Information and the arrow of time

Informatio n and the arro w of time Marcin Ostro wski Septem ber 25, 2018 Abstract This pap er is a discussio n ab out the relationship b etw een time and info rma- tion. W e argue that the direction of arro w of time is related to th e d irectivit y of information cop ying that occurs in Nature. 1 In tro duction Wh y does time run forward? This questio n, on the b order line o f Physics and Philos o - ph y , ha s b othered mankind fo r cen turies. According to some p eople, this is not a v alid question, since the impression of “ flow of time” can o nly b e a sub jective sensatio n of our conscio usness. The aim o f this work is not to determine the v a lidity of this thesis. How ev er, susp ecting that there may be something to it, we will seek an answer to another question: Why do we r ememb er the p ast and c an not r ememb er the futu r e? There m ust be some ph ysical r eason for this asymmetry . Even if th e mere passage of time is sub jective, suc h an asymmetry should not hav e a purely sub jective natur e, but m ust b e based on the laws o f Physics (Nature). Otherwise, living o r ganisms lik e ourselves, certainly would like to use the kno wledge about the f uture, since it is o f strategic importance for surviv al. A thirsty animal would simply go to w ater using a certain “memory of the future”, rather than head for the w atering hole, r emembering where it was y esterday , and having hop e that today it is in the same place (which may not always be true, for example, during intense dr ought). Another asymmetr y betw een the past and the future, can be expressed b y the question: Why c an we influ enc e the c ou r s e of fu t ur e events and we c an not influenc e the c ourse of events fr om the p ast ? Generally , it is b elieved that the rea son for this asymmetry is the second law of ther - mo dynamics, but the problem do es not s eem to be settled ent irely . Let us quote here a chapter fro m a bo ok b y Rudolf P eierls[1]: “The “arr ow of time” app e ars to b e in our minds. As long as we have no cle ar expla nation for thi s limitation, we might sp e culate whether the t ime dir e ction is ne c essarily u niversal, or wheter we c ould imagine int el ligent b eings whose time runs opp osite to ours so that, fr om o ur p oint of vie w, they c ould r ememb er the futur e and make plans for t he p ast.” Our main aim will b e to lo ok closer at this issue. 1 1 The problem of time is widely discussed i n the literature on the basis of ph ysics and phil osoph y (for example [2]-[5]). Ideas somewhat similar to our s can be found in ([6]). 1 2 The symmetry of time rev ersal and the mec hanics The problem of finding asymmetry b etw een the past and the future seems to b e partic- ularly dramatic if w e loo k closely at the la ws of mec hanics. B oth the Newton equations and the equations of mo der n physics tr eat e qually b oth directions of time. 2 What follows fro m the symmetry of the Newton’s equa tions? Cons ide r , for example, the g ame of billiards. Suppo se that in an airtight co ntainer we hav e N particles per fectly ela stically bouncing off the w alls a nd from eac h other (as in the ideal gas mo del). Ima gine that at a certain fixed time t 0 we k now the state of the sy s tem (i.e. the exact p osition and velo city of all particles in the cas e of a classic system). W e w ant to answer the following question: what will b e t he state of the sy s tem (gas), sa y in five min utes, i.e. at the time t 0 +5min? It turns o ut that the answer to this ques tion is just as easy (or difficult) as the answ er to the question: what was the state o f the gas five minutes ea rlier, i.e. at the time t 0 -5min. P redicting the future is thus as difficult as predicting the past. So w e hav e a full symmetry! One can give more exa mples of this k ind. T ak e a relatively simple mechanical system like the So lar System. When in 2011 we a sk: what will be the po sition of planets in the sky o n June 1 , 30 11, thus the answer to this ques tion is a s simple a s the answer (using an a ppropriate co mputer programme) to how it looked o n June 1, 1011 . 