The Arrow of Time

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THE ARROW OF TIME

Why does time never go backw�rd? The answer apparently lies not in the la,vs of nature, which hardly distinguish between past

and future, but in the conditions prevailing in the early universe

by David Layzer

t seems easy to distinguish the past have sprung from an exceedingly dense, scopic view of the world as a system de I from the future: memory provides us undifferentiated state in the finite past. generating toward complete disorder nor with a record of the past, but we All these processes have a quality in the microscopic view of the world as a have no certain knowledge of the future. common: they generate order, or infor changing but nonevolving system of in When events are interpreted according mation; they transform a simpler state teracting particles and fields is required to the most fundamental laws of physics, into a more complex one. In the phrase by fundamental phYSical laws. I submit however, the distinction between past of Sir Arthur Eddington, they indicate that both views derive instead from aux and future all but disappears. Intuitively which way "time's arrow" is pointing; iliary assumptions about the nature and we perceive the world as being extended they define what I shall call the "histori origin of the universe. I propose to re in space but "unfolding" in time; at the cal" arrow of time. place those assumptions with others that atomic scale the world is a four-dimen Paradoxically, the direction of time I believe are simpler and equally con sional continuum extended in both space can also be defined by a class of dia sistent with observation. The resulting and time. We assign special significance metrically opposite processes: those that model of the universe, although differ to a particular moment, the present, destroy information and generate disor ent from the one accepted by most physi which we view as the crest of a wave der. If I drop a lump of sugar into a cup cists, resolves the apparent contradiction continuously transforming potentiality of hot tea and stir , the spatial concentra between the historical and the thermo into actuality and leaving in its wake the tion of the sugar molecules, the orga dynamic arrows of time, and it reconciles dead past. Microscopic physics gives no nized motion of the tea and the differ both with the almost time-symmetric special status to any moment, and it dis ence in temperature between the tea character of physical laws at the micro tinguishes only weakly between the di and its surroundings represent macro scopic level. The theory implies that the rection of the past and that of the future. scopic information, or order. As the universe is unfolding in time but not un Our intuitive perception of the world sugar dissolves and the tea comes to rest raveling; on the contrary, it is becoming as unfolding in time cannot be dismissed and begins to cool, that information constantly more complex and richer in as being merely subjective. It has objec gradually disappears. The irreversible information. tive counterparts in a variety of proc processes that destroy macroscopic infor esses, including biological, geological mation (in this example molecular diffu Irreversibility and astronomical ones. There is evi sion, viscosity and heat conduction) are dence for it in the physiological proc manifestations of the second law of The historical and the thermodynamic esses underlying memory, in the growth, thermodynamics. This law states that all arrows of time both derive from proc development and differentiation of liv natural processes generate entropy, a esses that always have the same direc ing organisms, and in organiC evolution, measure of disorder. The irreversible tion; they are definedby events that can where random variation and natural se destruction of macroscopic order defines not be undone. What makes these proc lection have generated an immense and what I shall therefore call the "thermo esses irreversible? All phenomena can constantly increaSing variety of ever dynamic" arrow of time. ultimately be described as .the interac more highly organized living forms. The Neither the historical nor the thermo tions of elementary particles; if the laws earth's crust records the vicissitudes of dynamic arrow of time can be observed that govern those interactions do not 4.5 billion years of evolutionary change, at the microscopic level. The motion of distinguish between the past and the and the cratered surfaces of the moon, a single sugar or tea molecule generates future, what is the source of the irre Mars and Mercury preserve a chrono neither information nor entropy. "Order" versibility we observe in the macroscopic logical record of similar duration. Nor is a macroscopic concept, a property of world? mal stars, red giants, supernovas and systems made up of many particles; it One possibility is that the underlying white dwarfs represent stages in the evo has no meaning when it is applied to in microscopic laws are not in fact perfect lutionary life cycle of a single star. Final dividual atoms or molecules. In the phys ly time-symmetric. Evidence that a tem ly, the recession of distant galaxies sug ics of elementary particles the world poral asymmetry exists at the level of gests that the entire universe is a product changes but does not evolve. subatomic particles can be found in the of an evolutionary process: it seems to I shall argue that neither the macro- decay of the neutral K meson. One of

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THOUGHT EXPERIMENT concerning the diffusion of perfume cules spontaneously reassemble and condense in the bottle. In the reveals an apparent paradox: the process as a whole always pro· bottom drawings the same experiment is depicted in microscopic ceeds in the same direction, but it is defined by microscopic events detail. Individual molecules escape the surface and follow com· each of which is freely reversible. The bottle of perfume is opened plicated zigzag trajectories that take them to all parts of the room. in a hypothetical sealed room that cannot communicate with the This sequence of events could well proceed in reverse order, since outside world. The top series of drawings, when read from left to if every molecule reversed its direction, they would all retrace their right, shows molecules beginning to escape the surface of the liq. paths and return to the bottle. A molecule following a reversed tra· uid and gradually filling the room, until eventually all the perfume jectory would obey all the laws of physics, and indeed it would be has evaporated. When the drawings are read in the opposite direc· impossible to determine by examining the path of a single mole tion, they represent a process never 'seen in nature: all the mole· cule whether it was part of a forward experiment or a reversed one .

