cosmology and the arrow of time

Cosmology and the Arrow of Time

D. LAYZER

Harvard College Observatory, Cambridge, Massachusetts, USA

THE arrow of time manifests itself in three distinct ways: through the approach to equilibrium in closed systems containing a large number of interacting particles; through the phenomenon of memory in certain open systems; and through the expansion of the observ- able Universe. The approach to equilibrium in closed systems can occur through a wide variety of physical processes, including not only molecular transport processes like diffusion and conduction but also macroscopic processes like turbulence which convert ordered fields and motions into less highly ordered ones and ultimately into heat. All such processes generate entropy. By contrast, the accumulation and preservation of information about previous states of a system depend upon processes tha t generate information (negative entropy). As for the entropy changes associated with cosmic evolution, it can be argued tha t these too must be negative-- in spite of the fact tha t the Universe is, by definition, a closed system, l~or, as will be explained below, there are reasons for believing tha t the structure of the Universe was simpler in the past than it is now. Thus it is clear tha t the second law of thermodynamics does not afford a sufficiently broad framework for an understanding of the arrow of time in all its aspects.

I t is equally clear, however, tha t these aspects must be related, for irreversible processes in closed systems, memory in certain open systems, and the evolution of cosmic structure all define the same arrow. And there are indications tha t the three kinds of process are physically connected. For example, it is obvious tha t we cannot hope to understand why different systems show the same arrow so long as we confine our attention to isolated systems. Interactions with the environment play an essential par t in determining the initial conditions for closed systems, and some authors have suggested tha t they continue to play an essential par t in the subsequent development of what are regarded nominally as closed systems.

The present communication outlines a unified theory of the arrow of time. Although it requires a ra ther drastic departure from currently accepted cosmological ideas, this theory is consistent with theories of the kind tha t have been developed by van Hove (:) and Pri- gogine (2) to describe the approach to equilibrium in closed systems. In fact, it clarifies certain assumptions in these theories tha t have previously been regarded as ambiguous or obscure.

To fix ideas, let us recall how the transition from a reversible microscopic description to an irreversible macroscopic description is accomplished in non-equilibrium statistical mechanics. In the first place, it is necessary to suppose tha t the microscopic state of the system under consideration is specified by a probability distribution {10k} or, more generally, by a density matrix ~. The information associated with this description is defined by

I = S m . - S, (1)

10~* 2'/9

280 Cosmology and the Arrow of Time

where the entropy ~ is defined by

S = --~V'pk In Pk or S --- - Tr{o In ~}. (~)

Smax is the maximum value of S consistent with the macroscopic constraints on the system. Liouville's theorem, which is valid whenever there exists a well-defined Hamiltonian for the system, implies tha t S is a constant of the motion; the dynamical evolution of a closed system neither creates nor destroys information about its microscopic state. Thus if the approach to equilibrium is to be characterized by a loss of information (gain of entropy) at some level of description, there must occur a corresponding increase of information at other levels. I t is essential, therefore, to introduce a coarse-grained probability distribution {Pk} or density matrix 0. The probabilities Pk (= ~k~) refer to aggregates of microscopic states or of eigenstates of the density matrix. They must be chosen in such a way tha t the macroscopic variables tha t figure in a thermodynamic description of the system (e.g. the energy) do not vary appreciably within an aggregate. I shall give the name macroscopic information to the quanti ty i associated with {Pk} or ~, and the name microscopic information to the residual information Y = I - i . I t can be shown (a) tha t • is the expectation value with respect to the probability distribution {Pk} of the information associated with the conditional probability distribution for microscopic states in a given aggregate.

The use of a stochastic description and the introduction of coarse-graining do not in themselves disturb the symmetry between past and future tha t is inherent m the underlying microscopic description To prove tha t the macroscopic mformation decreases with time, one must make an assumption tha t explicitly distinguishes between the past and the future Van Hove (1) has shown tha t if the t tamiltonian has certain general properties it is sufficient to postulate tha t the nondiagonal elements of the coarse-grained density matrix vanish initially; the system then exhibits irreversible behavior at all later times. Similar theorems have been established in different formal contexts by a number of other authors. (2.4-6) All these theorems state tha t the macroscopic information will decrease monotonically with time if the microscopic information vanishes initially. The precise definitions of macroscopic and microscopic information depend, of course, on the formal context.

Thermal eqmlibrium may be defined as the state in which no macroscopic information at all is present, i.e. the state for which the coarse-grained entropy assumes its maximum value subject to the macroscopic constraints. The constraints (N~ = const, and (E~ -- eonst, are appropriate to the grand canonical distribution of Gibbs ~) (N and E are respectively the number of particles and the total energy, and the brackets denote averages over the coarse-grained probability dmtribution.)