3 Sym b ols of the past The fundamental differ ence b etw e e n the past a nd the future is the o ccurrence of the memory of pa st ev ents. These are the memories s tored in our minds, but also the conten ts of memor y of non-living facilities such as computers . Computer memo ry , like ours, k eeps the information reco rded in the past. So the inability to remem b er t he future is not only an ailmen t o f our bra ins but also brains of animals and memory of electronic equipment. What is memory ? It is no t only the memor y of h umans, animals or computer s . Also in the inanimate Nature there are pro cesses that can b e equated with the memory . W e can sp eak a bo ut memory when it is p ossible to rea d fr om a pro c e s s (a phenomenon or an ob ject) the informa tion ab out a no ther pro cess (a phenomenon or ob ject). Examples of this are: the trace of ra in imprinted on the ro ck millio ns of years befo r e, fossil b one s of animals such as dinosaurs , the sta rlight emitted thousands of years e a rlier, finally , historical memorabilia: photogra phs , drawings, chronicles, etc. Such ob jects will b e called “ symbols of the pas t” 3 . It ha ppe ns that in Nature we a re co nfronted with symbols that “spea k” of pa st, but we never meet s imila r symbols that “sp eak” of the future. On the ro ck we never see traces o f the rain which will b e ra ining in s ome time, we can not see pictures that will be taken in some time, and w e do not observe light from stars tha t after so me time will b e emitted. Of cour se, the past is no t always “well do cument ed”. Most pa s t e vents do not leave be hind “sym bo ls of the past” that are av ailable for us. Most of the ra infall after several ho urs do es not leav e a trace, a s well as most of the even ts ar e not photogra phed. This is es pecia lly a problem for histor ians studying ancient times, a rchaeologists and palaeontologists. 2 There is an exce ption, though, i n the world of elemen tary particles, namely the re is some asymme- try betw een past and future in we ak interactions. W e are talking about the vi olation of time rev ersal symmetry (T symmetry) during k aons deca y[7] -[8]. How ev er, it seems that in Earth’s conditions this phenomenon is to o marginal to b e able to lead to such a profound asymmetry betw een the past and the future. 3 This term is coined by the author solely for the purp ose of this work 2 4 Are there an y “sym b ols of the future”? The future do e s not offer us memor ies. Doe s this mea n that, unlike the pa st, w e do not hav e any knowledge ab out future even ts? Do we hav e s omething we might ca ll “symbols of the future”? The answer to this question is yes. “Sym b ols of the future” could rela te to o ur pro jections or plans fo r the future. F or example, when we or ganise a trip or a holida y , we can sc hedule when we will get out of home, ho w we will get to the station, which train we will catch, what we will visit ev ery day . Those plans, made by us a priori, are “symbols of the future” . With a bit of go o d orga nisation a nd luck, we can put them int o practice. It is tr ue tha t we do not alwa ys s ucceed, b ecause we can not predict everything entirely . Howev er, if our live is o rdered, then it o ften bec omes exceeding ly predictable (sometimes to our bitterness). There is , how ever, a crucial difference. In the case of “symbols of the pas t” w e just remember wha t has a lready happ ened, whereas in the case of “sy m b ols o f the future” we only provide o r plan, wha t will happ en. But we forget for a mo men t ab o ut the dif- ference b etw e e n memories (symbo ls of the past) and forecas ts (symbols of the future), treating them as if the difference in their perception w as only a sub jective impression of our brain. How ev er, “ symbols of the future” a re not cer tain. The plans that w e make (or our predictions) often do not work. But the pro blem is that “symbols o f the pa st” are not certain either. Their n umber may b e insufficien t or sometimes they can b e false. F o r example, if the offender inten tionally dis s ects false evidence to confuse the trail of the inv estigation. Are w e bac k to square one then? Uncertain “sym b ols of the past” do not guar antee that we remember the pa s t well and uncertain “sym b ols o f the future” make it imp oss ible to predict the future exactly . 5 Redundancy of the “sym b ols of the past” How ev er, there a re ob jective differences betw een “ s ymbols of the past” and “s y m b ols of the future”. W e could say that the uncertaint y of b oth have a completely “different calibre”. In the case of “ symbols of the future”, it is often enough to change one of them to ma ke the ent ire prediction fail. As an ex ample, re-co nsider the planning of a holiday travel. W e choo se a place and time, reserve the acco mmo da tion, chec k rail co nnection, buy a tick et in adv ance, infor m friends and take a leav e. All of these things (i.e. reserv ations stored in computer memory or in a prin ted form, record in our minds and the minds of our friends) a re examples of “s y m b ols o f the future”. But when one of them collapses , for example if we do not get a leav e within the pla nned per io d b