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© 1975 SCIENTIFIC AMERICAN, INC several possible decay modes of that par ber, 1969]. The apparent violation, how that define the historical and the thermo ticle seems to violate some symmetry of ever, is quite weak: it is observed less dynamic arrows. nature, and the usual interpretation of than 1 percent of the time. Moreover, K If the root of irreversibility is not to be the event is that the violated symmetry mesons are found only in experiments in found in the laws that govern microscop is time-reversal symmetry [see "Experi high-energy physics; they are not con ic events, it must derive from constraints ments in Time Reversal," by Oliver E. stituents of ordinary matter, and they on how those events take place. Laws Overseth; SCIENTIFIC AMERICAN, Octo- play no role in the macroscopic processes and constraints are complementary as-

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CONCEPT OF PHASE SPACE is employed to represent the dynamical bers for the specification of its state, since both position and state of a system of particles. Any particle can be described by a vector velocity have components on three axes . The corresponding (colored arrows), which defines its position and velocity. For a particle phase space must therefore have six dimensions. Since it is in a one·dimensional universe (a) two numbers sufficeto specify these not possible to construct a real space with more than three quantities, and the state of the particle can be represented in a phase dimensions, the phase space is represented here by a three· space having two dimensions. Every possible state of the particle cor· dimensional "slice" of the six·dimensional space. In a system responds to the location of some point in the phase space. A particle made up of many particles six numbers are required to speci. with freedom of movement in three dimensions (b) requires six num· fy the state of each particle, so that the corresponding phase

© 1975 SCIENTIFIC AMERICAN, INC pects of the physicist's description of na est. The laws define what is possible, the consider the motions of the planets in the ture. Laws describe the regularities un constraints what is actual or relevant. solar system. From Newton's law of grav derlying phenomena; they are few in The constraints can take the form of ini itation one could calculate all the past number and each applies over a wide do tial conditions, boundary conditions or and future positions of the planets, given main. Constraints serve to select from symmetry conditions. their positions and velocities at some ini the set of all events governed by a given As an illustration of how laws and tial moment. Newton's law explains why law the particular phenomenon of inter- constraints jointly shape phenomena, each planet moves in an elliptical orbit

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space must have a number of dimensions equal to six multi system of particles evolves (d), its dynamical state changes, and the plied by the number of particles. For example, a system of change is reflectedin the movement of its representative point in phase eight particles (c) could be represented by a point in a phase space. The path of the point in both the past and the future is entirely space having 48 dimensions_ All the information needed to determined by its initial position, and the dynamical history of the specify the state is embodied in the location of that single system of particles can therefore be predicted in complete detail. point, and every possible state corresponds to a unique point Moreover, the point in phase space can follow the same path in either in the 48-dimensional space. The axes of the three-dimensional direction, so that the motions of the particles are fully reversible. Once slice shown are chosen arbitrarily from among the 48. As a again three dimensions have been selected arbitrarily from among 48.

© 1975 SCIENTIFIC AMERICAN, INC PROBABILISTIC REPRESENTATION of a system composed of many perfume diffusion all the probability fluidis initially confined particles gives a more realistic portrayal of the system's behavior. Prob· to a small volume, since all the molecules of perfume are con· ability is represented by a fluid in phase space; the mass of fluid in any fined to the bottle. The form of the fluid is actually a hyper. region corresponds to the probability that the point representing the sphere having 6n dimensions, where n is the number of per· state of the system will be found in that region. In the experiment with fume molecules, but it is represented here by a sphere having

with the sun at one focus, why a line con must ultimately terminate in a set of con entropy assumes its maximum value for necting the sun to a planet sweeps out straints, including initial conditions, for given values of temperature and density. equal areas in equal times and why the the universe as a whole. It is in those Employing a formula first derived by squares of the planets' orbital periods are cosmological constraints that we can ex Ludwig Boltzmann and J. Willard Gibbs, proportional to the cubes of their orbital pect to find the seed of irreversibility. Shannon defined an entropy of infor diameters. It could explain those ob mation theory, which measures the un servations for any planetary system. On Information and Entropy certainty associated with statistical de the other hand, the law of gravitation scriptions of a system. The thermody does not explain why the orbits of the The processes that define the historical namic entropy of Kelvin and Clausius planets are nearly circular, why the or and the thermodynamic arrows of time and the statistical entropy of Boltzmann, bital planes nearly coincide or why all generate information and entropy re Gibbs and Shannon have identical math the planets revolve around the sun in the spectively. As Claude E. Shannon of the ematical properties: they are aspects of same direction. As Newton himself rec Massachusetts Institute of Technology a single concept. ognized, these regularities must arise showed in 1946, informatioI} is a prop Entropy and information are related from initial conditions. erty of statistical descriptions of physical by a simple conservation law, which To explain the regularities we would systems. It is measured in bits, or binary states that the sum of the information need a theory of the formation of planets. digits; one bit is the quantity of infor and the entropy is constant and equal to Such a theory could not supply the de mation needed to decide between two the system's maximum attainable infor tailed initial conditions of the solar sys equally likely possibilities. Information mation or entropy under given condi tem, but it would specify certain sta can also be regardcd as a property of tions. Expressed mathematically, the tistical properties of the primordial physical systems themselves, a measure law states: H + 1 = constant = Hm ax = systems from which planetary systems, of how highly organized they are. A Ima" where H (the Greek letter eta) and including our own, have evolved. This fundamental theorem proved by Shan 1 represent the actual values of entropy theory would itself proceed from partic non shows that the information content and information and Hmox and Imax are ular initial conditions, which would in of a system is the minimum number of the maximum possible values. Thus a turn exhibit statistical regularities invit bits needed to encode a complete statis gain of information is always compen ing theoretical explanation at a deeper tical description of the system. sated for by an equal loss of entropy. level. In that way we would be led to The concept of entropy is closely re Suppose some physical system has formulate a series of increasingly general lated to the concept of information. En eight (or 23) possible states; in binary cosmogonic problems, whose solutions tropy was first defined(by Rudolf Clau notation they could be labeled 000, 001, would yield increasingly general expla sius and Lord Kelvin) in the context of 010, 011, 100, 101, 110 and 111. The nations of the statistical regularities of thermodynamics, and it measures the specification of a particular state, for ex the astronomical universe. This hypo displacement of a system from thermo ample the one labeled 101, requires three thetical chain of cosmo gonic theories dynamic equilibrium; at equilibrium the binary digits, which is the quantity of

