (a remnant), or leave our universe altogether. So \"kept in the singularity\" becomes one of the rows in the table rather than a separate answer.",[393,3418,3419,3420,3423,3424,3428,3429],{},"The escape view gained strong support from the ",[402,3421,3422],{},"AdS\u002FCFT correspondence",", proposed by Juan Maldacena in 1997, in which a theory with gravity is exactly equivalent to an ordinary quantum theory without gravity on its boundary. ",[424,3425,3427],{"className":3426},[427],"[15]"," Since the boundary theory is unitary, black holes in that setting cannot destroy information. In 2004 Hawking publicly changed his position, and argued in 2005 that information is preserved. ",[424,3430,3432],{"className":3431},[427],"[16]",[393,3434,3435,3436,3439,3440,3444,3445,3449,3450,3453],{},"In 1993 Don Page showed what information escape would look like: the entanglement between the radiation and the remaining black hole should rise and then fall, following what is now called the ",[402,3437,3438],{},"Page curve",". ",[424,3441,3443],{"className":3442},[427],"[17]"," In 2019, several groups showed how to derive this curve from the gravitational calculation itself, using new techniques involving so-called \"islands\". ",[424,3446,3448],{"className":3447},[427],"[18]"," Many physicists regard this as major progress, but it is established only in simplified models, and exactly ",[406,3451,3452],{},"how"," the information gets out remains unclear.",[393,3455,3456,3457,3460,3461,3465],{},"The difficulty sharpened in 2012, when Almheiri, Marolf, Polchinski and Sully argued that if information escapes, an observer falling into an old black hole might meet a wall of high-energy particles, a ",[402,3458,3459],{},"firewall",", at the horizon, contradicting general relativity's prediction that nothing special happens there. ",[424,3462,3464],{"className":3463},[427],"[19]"," There is still no agreed resolution.",[393,3467,3468,3469],{},"Not everyone accepts that information must be preserved. In 2017 William Unruh and Robert Wald argued that information loss is a natural consequence of black hole formation and is not in conflict with any established principle. ",[424,3470,3472],{"className":3471},[427],"[20]",[410,3474,3476],{"id":3475},"evidence","Evidence",[3478,3479,3480,3487,3497,3507],"ul",{},[3481,3482,3483,3486],"li",{},[402,3484,3485],{},"The second law"," is among the best-tested principles in physics, confirmed across chemistry, engineering and biology.",[3481,3488,3489,3492,3493],{},[402,3490,3491],{},"Maxwell's demon and Landauer's bound"," have been realised with microscopic particles in the laboratory. ",[424,3494,3496],{"className":3495},[427],"[8,9]",[3481,3498,3499,3502,3503],{},[402,3500,3501],{},"Black hole horizon area"," behaves as the entropy analogy predicts: an analysis of the first detected black hole merger, GW150914, found the final horizon area larger than the sum of the initial ones, consistent with Hawking's area theorem. ",[424,3504,3506],{"className":3505},[427],"[21]",[3481,3508,3509,3511,3512,3516],{},[402,3510,381],{}," has never been observed from a real black hole; it is far too faint. Laboratory analogues, such as sound waves in a Bose–Einstein condensate that behave like a horizon, have produced radiation with the predicted features. ",[424,3513,3515],{"className":3514},[427],"[22]"," These test the mathematics, not black holes themselves.",[410,3518,3520],{"id":3519},"limitations-and-open-problems","Limitations and Open Problems",[3478,3522,3523,3529,3535,3541],{},[3481,3524,3525,3528],{},[402,3526,3527],{},"The low-entropy beginning is unexplained."," The second law depends on it, but no accepted theory says why the early universe was so ordered.",[3481,3530,3531,3534],{},[402,3532,3533],{},"The