ecause in the last minute some imp ortant issues come o ut, the who le trip will not take place. The fact that other symbols still supp o rt our inten tion is not relev ant here. In the case of the “s y m b ols of the past”, the situation is different. They are r edundant to each other and era s ing (or misrepre s ent ing), one o f them do e s not change the au- then ticity of the even t. F or exa mple, ima gine a ma n who was a par ticipant of a trip to Paris la s t y ear. How can w e be confident (say , after one year) that this man actually was on this trip? This is evidenced (among others) by: his memor y , the photos that he made using his camera and photos taken by others, showing him aga ins t a back- ground of the mo num ents of Paris. All these symbols affirm ea ch other and there is redundancy in them. Imagine that during this trip a cr ime was committed (say , theft) and it b ecomes of in terest to the p olice. Imagine tha t the pe r son in ques tion b eca me a susp ect in this ca se. Imagine a lso that after testimonies of all the participa n ts o f the trip and after analysis of pictures fro m their cameras, the following situatio n emer ged: 3 half of the pa rticipants said that this ma n w as actually on the trip, while t he other half strongly denies it. The man in question appears in half o f the pictures, but he do es not appear in the other half. Moreov er, the police repor ted another ma n who exactly a t the sa me time, was on a nother tr ip, say , in Rome, c la iming that the s uspe ct was there with him. Again, half of the pa r ticipants (and their pho tos) confirms that the susp ected participation in the trip, while the other half denies it. What would b e the s urprise of the detectives if the facts tur ned out that wa y . F ortunately , physical reality do es not play s uch tric ks on us. Even if there were some inconsistencies in the testimonies o f mem b ers o f b oth tours, we w ould b e more inclined to believe in a delibe r ate lie of one group aimed at providing the susp ect with an alibi than in any fluctuations in reality . Thu s, it is muc h easier to change the future by mo difying “symbols o f the future” than change the past by in tro ducing mo difica tio ns to “symbo ls of the past” . Just a tra in accident or a strike can hold the planned trip. In the case o f a past travel, it is no t enough to falsify or deface o ne picture. This should be done with a ll the photos o f all participants, with a recor d of memo ries of their brains, and man y symbols contained in the e nvironment. It is highly unlikely or imp os sible. Let us s umma r ise: “Symb ols of the p ast” in a r e du n dant manner mutu al ly c onfirm one version of p ast events. In the c ase of the “symb ols of the futur e” a change of even one of them often alters the c ourse of futur e event s. 6 Sym b ols of the past as copies of the information What are “symbols of the pa st”? T o answer this ques tion, we r efer here to a discuss ion of our ea r lier work [9]. W e identify there the information with the abstract quantit y that undergo es c o pying. W hat are “s ymbols of the past” if not just copies o f the information ab out the even ts of the past? “Symb ols of the p ast” ar e c opies of information ab out events of the p ast. There may b e ma ny symbo ls of one even t, for exa mple when we tak e a ph otog r aph of a n ob ject (e.g. the Eiffel T ow er). O n a sunny day , billio ns of photons are reflected from the E iffel T ow er every second. Thus, p otentially , by register ing the pho tons, it is po ssible to take millions of photos, i.e. copies of information ab out the ob ject. Even if we do no t do this, ther e will b e information co nt ained in the photons escaping into outer space a ll the time. So we hav e a huge redundancy . W e mean by it (fir stly) that symbols of the past (copies o f information) can b e numerous, and (sec ondly) that they are mutually non- contradictory . 4 This r edundancy occur r ed in “symbols of the past” leads to significant differences in the way we pe r ceive the future and the past. So wh y do w e remember the past? Because it is sufficient to k eep in mind one of the ma ny copies of information a bo ut the in cident to know that it happ ened. Then we do not need to know the state of the entire s ystem (as in billiards), to be able to judge ab out its past. Bet ween “sy m b ols of the past” we have a str ong correlation (redundancy). It is sufficient to know only a few of them (e.g. holiday photos), and the rest will overlap with them, no t co n tributing muc h to the case. Therefo re, despite the fact that comprehensive informa tion ab out even ts that happ ened a year ago is lo ca ted in an a rea with a radius o f 1 lig h t year, thanks to the r e dundancy such co mpr ehensive information is not often needed. Sometimes, it is enough to know a small percentage of all “ symbols o f the pas t” in order to k now what happened. 