Modern theories of irreversible processes m many-body systems provide insight into the detailed mechanisms responsible for the approach to equilibrium, as well as powerful methods for calculating transport coefficients. Moreover, their underlying assumptions are physically very plausible, especially as they apply to systems prepared in the laboratory. The initial absence of microscopic information, for example, can usually be attrJbuted to the way in which the system has been prepared. When they are applied to naturally occurring systems, however, these theories encounter certain conceptual difficultms, stemming from the lack of any prescription for distinguishing between macroscopic and microscopic information or for identifying the macroscopic variables tha t ought to figure in a macroscopic description. Indeed the distinction between the macroscopic and microscopic levels of description has been held by many writers to be purely subjective. I t is widely accepted that , although information about the detailed state of a complex physical system

D~v~ I~Yz~R 281

may be uninteresting or hard to get, in principle it can always be obtained. But if this view is correct, how can statistical considerations teach us anything about the arrow of time in naturally occurring systems~.

Attempts to resolve this difficulty have in the past proceeded along two different lines. Gold and others admit the possibility of a complete mier6scopic description of any finite

system and of the Universe as a whole. They seek to relate the arrow of time to the cosmic expansion, because of which the Universe acts as a sink for radiation. ~s~ Consequently a sys temtha t is open to the Universe cannot be in a state of thermodynamic equilibrium, and this is a necessary condition for it to show the arrow of time. Gold argues Cs~ tha t our sub- iective certainty tha t t ime "really goes" in one sense and not the other results from our being able to predict the past with much greater certainty than the future, given a knowledge of the present configuration. He interprets this asymmetry as a statistical effect related to the cosmic expansion: the world lines of the particles (including photons) tha t make up a complex system open to the Universe tend to diverge in the direction of the expansion, chiefly because photons are getting out but not in.

The second line of argument, which has been developed by Blatt and others, (9~ also attri- butes macroscopic irreversibility to interactions between ostensibly closed systems and their environment, but denies the possibihty of a complete microscopic description of any finite system. Blat t argues tha t interactions between a finite system and its environment, which can never be eliminated entirely, give rise to a stochastic contribution to the Hamiltonian of the system. The stochastic nature of the interactions is implicit in the definition of a finite system; the only purely mechanical system tha t exists is the whole Umverse. Once a stochastic contribution to the Hamiltonian of a system is admitted, the approach to equilibrmm can be demonstrated without recourse to coarse-graining,

The principal objections tha t have been raised to the arguments of Blat t and Gold stem from the belief t ha t the approach to equilibrium does not depend upon a cont/nu/ng inter- action between a system and its environment. I t is noteworthy tha t Blatt and Gold both regard the arrow of time as a property peculiar to proper subsystems of the Universe. According to both these v~ews the Universe as a whole admits a completely reversible microscopic description. According to the conventional view, on the other hand, an in~nite Universe represents the 1deal irreversible system, for the Poincar~ recurrence time associated with it is infinite, while tha t associated with any f ~ t e system is finite. The theory outlined below accepts "convent ional" versions of non-equilibrium statistical mechanics, but seeks to resolve the ambiguities inherent in their underlying statistical assumptions by establish- ing objective criteria for the distinction between the macroscopic and microscopic levels of description. These criteria are supplied by a picture of the Universe and its temporal development tha t differs from the one obtained through extrapolation of non-equilibrium statistical mechanics (1°~ (the Universe as an ideal isolated system tending toward equilibrium) and from the classical picture of the Universe as a perfect reversible machine

Instead of trying to develop a picture of cosmic evolution by extrapolating macroscopic laws, we shall follow the opposite approach and begin at the cosmological level. Perhaps the most fundamental difference between cosmology and macroscopic physics is tha t the latter must concern itself with an infinite variety of physical systems and physical conditions, while the former treats a single system and its evolution. Thus in microscopic physics it is natural, indeed unavoidable, to split up the mathematical description into a set of laws, which are the same for all systems under all conditions, and a set of auxiliary state- ments tha t serve to define particular states of particular physical systems. (n~ Because the laws do not make explicit reference to microscopic systems, they may be presumed to be valid also at the cosmological level though of course the possibility cannot be excluded


tha t some modifications may prove to be necessary. (For example, Newton's law of gravi- ta t ion does not apply to an infinite distribution of mat te r of constant mean density, but Einstein's does.) On the other hand, the auxiliary conditions tha t serve to define the Uni- verse must be regarded as being on an entirely different footing from those tha t serve to define macroscopic systems. Because the Universe is a unique system, the auxihary conditions t ha t define it must be accorded the same status as the physical laws Even if there existed, m some sense, a number of "poss ible" models of the Universe, there would be no reason to suppose tha t the physical laws tha t prevail in the Universe we inhabit would also prevail in the others.