meaning of entropy in cosmology is unclear."," Defining the entropy of the whole universe, including gravity, is not straightforward.",[3481,3536,3537,3540],{},[402,3538,3539],{},"The information paradox is not solved."," The Page curve calculations are strong evidence in simplified models, not a complete mechanism, and they do not describe real astrophysical black holes directly.",[3481,3542,3543,3546],{},[402,3544,3545],{},"Quantum gravity is missing."," Any final answer about the singularity requires a theory that no one yet has.",[410,3548,3550],{"id":3549},"order-on-borrowed-time","Order on Borrowed Time",[393,3552,3553,3554,3557,3558,3562,3563,3566],{},"Life seems to defy entropy. A seed becomes a sunflower whose seeds pack into precise spirals, a pattern connected to the ",[1354,3555,3556],{"href":333},"golden ratio",". Cells build intricate molecules from simple ones. In 1944 Erwin Schrödinger asked how this was possible, and answered that living things survive by exporting disorder to their surroundings: they take in low-entropy energy and give back heat. ",[424,3559,3561],{"className":3560},[427],"[23]"," The same principle constrains every theory of the ",[1354,3564,3565],{"href":36},"origin of life",". Order is not forbidden; it is paid for.",[393,3568,3569,3570,3572],{},"Entropy therefore connects things that seem unrelated: the efficiency of engines, the direction of time, the price of forgetting, the security of cryptography, and the fate of whatever falls into a black hole. Maxwell's demon, which looked like a loophole, turned out to reveal that information is physical. Laplace imagined a demon who knew everything and so could predict everything, a figure behind ",[1354,3571,177],{"href":176},". Maxwell's demon teaches the complementary lesson: even perfect knowledge would have to be stored, and eventually erased, and that has a cost.",[393,3574,3575],{},"What we know is that entropy rises and that information carries a thermodynamic price. What physics suggests, though it has not been confirmed, is that information is never truly destroyed, not even by black holes. What remains unresolved is how a black hole gives it back, and why the universe started with so little entropy that all of this could happen at all.",[390,3577,3579],{"id":3578},"references","References",[393,3581,3582,3585,3586,3589,3590,3593,3594],{},[402,3583,428,3584],{}," Clausius, R. (1865)."," “Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie.”\n",[406,3587,3588],{},"Annalen der Physik",", ",[402,3591,3592],{},"201","(7), 353–400.\n",[1354,3595,3596],{"href":3596,"rel":3597},"https:\u002F\u002Fdoi.org\u002F10.1002\u002Fandp.18652010702",[3598],"nofollow",[393,3600,3601,3604,3605,3608,3609],{},[402,3602,858,3603],{}," Uffink, J."," “Boltzmann’s Work in Statistical Physics.”\n",[406,3606,3607],{},"Stanford Encyclopedia of Philosophy",".\n",[1354,3610,3611],{"href":3611,"rel":3612},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Fstatphys-Boltzmann\u002F",[3598],[393,3614,3615,3618,3619,3608,3621],{},[402,3616,1351,3617],{}," Callender, C."," “Thermodynamic Asymmetry in Time.”\n",[406,3620,3607],{},[1354,3622,3623],{"href":3623,"rel":3624},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Ftime-thermo\u002F",[3598],[393,3626,3627,3630,3631,3608,3633],{},[402,3628,1380,3629],{}," Maroney, O."," “Information Processing and Thermodynamic Entropy.”\n",[406,3632,3607],{},[1354,3634,3635],{"href":3635,"rel":3636},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Finformation-entropy\u002F",[3598],[393,3638,3639,3642,3643,3589,3646,3649,3650],{},[402,3640,1403,3641],{}," Szilard, L. (1929)."," “Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen.”\n",[406,3644,3645],{},"Zeitschrift für Physik",[402,3647,3648],{},"53",", 840–856.