three dimensions (a). As the system evolves, the probability gated and more numerous (c) as the number of possible states of the fluid must migrate to more distant regions of the phase space, system increases. Eventually the entire hypervolume is filled with small but because the trajectories of all the particles are determined branches of fluid, although the total volume of fluid remains constant. the fluid is incompressible; it cannot expand as a gas does. From a macroscopic point of view the distribution of fluid now seems Instead it puts out "fingers" (b), which become more elon. uniform, although it is far from uniform when it is examined closely (d).

information associated with the descrip described with a finite quantity of thousands of other particles in the air, tion "The system is definitely in the state information. and with the sides of the container and 10l." The uncertainty or entropy asso In order to understand the connection the walls of the room, each time chang ciated with this description is evidently between cosmology, entropy and infor ing its speed and direction. At length we zero. At the other extreme, if we had no mation let us conduct a simple thought find it, still in motion, in a distant part information about the state of the sys experiment. In one corner of a room in of the room. If such a film were viewed tem, we would be compelled to assign which the air is perfectly still I open a in reverse, we would see the perfume equal probabilities to each of the eight bottle of perfume. A little later my part molecules retracing their complicated possible states. In this case the informa ner in the experiment, standing in the trajectories, converging on the perfume tion is zero. Since the sum of the en opposite corner, reports that he can smell bottle and there uniting to form a liquid. tropy and the information in the system perfume. Molecules of perfume have evi If we singled out a particular molecule, is constant, the entropy must now be dently escaped from the surface of the we would find that its trajectory obeyed three bits. In general, if a system has 2r liquid and, colliding with other mole all the laws of physiCS, since the laws possible states, where T is an integer, the cules and following complicated zigzag that govern molecular motions are sym maximum quantity information or en paths, have made their way across the metric with respect to time reversal. tropy is equal to the logarithm to the room. If we wait long enough, all the Nothing about the trajectories of indi base 2 of 2r, or T. perfume will evaporate and perfume vidual molecules would enable us to dis molecules will be found distributed uni tinguish between the actual film and the A Thought Experiment formly throughout the room [see illustra reversed one. Why, then, would we be tion on page 57]. reluctant to accept the reversed film as a For real systems the number of pos Experience and the second law of record of a real event? sible states can be very large, but it is thermodynamics tell us that the process The obvious and conventional answer not infinite. The number of states, and is irreversible: no matter how long we is that in the reversed film the initial con therefore the maximum quantity of in wait, th e perfume molecules will not ditions are exceedingly special. At the formation, is limited by the uncertainty spontaneously reassemble in the bottle. beginning of the reversed film each one principle formulated by Werner Heisen In principle, however, such an event is of an enormous number of molecules is berg. The principle states that there is an not impossible. Imagine that the entire on a trajectory that will eventually lead irreducible uncertainty in our knowledge experiment could be recorded on film it to converge on a certain small volume of the position and momentum of any in microscopic detail, so that we could of space (the perfume bottle) to the ex real particle; the state of the particle can follow the motions of each perfume mol clusion of all other similar volumes (the not be specified with greater precision ecule individually. We might see a par rest of the room). Such an initial state is than our uncertainty allows. As a conse ticular molecule in the liquid accelerated extremely improbable, and that has quence of the uncertainty principle any by a collision so that it is able to escape often been taken as sufficient explana finite physical system can be completely from the surface. It then collides with tion of the irreversibility of thermody-