There is another reason for according the cosmological auxiliary conditions the status of laws. The infinite var ie ty of auxiliary conditions tha t can be applied to macroscopic systems is a consequence of the fact tha t we are interested in the properties of individual systems. As soon as we shift our at tent ion from individuals to classes (for example, from individual stars to the whole class of stars), the choice of suitable conditions becomes much narrower. Thus if we consider the stars as a whole, we must regard the distribution of such parameters as mass, age and chemical composition, which are freely disposable when we t reat individual stars, as subject to definite statistical laws which would be supplied by a theory of star formation. Ttus theory m turn would contain some disposable parameters , for example parameters specifying the properties of protogalaxies, whose statistical distribution would be determined by a theory for the formation of galaxies. Ultimately, statistical laws governing the distributions of and correiatlons among the parameters tha t specify all the macroscopic properties of astronomical systems should emerge from a sufficiently complete theory of cosmic structure and evolution. But there is no reason why such a theory should itself contain disposable parameters .

I f we regard the auxiliary conditions t ha t define the Umverse as being on a p a r w i t h t h e ordinary laws of physics, we may employ the usual empirical criteria of simplicity and economy as guides to their formulation.

The simplest assumption about the Universe tha t is consistent with current physical theoriesis the so-called cosmological principle, which states tha t the spatial structure of the Universe is statistically homogeneous and isotropic: no average proper ty of the spatial distribution of ma t t e r and motion serves to define a preferred position or a preferred direction in space. The cosmological principle has two well-known consequences. In the first place, i t leads to a unique decomposition of space- t ime into space and time.(l~ Thus the cosmologi- calprinciple re-establishes absolute simultaneity. The reason is tha t the properties of homo. genelty and lsotropy (and the metric tha t expresses these properties) are not invariant under Gahlean transformations. I n the second place, the cosmological principle, together with Emstein 's theory of grawtation, implies t ha t the Universe is expanding from or contracting toward a singular state of (formally) infinite density. (18~ Thus the cosmological principle implies tha t the temporal structure of the Universe is anisotropic. Cosmic t ime is both absolute and directed.

In the past i t has been assumed that , even though the spatial structure of the Umverse may be homogeneous and isotroplc in a statistical sense, a complete description of the con- tents (or geometry) of space a t a given instant of cosmic t ime would distinguish between different positions and different directions, orang to the presence of local irregularities. For a distribution tha t is either finite or bounded, this s ta tement is both obwous and trivial. For an infinite and unbounded cl~sicat (as opposed to quantum mechanical) distribution, i t is less obvious but still true. For an infinite and unbounded distribution tha t can be characterized by a countably infinite set of occupation numbers it need not, I suggest, be true a t all.

DAvm I~YzE~ 283

Consider, for example, a statistically homogeneous distribution of points along an infinite straight line t ha t has been divided into cells of equal length. The distribution is specified by an infinite but countable sequence of occupation numbers, which are integers. Because the distribution is infinite, we infer from the ergodic theorem of Birkhoff and Khinchin tha t any of its statistical properties can be ascertained to any desired degree of accuracy through sufficiently extensive measurements, provided only tha t the correlation distance is finite.C14) Suppose t ha t we are now given a second statistically homogeneous chstribution of points along a line divided in the same way, and are asked to decide whether or not the two distributions are identical. Let us assume tha t we are allowed to carry out a countably infinite series of operations. This will enable us to decide whether the two distributions have the same statistical properties, because a complete set of moments and correlations of the occupation numbers is countahly infinite, and the value of each such proper ty can be found through a countably infinite series of measurements. Suppose tha t the two distributions turn out to have the same statistical properties. Can we discover any non-statistical differences between theme.