4 W e observe i t esp ecially when we read different bo oks on the same s ub ject. Eac h of them br ings less and less to the case, un til, finally , in the next one, most of the facts are duplicate s of what w e ha ve read elsewhere (maybe with the exception of a f ew paragraphs) 4 F or the future, it is no t the case. W e must know the state of the entire system to b e able to predict anything ab o ut its future. T o know what w ill happ en exactly in a year, you need to know the state o f all particles from the area with a ra dius o f 1 light year. Each individual c hange may yet sp oil everything (change the course o f histor y). The differ enc e b etwe en the p ast and the futu r e se ems to st em fr om the dir e ction of c opying information. F or some r e ason the information ab out p ast events is c opie d into the futur e, n ot vic e versa. Symbols o f the past, a s we said earlier, are copies of infor mation a bo ut past e vents. But most im p ortantly , they are easily readable c o pies. What do we mean b y that? Referring to the unitarit y of time ev olution in Quantum Mechanics, w e can conclude that if w e burn a b o o k , then the smoke and as h formed after combustion will include the en tire information that w as con tained in it. The problem is that there does not exist (and probably will no t ex ist) a technical metho d of recons tructing this information. Thu s, burning do cuments is still the be s t wa y to get rid of inconv enient evidence. The situation is different in the ca se of copies that are easy to r ead. Light r eflected from the printed s he e t carries the information about its conten ts and is an example of a copy eas y to rea d. Why? Because w e only need the lens a nd the CCD to repro duce from a part of the reflected light the form of printed text. This is what we call the “copies ea sy to read”. F or some reaso n the past offers them for us, and the future do es not. Unlike “symbols of the pa st”, “sym b ols of the future” are not copies o f the information about future ev en ts, but they are just our assumptions, inten tions and sometimes only dreams. When we remember the past we are not using unitar it y of time evolution. If we did, then we could rea d a b o ok, watch ing the smo ke emitted from burning it. What hu man bra in do es is reg ister easy to r ead “symbols o f the past” a nd memorise them by b ecoming their databa se. Most impo rtantly , the wa y the symbols are stored (i.e., memory org anisation) is that they ar e still easy to r ead. Memory of the past do es not lie in ca pturing the current state of the par ticle s and simulating w her e they must hav e bee n in the past. The br ain is not able to do that and nev er will b e (it would hav e to hav e something of a demo n as the Maxwell’s demon). If that were the cas e, then remembering the past and predicting the future would b e equally e a sy for it. Then, for the brain, the past would not be different from the future. Let us now turn to the second question: “Why can we change the future and we c an not change the past?” W e can not change even ts from the pa st b ecaus e w e will never erase all copies of the informa tion ab out it. Esp ecially when those copies a re num erous . F or the future, the situatio n is differen t. W e do not need to r emov e multiple copies of information a bo ut future even ts b e cause they do not yet ex ist. The result is that it is m uch ea sier to modify the future than the past. 7 Irrev ersibilit y Copying of information is a reversible pro cess . 5 Therefore, we should be a ble to era se the past, if w e can reverse the copying pro ce s ses and dele te all copies in which infor- 5 W e can ill ustrate it by the si m plest example. Consider tw o lines of nu mbers. In the first line we ha ve: “010111 00011” while in the second line there are only zeros. The first line is treated as data that we wan t to copy , the second is a medium prepared for copying. Zeros mean that the medium is clean as a white sheet of pap er r eady to wr ite on. Cop ying will result in a situation where i n b oth lines there wil l b e the same s tring: “01011100011”. Is it p ossible to reve rse this cop ying? That is , is it p ossible to reconstruct the situation pri or to copying? Of course it is p ossible. It is enough to wr ite only zero es in the b ottom li ne. The case would look differently if we did not hav e a clean medium i n the form of lines of zeros. Then its preparation would require erasi ng the original recording (noise), and therefore it would b e irreversible. 