INFORMATION THEORY provides a quantitative interpretation of number of cells. As the system of particles evolves, probability the distribution of probability fluid.The phase space is divided into fluid occupies progressively more cells (b), and eventually the small cells of equal hypervolume, and in the initial state all the fluid is distribution becomes uniform; each cell contains an equal assumed to be confined to one cell (a). The information required to volume of fluid(c). The state of the system is then indeter· specify that distribution is equal to the logarithm to the base 2 of the minate, and no information is required to specify it. When the

namic processes. It is possible to pursue the particle is completely determined by our thought experiment is represented the matter further, however, and ask its initial conditions. Similarly, in phase by a unique trajectory in a phase space what makes thoseinitial conditions so space the entire curve is determined by having 6n dimensions, where n is the unlikely. its starting point. Moreover, the path of number of perfume molecules. (Typical the point inphase space cannot intersect ly n is very large; ifthere is just a gram Phase Space itself or form branches (although it can of perfume, it is about 6 X 1020.) The describe a closed curve). If it did inter trajectory links the points that represent In order to discuss the question we sect itself, there would be a state of the the initial and finalstates of the experi need a convenient means of represent particle (the state at the point of inter ment, but if we were to examine those ing the changing dynamical state of a section) with more than one possible points, we could make no qualitative dis system contailling a vast number of par subsequent state, and the dynamical his tinction between them. Each point fol ticles. The concept of phase space meets tory of the particle would not be unique lows as a consequence of the other, and this need. ly determined. the description of the motions of the The dynamical state of a single parti We can use the same technique to de molecules is fully reversible. cle is completely described by its posi scribe a closed system of many interact The analYSis of our thought exper tion and its velocity. To express those ing. particles. The dynamical state of a iment in phase space seems to abolish quantities we need six numbers: three system of n particles is specified by 6n the arrow of time and to imply an abso coordinates for the position and three numbers: the three position coordinates lute determinism that leaves no room in components of the velocity. In Cartesian and the three velocity components of the future for novelty. That description, coordinates the numbers correspond to each of the n particles. We may think of however, is unrealistically precise. It as position and velocity along the x, y and these numbers as the coordinates of a sumes the existence of much more infor z axes. We may think of the six numbers point in a space having 6n dimensions; mation about the system of perfume as the six position coordinates of a point to describe the system we must specify molecules and air molecules than we in a space having six dimensions, the par the location of a single pOint in that could possibly acquire. We do not know ticle's phase space. To every point in the phase space. Again, the dynamical his the precise initial positions and velocities phase space there corresponds a unique tory of the system is represented by a of the 6 X 1020 perfume molecules, even dynamical state of the particle in real curve in phase space that is completely within the limitations of the uncertainty space, and as the particle moves in real and uniquely determined by its starting principle. We know only that all of them space its representative point traces a point. Because of collisions and other are ini tiall y confined to a certain small curve in phase space. If we know the interactions between the particles the volume, the bottle. As a result we cannot particle's position and velocity at any curve in phase space may have a com specify the precise coordinates of the one moment, we can predict all its subse plicated or irregular shape but it cannot system's representative point in phase quent motion with arbitrary precision; branch or intersect itself. space; all we can say is that it must lie in other words, the dynamical history of The diffusion of perfume molecules in somewhere within a certain small vol-

phase space is examined at finer scale, however, it is found the base 2 of the number of cells . Information in the initial condition that the distribution of fluid is far from uniform. By dividing about the macroscopic state of the system has been converted into in· each cell into many smaller cells (d) one can show that the formation about the microscopic state. It can be proved that if mi· amount of information needed to specify the state of the entire croscopic information is initially absent, all the macroscopic infor· system remains unchanged; it is still equal to the logarithm to mation in the system will be converted into microscopic information.

ume, or "hypervolume," of 6n-dimen tainty principle, and that number cannot From a macroscopic point of view it ap sional space. change as long as each state defines a pears to be distributed uniformly, but at To represent this information let us re unique dynamical history. microscopic scale its distribution is high place the point in phase space by a blob From these considerations we can ly nonuniform. of imaginary fluid that uniformly fills the conclude that the probability fluid is con small hypervolume corresponding to our tinuous and incompressible; it behaves Flow of Information actual knowledge of the initial state. The more like a liqUid than like a gas. It ex imaginary fluid represents probability, pands into the hypervolume not by The distinction between a uniform and the mass of fluid in any region of the changing its density, as a gas would, but and a nonuniform distribution of prob phase space represents the probability by sending out "fingers" that grow longer ability fluidrepresents a qualitative dif that the dynamical state of the system and narrower and more numerous as the ference in the information content of corresponds to a point within that region system evolves. Gibbs compared the the system. In order to measure the in [see illustration on pages 60 and 61]. process to the manner in which India ink formation we must divide the accessible How does the probability fluid spread slowly spreads in still water. region of phase s pace into small cells of in phase space as the perfume molecules As the probability fluid extends fingers equal hypervolume; for convenience we diffuse in physical space? One might at smaller and smaller scales, the total shall divide it into 2r cells, where r is an guess that it would simply expand in all hypervolume occupied by the fluidre integer [see illustration on these two directions, just as the perfume does, and mains constant but the shape of the oc pages]. Initially all the probability fluid ultimately fill all the available hypervol cupied region grows steadily more com is confinedto one of the cells. The infor ume more or less uniformly. Actually it plex. After enough time has passed the mation required to specify that state is behaves quite differently. fluid will appear to be distributed uni simply the number of binary digits need Because the motions of the perfume formly throughout the entire hypervol ed to designate a particular cell. The re molecules are completely determined by ume; when the fluid is examined at a quired number of bits is the logarithm to their initial state (even if we do not know very small scale, however, it will be the base 2 of the number of cells, or log2 that state), the probability fluidmust re found that the distribution is still far 2r = r. Thus the initial state of the main a single, continuous blob. If it were from uniform. In this description of our thought experiment can be represented to break up into two or more separate thought experiment we have found a by r bits of information. blobs, there would be a dynamical his striking difference between the initial In the final state, when the fluid is to,ry represented by a branching trajec state and the final state. At the outset distributed uniformly among the cells, tory, which we have seen to be impos the probability fluid is confined to a each of the 2r cells contains the same sible. Moreover, the volume of the blob small region of phase space, which it oc volume of probability fluid. At that level cannot change, because the volume is cupies uniformly; the rest of the hyper of description the final state is complete proportional to the number of distin volume is empty. In the final state the ly indeterminate and no information is guishable states allowed by the uncer- fluid occupies the entire hypervolume. needed to specify it. In the evolution of