Because the £hstributions are unbounded, they can be matched in a countably infinite number of ways. Now it is clear t ha t any finite s tretch of the first distribution can be matched in the second distribution, because every finite sequence of occupation numbers has a finite probabil i ty of occurrence. Whether the two distributions can always be matched along their entire lengths, I have not been able to determine. I n the present context, however, this question is not of crucial importance, for in an expanding Universe of the Fr iedmann type the visible Universe is always finite. Thus it appears to be possible to construct infinite and unbounded distributions whose statistical description is also a complete description, in the sense tha t it contains all the information about the distribution t ha t can be gained through measurements within a finite volume. Not all distributions satisfying the cosmological principle have this property, however. For the sake of defi- niteness I shall designate as the strong cosmological pmnciple the assumption t ha t the Universe admits a complete statmtical description tha t does not distinguish between different positions and different directions in space.

The strong cosmological principle seems at first sight not to be applicable to a Fnedmann Universe of fimte volume. Such a Universe, however, is also finite in t ime, it expands to a state of finite minimum mean density and then contracts toward the singular state of infinite density. Model Universes of this kind are usually pmtured as going through an endless succession of expansion-contraction cycles, but this picture has two unphysical features. In the first place, there does not yet exist any convincing way of connecting solutions of the gravitat ional field equations on the two sides of a singularity, nor does there seem to be any way of avoiding the singularity through reasonable assumptions about the form of the mat te r -energy tensor a t high densities. (is) In the second place, it will be argued below tha t in the initial phases of the expansion the Universe is in a state of thermodynamic equilibrium and tha t the initial and final states are identical. This would make it physically meaningless to arrange expansmn-contraction cycles in any sort of linear sequence. Thus, instead of a single cyclic Universe, we must envisage something hke a Glbbsensemble of Universes tha t are finite in both space and time, each one corresponding to a particular realizatmn of a statistical description satisfying the strong cosmological principle. The fact tha t we can carry out observations tha t pertain to only one member of this ensemble is analogous to the fact tha t in an infinite Fr iedmann Universe we can only make observations within a finite region. These considerations suggest tha t the strong cosmological principle does not exclude the possibility of a finite Universe; but this question needs further attention.


The strong cosmological principle imphes tha t a certain kind of information, which we now identify with microscopic information, is not present in a complete description of the Universe. Let us now t ry to obtain a more precise idea of what constitutes microscopic information.

We first note tha t the temporal development of a system under known dynamical laws does not alter the quanti ty of information associated w~th it. Now consider the state of the Universe at epochs very near the beginning of the expansion when the mean density is very high. Because the relaxation time varies reversely as the square of the density, while the characteristic time associated v~th the cosmic density varies as ~-1/2 or ~-2/a, depending on whether mat ter or radiation dominates, (la) there must be a range of densities for which local thermodynamic equilibrium prevails to a high degree of approximation. Numerical estimates place the lower limit of the density well below tha t of nuclear mat ter for a wide range of assumptions about the temperature and composition. In this phase of the expansion, a complete statistical description of the Umverse requires a comparatively small number of parameters, including the radius of curvature, the temperature at some epoch, and the lepton-baryon ratio. Other reformation about the local physical state comes m the usual way from equihbrinm statistical mechamcs.

In principle i t is now possible, if the relevant physical laws are known, to calculate a temporal sequence of cosmic states in the direction of decreasing mean density. At sufficiently low densities the conditions necessary for the maintenance of thermodynamic equilibrium can no longer be met and the distribution will take on a non-equihbrinm character. Thus as the Universe expands it becomes increasingly complex, more and more parameters are needed to describe its current state, though the information associated with the descmptlon stays constant. Calculations of nucleogencsis ¢17) in the early stages of the expansion afford an excellent illustration of th~s process. The apparent failure of the second law of thermodynamics is a direct consequence of the cosmic expansion, which also leads to other departures from physical laws relevant to macroscopic systems, such as the laws of conservation of energy and momentum.

There does not as ye t exist a detailed theory for the formatmn of galaxies and other astronomical systems through the growth of density fluctuations. (is) However, theoretical arguments have been advanced tha t this process is possible under conditions hke those postulated here. (19) Thus at the present time it would appear to be a tenable hypothesis tha t the Universe has evolved from an ~mtial state of thermodynamic equilibrium and tha t the complex structure we observe today has developed gradually through processes governed by existing physical theories. This hypothesrs furmshes the presemption we have been seeking for distinguishing between the macroscopic and microscopic levels of description: the variables tha t figure in a complete statistical description of the Universe and its evolution are to be regarded as macroscopic Microscopic reformation is accordingly always absent, the specific entropy of the Universe at any given instant has the maximum value consistent with the macroscopic description.