5 mation ab out an event is recorded. 6 These types of exp er iments ca n b e perfo rmed in case of simple systems (i.e. consisting of a small n umber of particles). They ar e the so called quantum erasure exp er imen ts [10]-[11]. They ar e based on the reversibility of unitary evolution of an isolated quantum mechanical sys tem. The problem emerges when so me witnesses of the ev ent (particles c arrying the information about it) “es- cap e from the crime scene”. Then the era sure of the even t is not po ssible. This is a particular pr o blem in the ca se of photons. They hav e a n “ unplea sant” habit o f esc a p- ing with the highest sp eed in na ture. If such a photon escap es in to the empt y spa ce (v acuum) no thing is a ble to ov ertake it and fo r ce it to tur n back (unless during e arlier preparatio n for the exp eriment, a sufficien t n um b er of mirrors hav e b een set). When an even t happ ens, for example, o n the sur face of the E arth o n a clea r day , the photons reflected every s e cond from our b o dy have virtually no ba r riers to fly aw ay in to spa ce. Each of them carries a small piece of information ab out us (ab out our po sition, a p- pea rance, etc.). It is true tha t ea ch of them is running in a slightly different direction, and carries in itself to o little use ful information. How ever, would it not b e possible in this situation to fo cus the photons and collect back the information carr ied b y them? T o answer this question let us loo k at the equation of r esolution o f ligh t micr oscop e: d = λ sin α . (1) The most important mor al o f Eq. (1) is that the resolution d of such a microscop e is of the o rder of a wav elength λ which is emitted by the observed ob ject (sp ecimen). F or this purp ose , the ap er ture ang le α m ust b e of the or de r of 90 0 i.e. the photons reflected from an ob ject must b e collected by a lens with the widest po s sible rang e o f angles. But that is not what we mean here. The mo s t imp or tant thing we should note is that in the formula (1) there do es not exist the distance betw ee n the micro scop e lens and the observed ob ject. This distance does not a ffect the r esolution in an y wa y . This means that rega rdless o f whether it is of the order of millimetres or light y ears, resolution that we get is the same (we omit the p ossible o pacity of the medium which at large distances can intro duce significant distortions ). O bviously , for a large distance betw een an ob ject and a lens, in order to obtain a sufficient angle of ap er ture, the lens needs to b e sufficient ly large. Ho wev er, this is basically a tec hnical pro ble m. 7 W e certainly do not cla im that this t ype of exper imen t is feasible. This should be regar ded mor e a s a thought exp eriment, showing that information about even ts from distant past, which did not leav e a trace on E arth, can exist in space and, theoretica lly , can b e read. Thus, those even ts can not b e e r ased (re versed). Surely , this can not b e done lo cally , b eing just on Earth. Therefore, such an event app ears ir reversible. 8 8 A question ab out the essence of time What hav e we b een able to do? W e hav e r eplaced tw o questions of a r ather s ub jectiv e form (“Wh y do we remember the past and do not remem b er the future?” and “Wh y can we only change the future?”) with one: Why is the information c opie d only in one dir e ction in time? 6 It’s a bit like in the mo vies crim e: perfect crime w hen is p ossible to di spose of all wi tnesses of crime. 7 F or example, if we take the r esolution d = 1 mm observing at wa v elengths λ = 500nm, then we ha ve f rom Eq. (1) sin α = 5 ∗ 10 − 4 , which gi ves for a distance of observ ation equal to 100 l igh t-year lens r adius equal to 0.05 li gh t-yea rs. A suitable gravitational lens e.g. in the f orm of prop erly shap ed transparen t dust gi ving an appropriately shaped gravit y field should work. 8 Obv iously w e can not assume in adv ance that w e are not part of some gr and time erasure ex- peri men t, run on a cosmic scale, in which a large num b er of prop erly arranged mirr or s c hanges the direction of mov emen t of all the photons, and the ev en t is erased. Ho wev er, this seems quite unlikely . 