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© 1975 SCIENTIFIC AMERICAN, INC the system all the information contained of the cell's volume. Although the den has merely been converted into micro in the initial state seems to have disap sity of the fluid has not changed, the scopic information in the finalstate. peared. shape of the occupied region is now very This conclusion can be made com If we examine the distribution of the complex. By dividing the cell into suffi pletely general and precise. No matter fluid on a finer scale, however, we can ciently small "microcells" it can be shown how we choose to partition the phase discover where the information has gone. that the information required to specify space into "macrocells," we can define If each cell contains an -equal volume of the distribution of the fluid in the entire macroscopic information as the informa fluid and the total volume of fluid has region of phase space is again log2 2r = T. tion needed to specify the set of prob not changed, then within each cell the The macroscopic information present in abilities associated with these macro probability fluid must occupy o nly 1/2r the initial state has not disappeared; it cells; the information needed to specify

a b /

RANDOM PERTURBATIONS from outside a system of particles and particles inside a container can interact gravitationally with tend to dissipate microscopic information. In a system that cannot distant matter. As a result random disturbances destroy all in· communicate with its environment (a) the paths of all particles are formation about the microscopic state of the system. Because the forever determined, and the probability fluid in the system's phase future state of the system can no longer be predicted from its pres· space is incompressible. No real system, however, is completely ent state the probability fluidis no longer incompressible (b); it isolated. Heat is communicated through the walls of any container, expands like a puff of smoke to fill the entire region of phase space.

© 1975 SCIENTIFIC AMERICAN, INC c the distribution of fluid within the mac of the initial states of closed systems: The a b rocells we define as microscopic. As the entropy of a closed system will increase closed system of molecules evolves, the only if macroscopic information is ini total quantity of information needed to tially present in the system and micro specify the distribution of probability scopic information is initially absent. fluid in the system's phase space remains constant, but macroscopic information Random Perturbations can be converted into microscopic in formation and vice versa. Those special initial conditions may What do these two kinds of informa provide an explanation of the thermo tion represent? We can identify macro dynamic arrow of time, but it is hardly scopic information with our knowledge a satisfying one. Why are those par of the statistical properties of the sys ticular initial conditions regularly satis tem, and microscopic information with fied in nature? Microscopic information our detailed knowledge of the individual seems easy enough to generate. Why molecules. In particular, microscopic in does it appear only in the final states of formation represents our knowledge of natural systems and not in their initial correlations between the velocities of states? What does it mean to say that particles. In our thought experiment microscopic information is absent from microscopic information was initially ab a certain state? One can always acquire sent, because in the initial state there such information by expending enough were no correlations between molecular energy. Finally, what significance can velocities; knowledge of the velocity of we attach to the distinction between the one molecule would not have enabled us macroscopic and the microscopic level to predict the velocities of any other of description? A plausible way of deal molecules. As the system evolved, colli ing with these questions was described sions created correlations between parti in 1912 by the French mathematician cle velocities, and all the macroscopic Emile Borel. In recent years Borel's ar information present was ultimately con gument has been rediscovered and elab verted into the microscopic information orated by John M. Blatt, by Peter G. represented by those correlations. Bergmann and Joel L. Lebowitz and For certain kinds of physical systems by Philip Morrison. and under certain initial conditions it can Our conclusion that the microscopic be shown that this process is inevitable. information of a system increases as the If microscopic information is initially ab macroscopic information decreases is sent from a closed system composed of valid only for closed systems, that is, sys many interacting particles, then the in tems that do not communicate with their formation needed to specify the system's surroundings. Borel showed that no fi macroscopic state must decrease steadily nite physical system can be considered until it has all been converted into mi closed. For example, consider the room croscopic information. Since 1946 theo in which we conducted our thought ex rems with this general form have been periment on the diffusionof perfume. established (for particular classes of Even if the room has no door or win physical systems and for particular defi dows, and even if the walls are insulated nitions of microscopic information) by Nikolai Bogolyubov, Leon C. P. van Hove, Ilya Prigogine, Radu Balescu, Mark Kac and others. "TOY UNIVERSE" consists of a straight line extending infinitelyin both directions Because macroscopic information in and divided into small domains that are variably decreases in those situations either occupied (dark squares) or empty where thermodynamic entropy increases, (open squares). If the distribution of it is tempting to define thermodynamic squares obeys certain statistical properties, entropy as negative macroscopic infor it can be shown that the toy universe con· mation. In fact, such a definition leads tains no microscopic information. The de· directly to the equation presented ear tailed properties of particular sequences of lier: H + [= Hmax = [max. We now in squares, for example, have no meaning. terpret H as thermodynamic entropy Such properties cannot distinguish one rep resentation of the universe from another and [ as macroscopic information. The (a, b), since any sequence of any finite length entropy is then always positive or zero, can be found somewhere in all infinite rep and if the maximum entropy remains resentations. Nor can a particular sequence constant, as it must in a closed system, define a unique position in a single repre then the entropy must increase as the sentation, since the same sequence is certain macroscopic information decreases. We to be found elsewhere (c). The argument have now traced the origin of the ther can be extended to the real universe, which modynamic arrow of time to a property satisfies the required statistical conditions.