\n",[1354,3651,3652],{"href":3652,"rel":3653},"https:\u002F\u002Fdoi.org\u002F10.1007\u002FBF01341281",[3598],[393,3655,3656,3659,3660,3589,3663,3666,3667],{},[402,3657,1623,3658],{}," Landauer, R. (1961)."," “Irreversibility and Heat Generation in the Computing Process.”\n",[406,3661,3662],{},"IBM Journal of Research and Development",[402,3664,3665],{},"5","(3), 183–191.\n",[1354,3668,3669],{"href":3669,"rel":3670},"https:\u002F\u002Fdoi.org\u002F10.1147\u002Frd.53.0183",[3598],[393,3672,3673,3676,3677,3589,3680,3682,3683],{},[402,3674,1854,3675],{}," Bennett, C. H. (1982)."," “The thermodynamics of computation—a review.”\n",[406,3678,3679],{},"International Journal of Theoretical Physics",[402,3681,1778],{},", 905–940.\n",[1354,3684,3685],{"href":3685,"rel":3686},"https:\u002F\u002Fdoi.org\u002F10.1007\u002FBF02084158",[3598],[393,3688,3689,3692,3693,3589,3696,3698,3699],{},[402,3690,1866,3691],{}," Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E., & Sano, M. (2010)."," “Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality.”\n",[406,3694,3695],{},"Nature Physics",[402,3697,3165],{},", 988–992.\n",[1354,3700,3701],{"href":3701,"rel":3702},"https:\u002F\u002Fdoi.org\u002F10.1038\u002Fnphys1821",[3598],[393,3704,3705,3708,3709,3589,3712,3715,3716],{},[402,3706,1871,3707],{}," Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., & Lutz, E. (2012)."," “Experimental verification of Landauer’s principle linking information and thermodynamics.”\n",[406,3710,3711],{},"Nature",[402,3713,3714],{},"483",", 187–189.\n",[1354,3717,3718],{"href":3718,"rel":3719},"https:\u002F\u002Fdoi.org\u002F10.1038\u002Fnature10872",[3598],[393,3721,3722,3725,3726,3589,3729,3732,3733],{},[402,3723,1891,3724],{}," Shannon, C. E. (1948)."," “A Mathematical Theory of Communication.”\n",[406,3727,3728],{},"Bell System Technical Journal",[402,3730,3731],{},"27","(3), 379–423.\n",[1354,3734,3735],{"href":3735,"rel":3736},"https:\u002F\u002Fdoi.org\u002F10.1002\u002Fj.1538-7305.1948.tb01338.x",[3598],[393,3738,3739,3743,3744,3608,3746],{},[402,3740,3741,3742],{},"[11]"," Adriaans, P."," “Information.”\n",[406,3745,3607],{},[1354,3747,3748],{"href":3748,"rel":3749},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Finformation\u002F",[3598],[393,3751,3752,3755,3756,3589,3759,3762,3763],{},[402,3753,2370,3754],{}," Bekenstein, J. D. (1973)."," “Black Holes and Entropy.”\n",[406,3757,3758],{},"Physical Review D",[402,3760,3761],{},"7","(8), 2333–2346.\n",[1354,3764,3765],{"href":3765,"rel":3766},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.7.2333",[3598],[393,3768,3769,3772,3773,3589,3776,3779,3780],{},[402,3770,2375,3771],{}," Hawking, S. W. (1975)."," “Particle creation by black holes.”\n",[406,3774,3775],{},"Communications in Mathematical Physics",[402,3777,3778],{},"43",", 199–220.\n",[1354,3781,3782],{"href":3782,"rel":3783},"https:\u002F\u002Fdoi.org\u002F10.1007\u002FBF02345020",[3598],[393,3785,3786,3789,3790,3589,3792,3795,3796],{},[402,3787,3324,3788],{}," Hawking, S. W. (1976)."," “Breakdown of predictability in gravitational collapse.”\n",[406,3791,3758],{},[402,3793,3794],{},"14","(10), 2460–2473.\n",[1354,3797,3798],{"href":3798,"rel":3799},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.14.2460",[3598],[393,3801,3802,3805,3806,3589,3809,3811,3812],{},[402,3803,3427,3804],{}," Maldacena, J. (1998)."," “The large N limit of superconformal field theories and supergravity.”\n",[406,3807,3808],{},"Advances in Theoretical and Mathematical Physics",[402,3810,1439],{},", 231–252.\n",[1354,3813,3814],{"href":3814,"rel":3815},"https:\u002F\u002Fdoi.org\u002F10.4310\u002FATMP.1998.v2.n2.a1",[3598],[393,3817,3818,3821,3822,3589,3824,3827,3828],{},[402,3819,3432,3820],{}," Hawking, S. W. (2005)."," “Information loss in black holes.”