6 The adv a ntage of a question p os e d in such a way (in comparison with the previous t wo) is that it has a ph ysical natur e a nd is not related to fields such as philos ophy or psychology . A t this point we could actua lly finish the discuss ion, bec a use the answ er to this question is already a case for a completely different s tory . But we will allow ourselves to sp eculate and pr esent a some hypothesis . Making multiple c o pies o f information is p oss ible, if we hav e sufficient num ber of free media to r ecord it. In our world, these media ar e particles (atoms, electr ons, photons) and their conglomer ates (condensed phase). How ev er, the total num ber of particles in the Univ erse inc r eases over time. In particula r, this a pplies to pho tons. 9 Photons ar e very imp or tant, as they arrive from the Sun to the Earth, drive the ma jority of pro c esses on the pla net and do not allow multiple subsystems to a chiev e the thermodyna mic equilibrium (this applies particularly to biological subsystems). Where do these photo ns arise? The y are produced by ther monuclear fusion reactions o ccurring in the cor e of the Sun. The merger of four protons in helium nucleus forms a few g amma-ray photons with the energ y of the order o f MeV. Howev er, be fore these photons rea ch the photo s phere, they are r ep eatedly , inela stically scattered, dividing int o a the num ber of secondary photons with low er energy . The surface of the Sun (photosphere) emits photons of a verage energ y of the order 2 eV (equiv a lent to y ellow light). The ov erall ba lance of g ains and losses for particles asso ciated with the act of pro ducing one helium nucleus is as follows 10 : 4 1 1 H + 2 e − → 4 2 H e + 2 ν e + 10 7 γ (2) In a world where there is a 10 7 additional photons (rig h t side of E q . 2 ) ther e are muc h more media to store many of the new symbols of the past, while it is difficult to imagine this in a world in which ther e are six particles (left side of E q. 2). It is natura l that in states with a greater num ber of par ticle s there will a pp ea r the infor mation (symbols of the past) ab out states wher e there were fewer such par ticles, but not the o ther wa y round. Then we identify in a natural wa y states with a sma ll num b er of particles as a remembered past, a nd sta tes w ith more pa r ticles as the unknown future. So we can say that the future embraces the past, while the past do es not embrace the fut ure. Thu s, the moment which corre s po nds to the situation on the left side of the eq uation corres p o nds to the past, while the r ig ht to the future. One might ask, “Why did the Univ erse b egin with the state of the (relatively) sma ll nu mber o f pa r ticles having a high density and energy?” But this question is ill- po sed. W e o nly need to a sk why s uch a state was among the states whic h were assumed b y the Universe. Because if such a sta te is one of the p ossible s tates of the Universe, it naturally will be the b eginning of time. If we identify (naiv ely) en tropy S with the n umber of par ticles in the Universe, this might justify the conclusion that it is not t he entrop y that incr eases as a re sult of the passage o f time, but conv ersely , the sub jective feeling of time passage follo ws the direction of en tropy incr ease. 9 Chaos and the direction of the arro w of time Sometimes it is suggested that the direction of the arr ow of time can b e asso ciated with the existence of chaos [1 2]. It is also claimed that the transitio n from the des crip- 9 In the case of other particles such as electrons or protons the si tuation is different . Their existence is sub ject to principles of conserv ation (of c harge, of lepton num ber, of baryon num ber). 10 What ma y arouse some concern i s the presence of t wo electrons on the substrate. Of course they are not directly inv olve d i n the cycle p-p, but they annihilate with t wo posi trons pro duced in the first reaction of the cycle. 7 tion bas e d on tra jectories to the statistica l one (based on a pro bability dis tribution) guarantees the distinction of o n directions of time. F or exa mple, acco r ding to [12], ra n- dom walking desc r ib es an unidirectional increase of blurring of the particle’s p osition tow ards the future. In our opinion, it is exactly the o ppo site! Chaos not only do es not deter mine the arrow o f time, but in fact it blurs the distinction betw een the past and the future. If all systems in na ture were c haotic, the difference b etw een the past and future would be erased as a result o f the destruction of all ”symbols of the past” . 