© 1975 SCIENTIFIC AMERICAN, INC and made very thick, the system of mole hensively. Whether such a description age density along an infinite straight cules cannot be isolated from the rest of would contain a finite quantity or an in line. We can estimate the statistical the universe. Perfume and air molecules finite quantity of information would de properties of this one-dimensional uni must collide with the walls, which are pend on whether the volume of the uni verse with arbitrary precision; for exam also in contact with the outside world. verse is finite or infinite. (Relativistic cos ple, we can estimate the mean number More important, it is impossible in prin mology admits of both possibilities.) The of points per unit length to any desired ciple to shield the molecules from gravi universe, however, has certain distinc accuracy by taking averages over ever tational interactions with distant matter. tive properties not shared by its subsys longer line segments. Can we specify The effects of such interactions are ex tems. In pal'ticular every finite subsys any nonstatistical, or microscopic, prop ceedingly small, but they are not trivial. tem of the universe is bounded, but the erties of the toy universe? What would Borel calculated that the change in universe itself, whether it is finiteor in constitute a microscopic property? Sup gravitational potential caused by dis finite, is assumed to be unbounded. pose we are given two representations of placing one gram of matter by one centi Moreover, it seems to conform to what I the toy universe, identical in all their meter at the distance of the star Sirius shall call the strong cosmological princi statistical properties. I assert that we de would, in the course of one microsecond, ple, which states that no statistical prop fine a microscopic property if we find substantially alter the microscopic state erty of the universe defines a preferred some way to distinguish between the of a macroscopic volume of gas. position or direction in space. The (ordi representations, since the only kind of The unavoidable interaction of a nom nary) cosmological principle, so named information on which such a distinction inally closed system with the rest of the by Albert Einstein in 1916, states that could be based is nonstatistical, and universe operates as a small random per the spatial distribution of matter and hence microscopic, information. turbation that destroys correlations be motion in the universe is homogeneous In order to represent the influence of tween the velocities of particles. The and isotropic apart from local irregulari the uncertainty principle we must divide perturbation therefore dissipates micro ties; the strengthened version adds that the one-dimensional universe into cells scopic information, and it perpetually the local irregularities themselves are of equal length, the length representing re-creates the initial condition needed to statistically homogeneous and isotropic. the precision with which the position of ensure the decay of macroscopic infor The strong cosmological principle is a a single particle can be specified. If we mation and the growth of thermodynam direct descendant of Copernicus' thesis then specify the number of particles oc ic entropy. Because the system is no that our planet does not occupy a priv cupying each cell, the toy universe is longer isolated, its dynamical history is ileged position in the cosmos. It is the represented by an infinite, doubly open no longer completely determined. The simplest and most comprehensive sym ended sequence of "occupation num probability fluid in phase space is no metry postulate one can make without bers." Microscopic information is now longer incompressible; it expands like a contradicting observational evidence or defined as information that would en puff of smoke to fill the available hyper established physical laws. It has an un able us to distinguish between two such volume [see illustration on page 64]. In expected consequence directly related to sequences of occupation numbers hav the real world, then, macroscopic infor our search for the origin of the thermo ing the same statistical (or macroscopic) mation decays into microscopic informa dynamic arrow of time. I shall argue that properties. We might try to establish tion, but the microscopic information is the strong cosmological principle implies that two sequences are differentby dissipated by random perturbations. that microscopic information about the matching them up, cell for cell, along universe is objectively absent: it cannot their entire length. Since neither se The Cosmological Principle be acquired or specified. This limitation quence has a beginning or an end or any of our knowledge represents a kind of other preferred point, however, there are Borel's argument hinges on the as cosmic indeterminacy, related to but dis infinitely many ways to align them. It is sumed randomness of interactions of tinct from the indeterminacy demanded impossible in principle to complete an supposedly closed systems with the rest by Heisenberg's uncertainty principle. It infinite series of tasks , so that by this of the universe. If the positions and ve is a property of the universe as a whole method we could never prove the im locities of all perturbing particles were but not of bounded subsystems, for possibility of a match. known, we could expand the definition which microscopic information can be Alternatively, we could try to prove of a closed system to include the perturb freely specified or acquired. that the two sequences are identical. ing particles. That larger system, how The notion of cosmic indeterminacy First we would select from one line of ever, would itself be subject to exter can be illustrated by considering a "toy occupation numbers a sub-sequence of nal perturbations. Ultimately we would universe" of pointlike particles distrib arbitrary length, then we would search need to include the entire universe in uted randomly but with uniform aver- for an identical sub-sequence in the oth- our description. Given a complete mi croscopic description of the universe (within the limitation imposed by the uncertainty principle) there could be no EVOLUTION OF THE UNIVERSE represents a growth of macroscopic information. In a qualitative distinction between the two model devised by the author and his students the initial state of the universe is assumed to directions of time, for such a description be devoid of all information and structure. In the period immediately following the "big bang" the universe is in thermodynamic equilibrium, maintained by the rapid inter· would be symmetric with respect to time (a) action of particles and radiation. After expanding for about 15 minutes, the universe crystal. reversal. Is it possible, however , even in lizes, or freezes, into an alloy of metallic hydrogen and helium (b) . Because of the continuo principle, to compile such a description? ing cosmic expansio n this solid universe shatters into fragments of approximately planetary Every finitephysical system admits of mass (e), which form a "gas" in the sense that they interact frequently and randomly much a complete microscopic description con like the molecules of an ordinary gas. In the planetary gas density fluctuations eventually taining a finite quantity of information, develop ( d) as groups of several fragments adhere; the fluctuationsgrow at larger and and it may seem that the universe as a larger scales as groups of fragments themselves aggregate (e). Eventually a hierarchy of whole could also be described com pre- structures is formed, corresponding to stars, galaxies and clusters of galaxies seen today (/).