\n",[406,3823,3758],{},[402,3825,3826],{},"72",", 084013.\n",[1354,3829,3830],{"href":3830,"rel":3831},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevD.72.084013",[3598],[393,3833,3834,3837,3838,3589,3841,3844,3845],{},[402,3835,3443,3836],{}," Page, D. N. (1993)."," “Information in black hole radiation.”\n",[406,3839,3840],{},"Physical Review Letters",[402,3842,3843],{},"71",", 3743–3746.\n",[1354,3846,3847],{"href":3847,"rel":3848},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.71.3743",[3598],[393,3850,3851,3854,3855,3589,3858,3861,3862],{},[402,3852,3448,3853],{}," Almheiri, A., Hartman, T., Maldacena, J., Shaghoulian, E., & Tajdini, A. (2021)."," “The entropy of Hawking radiation.”\n",[406,3856,3857],{},"Reviews of Modern Physics",[402,3859,3860],{},"93",", 035002.\n",[1354,3863,3864],{"href":3864,"rel":3865},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FRevModPhys.93.035002",[3598],[393,3867,3868,3871,3872,3589,3875,3878,3879],{},[402,3869,3464,3870],{}," Almheiri, A., Marolf, D., Polchinski, J., & Sully, J. (2013)."," “Black holes: complementarity or firewalls?”\n",[406,3873,3874],{},"Journal of High Energy Physics",[402,3876,3877],{},"2013",", 62.\n",[1354,3880,3881],{"href":3881,"rel":3882},"https:\u002F\u002Fdoi.org\u002F10.1007\u002FJHEP02(2013)062",[3598],[393,3884,3885,3888,3889,3589,3892,3895,3896],{},[402,3886,3472,3887],{}," Unruh, W. G., & Wald, R. M. (2017)."," “Information loss.”\n",[406,3890,3891],{},"Reports on Progress in Physics",[402,3893,3894],{},"80",", 092002.\n",[1354,3897,3898],{"href":3898,"rel":3899},"https:\u002F\u002Fdoi.org\u002F10.1088\u002F1361-6633\u002Faa778e",[3598],[393,3901,3902,3905,3906,3589,3908,3911,3912],{},[402,3903,3506,3904],{}," Isi, M., Farr, W. M., Giesler, M., Scheel, M. A., & Teukolsky, S. A. (2021)."," “Testing the Black-Hole Area Law with GW150914.”\n",[406,3907,3840],{},[402,3909,3910],{},"127",", 011103.\n",[1354,3913,3914],{"href":3914,"rel":3915},"https:\u002F\u002Fdoi.org\u002F10.1103\u002FPhysRevLett.127.011103",[3598],[393,3917,3918,3921,3922,3589,3924,3927,3928],{},[402,3919,3515,3920],{}," Steinhauer, J. (2016)."," “Observation of quantum Hawking radiation and its entanglement in an analogue black hole.”\n",[406,3923,3695],{},[402,3925,3926],{},"12",", 959–965.\n",[1354,3929,3930],{"href":3930,"rel":3931},"https:\u002F\u002Fdoi.org\u002F10.1038\u002Fnphys3863",[3598],[393,3933,3934,3937,3938,3941,3942],{},[402,3935,3561,3936],{}," Schrödinger, E. (1944; Canto edition 1992)."," ",[406,3939,3940],{},"What Is Life?","\nCambridge: Cambridge University Press.\n",[1354,3943,3944],{"href":3944,"rel":3945},"https:\u002F\u002Fdoi.org\u002F10.1017\u002FCBO9781139644129",[3598],{"title":3947,"searchDepth":93,"depth":93,"links":3948},"",[3949,3950,3955,3961,3962,3968,3969,3970],{"id":412,"depth":93,"text":413},{"id":841,"depth":93,"text":842,"children":3951},[3952,3953,3954],{"id":846,"depth":152,"text":847},{"id":952,"depth":152,"text":953},{"id":1336,"depth":152,"text":1337},{"id":1361,"depth":93,"text":1362,"children":3956},[3957,3958,3959,3960],{"id":1365,"depth":152,"text":1366},{"id":1395,"depth":152,"text":1396},{"id":1615,"depth":152,"text":1616},{"id":1858,"depth":152,"text":1859},{"id":1883,"depth":93,"text":1884},{"id":2333,"depth":93,"text":2334,"children":3963},[3964,3965,3966,3967],{"id":2337,"depth":152,"text":2338},{"id":2362,"depth":152,"text":2363},{"id":3316,"depth":152,"text":3317},{"id":3338,"depth":152,"text":3339},{"id":3475,"depth":93,"text":3476},{"id":3519,"depth":93,"text":3520},{"id":3549,"depth":93,"text":3550},"md",[297,374,375,376,377,378,379,310,380,381,382],{},true,{"title":371,"description":372},"evidence\u002Fentropy","eQTQmF5xOIvkb4pNt1bdpKryRqJvNy_En98CkJfTkog",1789762626428]