11 W e could take the point of vie w presented in [12] if we a ccepted the existence of ob jectiv e probability of r andom ev ents, asso ciated with low ent ropy a t the b eginning of the Universe and then e volving in the direction of the entrop y’s growth. How ever, we assume the ” sub jectiv e” interpretation [9]. W e believe that the probability distribution is our knowledge ab o ut the curr ent state o f a rando m pro cess . The increase of e n tropy with time means that our prediction abo ut the future state becomes less accur ate. How ev er, we can make a predictio n ab out the pa s t in the same way . W e can define the ent ropy decreas ing in time, which mea ns that our forecast of the state of the pro cess for mor e dista n t past also b eco mes less a ccurate. The autho r of [12] admits that in the ma thematical equations descr ibing a random pro cesses there ar e co mpo nents corr esp onding bo th to the inc r ease and to the decreas e of entropy over time. But the la tter a re cut arbitrar ily as having no r e fer ence to reality . How ev er, such an approach is an arbitra r y impo sition o f one direction of time, a nd can not b e considered a s a proo f of irreversibility of pr o cesses in nature. Wh y ar e weather forecasts not made for the pa s t? The reaso n is not b ecause the probability is asymmetric in time, but b ecause such foreca sts are unnecessa r y (there are ”symbols of the past”). If s uch forecasts were needed, they would b e equally difficult to cre a te and their results would b e sometimes equally surpr ising as those for the future. 12 Contrary to the tendencies a dvocated by so me authors , neither the co ncept of proba- bilit y nor the logic distinguish any direction of time. The dire ction of the a rrow of time on Ea rth is determined mainly b y the direction of the Sun’s r ays. It is an ir reversible, but not a chaotic pro cess. References [1] R. Peierls, “ Surprises in Theoretical Physics”, Princeton Universit y Press (1979), section 3.8 ”Irreversibility”; 11 As a practical example, let us consider suc h ”symbols of the past” as the fossi ls from the earli er geological epo chs. It turns out that fossi l s older than 200million ye ars are found only in the sediments underlying the con tinen tal crust, but never in the oceanic crust [ 13]. Thi s i s caused by the participation of the o ceanic crust i n the con vec tion of the Earth’s mantle. The oceanic crust i s formed constan tly in the spreading zones and destroy ed (melted) i n the subduction zones. Thus, the age of the oldest parts of the o ceanic crust is up to 200 mill i on y ears. In the case of continen tal crust, the situation is differen t! It do es not participate directly in the con ve ction. Its fr agments are only moving only across the surface, but are not dragged in to the Earth and melted. Thanks to this, in the sediment s accum ulated on its surface, it stores evidence of biological l ife which is muc h older than 200 mi llion years. W e can sa y that the contine ntal crust is like a hard drive which stores data about the geological history of the planet. W ell, but what do es i t all hav e to do wi th c haos? The answer is that conv ection is an example of a c haotic process [13]. If the entire Earth’s crust was inv olve d in this pro cess, our understanding of li fe, for example, in the Cam brian, w ould b e as scarce as our knowledge ab out the l i fe on Ear th in the next 500 mill ion y ears. 12 Do you r emem b er the movie ”50 First Dates”? Just imagine what would happ en if all human s (and other data carr iers) had the same ai l men t as the main female characte r, and the rain puddles from the day bef or e dri ed up v ery quickly (high sp eed of c haotic phenomena ). 8 [2] C. Kiefer, Do es Time Exist i n Quantum Gr avity? , ht tp://www.fqxi.o rg/communit y/forum/to pic/265 ; [3] C. Rov elli, F or get Time , htt p://www.fqxi.o rg/co mm unity/forum/topic/227 ; [4] J. Barbo ur, The Natu r e of Time , http://www.fqxi.org/communit y/forum/to pic/360 ; [5] C. Ca llendar, What Makes Time Sp e cial , http://www.fqxi.org/communit y/forum/to pic/302 ; [6] D. Z. Alb ert, “Time a nd chance”, Harv ard Univ ersity Press , (20 00); [7] P . Davies, “Ab out Time: E instein’s Unfinished Rev olution”, Or ion Pro ductions (1995); [8] M. Gardner, Can Time Go Backwar d? , Science Amer ic an 216 , No. 1, page 6 (1967); [9] M. Ostrowski, Information and Interpr etation of Quantum Me chanics , ht tp://xxx .la nl.gov/abs/101 1.5039 ; [10] P . G. Kwait, A. M. Steinberg , and R. Y. Chiao , Observation of a ,,Quantum Er aser”: a R evival of Coher enc e in a Two-Photon In terfer enc e Exp eriment , Phys. Rev. A, 45 , 7729, (19 92); [11] X. Y. Zhou, L. Mandel, Induc e d Coher enc e and Indistinguishabil ity in Optic al Interfer en c e , P hys. Rev. Lett., 67 , 318, (1991); [12] I. Pr igogine, “The end of Certainnt y . Time, Cha o s and the New L aws of Na tur e”, The F ree P ress, New Y ork (19 9 7); [13] L. Czecho wski, “T ektonik a plyt i kon wek cja w plaszczu Ziemi” , Wydawnict wo Nauko we P WN, W a rszaw a (19 94); 9