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© 1975 SCIENTIFIC AMERICAN, INC er line. In an infinite line any sub-se is in some sense an isolated system, why are generated as the universe evolves quence of finite length occurs infinitely has it not settled into equilibrium? One away from a hypothetical state of local often. The law of large numbers guaran answer, favored by many cosmologists, thermodynamic equilibrium. tees that our search must succeed after is that the cosmological trend is in fact Is it reasonable to suppose the uni a finite number of trials. Moreover, we toward equilibrium but that too little verse ever was (or ever will be) in local will succeed no matter how long a sub time has elapsed for the process to have thermodynamic equilibrium? In order to sequence we choose, provided only that reached completion. Fred Hoyle and answer this question we must compare its length is finite. The two infinite se J. V. Narlikar have written: "In the 'big the rates of equilibration processes quences are operationally indistinguish bang' cosmology the universe must start (those that generate entropy) with the able; ifthey were not, it would be with a marked degree of thermodynamic rate of cosmic expansion or contraction. possible to exhibit at least one sub-se disequilibrium and must eventually run Neither rate is constant. As we proceed quence of one that could not be dupli down." I shall argue that this view is backward in time toward the big bang cated in the other. Thus there is only fundamentally incorrect. The universe is the rate of expansion increases, and at one infinite sequence of digits with the not running down, and it need not have the origin of time-the cosmological sin statistical properties that define the toy started with a marked degree of dis gularity-the expansion rate is infinite. universe. Two representations of the uni equilibrium; the initial state may indeed The rates of equilibration processes also verse with identical statistical properties have been wholly lacking in macroscopic increase, however, as we approach the are indistinguishable. Since microscopic as well as microscopic information. cosmological singularity, because en information, by definition, is what could Suppose that at some early moment counters between particles become more make a distinction possible, we must local thermodynamic equilibrium pre frequent with increasing density and conclude that it is objectively absent. vailed in the universe. The entropy of temperature. In fact, in the period im The argument can readily be extended any region would then be as larg e as mediately following th e singularity the to infinite models of the real, three-di possible for the prevailing values of the rates of equilibration processes are much mensional universe that satisfy the strong mean temperature and density. As the higher than the rate of cosmic expansion. cosmological principle and the addi universe expanded from that hypotheti As a result the big bang is an exceeding tional requirement that the scale of local cal state the local values of the mean ly gentle process; local equilibration structure be finite. The pattern of stars density and temperature would change, processes easily keep pace with the and galaxies visible from the earth is so and so would the entropy of the region. changing macroscopic conditions of tem complex and distinctive that it would For the entropy to remain at its maxi perature and density during the first seem to define our position in the uni mum value (and thus for equilibrium to fraction of a microsecond. It is only for verse as uniquely as a thumbprint de be maintained) the distribution of en this brief initial phase in the evolution fines its owner. In an infinite, statistically ergies allotted to matter and to radiation of the universe that local thermodynamic homogeneous and isotropic universe, must change, and so must the concen equilibrium can be assumed, but from however, it is certain that the same pat trations of the various kinds of parti that assumption it follows that the ex tern of stars and galaxies recurs repeat cles. The physical processes that medi pansion of the universe has generated edly. If our universe satisfiesthe strong ate these changes proceed at finite rates; both macroscopic information and en cosmological principle, its meaningful ifthese "equilibration" rates are all much tropy. Thus the cosmological arrow , the properties are all statistical and its mi greater than the rate of cosmic expan historical arrow and the thermodynamic croscopic state is completely indeter sion, approximate local thermodynamic arrow all emerge as consequences of the minate. Since the time of Newton it has equilibrium will be maintained; ifthey strong cosmological principle and the been implicit in cosmological thinking are not, the expansion will give rise to assumption that local thermodynamic that the universe, in principle, admits of significant local departures from equi equilibrium prevailed at or near the ini a complet e microscopic description. We librium. These departures represent tial singularity. Remarkably, neither of can now see that that need not be so . If macroscopic information; the quantity of these assumptions refers directly to time there is enough symmetry in the uni macroscopic information generated by or temporal processes. verse, there is no place for microscopic the expansion is the difference between One final question remains if this for information. the actual value of the entropy and mulation is to be considered plausible: the theoretical maximum entropy at the Does the cosmic expansion generate the The Origin of Macroscopic Information mean temperature and density. particular kinds of macroscopic informa This argument does not depend on tion that characterize the universe to We have seen that the thermodynamic the cosmic expansion as such but on the day? It is possible that some of the arrow of time results from the absence of finite rate at which density and tempera information was present from the out microscopic information and the pres ture can change. The conclusion would set, perhaps in the form of density fluc ence of macroscopic information in the be the same if the universe were con tuations. The question cannot yet be initial states of closed systems. We found tracting from a state of equilibrium in answered with confidence, but it is im that microscopic information is objec stead of expanding: if the rate of con portant to note that there is no theoreti tively absent in a universe satisfying the traction were greater than the rates of cal necessity for st,ructure in the initial strong cosmological principle; on the those processes that maintain thermo state. My students and I have developed other hand, we have found no reason dynamic equilibrium, both macroscopic a model of the evolution of the universe why macroscopic information should not information and entropy would increase. that begins with a state of complete also be lacking. Indeed, the complexity The result therefore does not fix the di thermodynamic equilibrium at zero tem Of the astronomicaluniverse seems puz rection of the cosmological arrow of time perature [see illustration on preceding . zling. Isolated systems inevitably evolve with respect to the direction of the ther page]. Hence it is at least possible that toward the featureless state of thermo modynamic arrow. It does establish that the astronomical universe, with all its dynamic equilibrium. Since the universe macroscopic information and entropy richness and diversity, has evolved from

© 1975 SCIENTIFIC AMERICAN, INC a state wholly devoid of information and structure. If we postulate the existence of such a primordial state, we can even dispense with the separate assumption of the strong cosmological principle. The about ho� once again, statistical homogeneity and isotropy of an unusual source the universe follow from the fact that all has sp awned a known physical laws are invariant under spatial translation and rotation. world-shaking idea.

Novelty and Determinism

We have now traced the thermody namic arrow and the historical arrow to their common source: the initial state of the universe. In that state microscopic information is absent and macroscopic information is either absent or minimal. The expansion from that state has gen erated entropy as well as macroscopic structure. Microscopic information, on the other hand, is absent from newly formed astronomical systems, and that is why they and their subsystems exhibit the thermodynamic arrow. This view of the world evolving in time differs radically from the one that has dominated physics and astronomy since the time of Newton, a view that finds its classic expression in the words of Pierre Simon de Laplace: "An intelli gence that, at a given instant, was ac It's happened before: education, aids to quainted with all the forces by which na When two bicycle repair the handicapped, ture is animated and with the state of the men from Dayton opened government, medi bodies of which it is composed, would a whole new world at cine and just about ifit were vast e nough to submit these Kitty Hawk. And when an everything else. eccentric Balkan immi data to analysis-embrace in the same It works, as we said, grant made Edison's light by digital input-with formula the movements of the largest bulb practical by invent about the same amount bodies in the Universe and those of the ing alternating current. of computer power that it lightest atoms: nothing would be uncer And when a little known takes to run a teletype, tain for such an intelligence, and the fu company that made film this small (10 lb.) device ture like the past would be present to its for high altitude photog produces an unlimited eyes." raphy came down to vocabulary. Program In Laplace's world thereis nothing earth with a new kind of mable inflection (pitch) , that corresponds to the passage of time. copier that's now known rate and volume permit around the globe. . the production of natural For Laplace's "intelligence," as for the And now, one of the and intelligible output _God of Plato, Galileo and Einstein, the most important advances with the subtleties of past and the future coexist on equal ever in electronic com rhythm and patterning terms, like the two rays into which an munication has come typical of human speech. arbitrarily chosen point divides a straight from-of all places-a We call this amazing line. If the theories I have presented nut and bolt company! development VOTRAX, here are correct, however, not even the The nut of this break and we have combined it ultimate computer-the universe itself through is an ingenious with other electronics to ever contains enough information to voice synthesizer which create an entire family of operates from a digital products, including limited completely specify its own future states. input and emits clear vocabulary and multi The present moment always contains an human speech through lingual systems, which we element of genuine novelty and the fu solid state circuitry. will be telling you about ture is never wholly predictable. Be Coupled with a computer as time goes by. In the cause biological processes also generate or other digital devices, meantime, why not write information and because consciousness it opens an entire new us and ask for details? enables us to experience thoseprocesses technology in commu ni Thank you . . . from a directly, the intuitive perception of the cations for business, very unusual company world as unfolding in time captures one of the most deep-seated properties of the Vo cal Interface Division universe. Federal Screw Works . 500 Stephenson Highway . Troy, Michigan 48084

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