This distinction between the macroscopic and microscopic levels of description does not correspond to any fixed distinction between levels of, say, the B B G K ¥ hierarchy. In some cases the line may be drawn between the hydrodynamic and the kinetic levels; m others, it may he deeper. For any given physical system the decision depends on the relative rates of the processes tha t have figured m the formation or preparation of tha t system as com- pared with the rates of relaxation associated with various levels of the hierarchy For example, if Bogoliubov's conjecture (e) is correct, the macroscopic level of description for a dilute gas should normally correspond to either the hydrodynamic or the kinetic level. But to settle the question for a particular system one would need to know its provenance and

DAvra LAyzz~ 285

history. Thus it is always possible, in principle, to prepare a system in such a way that the macroscopic level contains detailed information about microscopic properties of a system, as in the well-known spin-echo experiment. (2°)

The preceding considerations supplement the arguments given by Boltzmann and Gibbs to explain the approach to equilibrium in isolated systems. They provide an objective basis for such assumptions as the ~tosszahlan~atz and the assumption of equal a ~r/ori probabilities in phase space. Moreover, they provide a formal prescription for determiniug the conditions under which such assumptions are vahd. ~inally, they explain why irreversible processes in different isolated systems define the same direction m time and why this direction coincides with the direction of cosmic evolution. ~or in a given system the arrow of time, as defined by irreversible processes taking place in the system, points away from the "initial" state. The initial state of a system may be defined as the one that separates the history of the system into a part during which it is effectively isolated from its environment and a part during which it interacts strongly with its environment. Because cosmic evolution is a branching process---a process of increasing local differentiation--the arrow of time as defined by naturally occurring systems coincides with that defined by cosmic evolution.

Although the anisotropy of temporal processes in isolated systems is a consequence of the amsotropy of cosmic evolution, the nature of the anisotropy is fundamentally different in the two cases. Macroscopic systems tend toward a state of thermal equilibrium, which is essentially a state of timelessness Cosmic evolution, on the other hand, is essentially a historical process, fashioning the present configuration from materials supphed by the past. I t is a consequence of the strong cosmological principle that a given configuration of the Universe contains precisely enough information to determine succeeding configurations. I t does not, however, contain enough information to determine configurations through which it has already passed, though it contains actual traces of such configurations. That a complete description of the present configuration of theUniverse does not enable us,in principle, to calculate past configurations is an immediate consequence of the statistical character of the descraption. For example, from a knowledge of its present configuratmn, it is impossible to reconstruct the history of a system that has undergone irreversible changes. Thus, at the cosmological level, the future is uniquely characterized by its predictability and the past is uniquely characterized by the fact that a partial record of it is contained in the present configuration.

Bergson pointed out that no absolute significance can be attached to the passage of time in the Newtonian world picture (~1) A reversible microscopm description of the Universe does not in fact describe a temporal succession of spatial configurations, but rather a four- dimensional world in which certain necessary relations obtain among distinct spatial (or space-like) hypersurfaces. In the present world picture the passage of time does have an absolute character. Cosmic evolution generates a uniquely defined temporal sequence of eonfigurations of increasing complexity. There is no way of arriving at a complete descrip- tmn of a given configuration except by traversing the sequence of preceding configurations. The present configuration contains traces of the past but none of the future. Although the future is determined by the present, it cannot be said to exist until it actually becomes the present. For by virtue of the ever-increasing complexity of the cosmic distribution, more reformation is needed to specify any given future state than is needed to specify the present state. Thus it is impossible for future configurations to be m any sense prefigured in the present configuration; the additional information is generated by the process of cosmic evolution itself.

These aspects of cosmic evolution have obvious parallels in biological evolution and in the experience of time in individual biological systems. These parallels suggest that the


properties of biological t ime are most directly related to the growth of information in some par t of the system rather than to the irreversible processes contemplated in statistical mechanics, though these, of course, must also be going on in the system. Schroedinger has suggested tha t this growth of information is the essential feature of life. (~2) Of course, the total ent ropy of a biological sys tem need not be constantly decreasing, i t is only necessary t ha t information should be generated in some par t of the system. Essentially the same idea was expressed by Bergson. (~1) "Wherever something is ahve, there is open, somewhere, a register in which t ime is being inscribed."

Although the continual generation of new information is characteristic of both biological and cosmic processes, the underlying causes are completely different. Cosmic evolution is a consequence of the cosmic expansion; biological evolution and the phenomenon of life depend on interactions between bmlogical systems and their environment- - in particular, on the ability of biological systems to extract reformation from their environment. That information is there, however, because i t has been generated by cosmm evolution, Thus the two kinds of process are in fact closely linked.

I thank Professor O. Klein for stimulating conversatmns during the final stages of the preparat ion of this communication, which was completed a t Stockholm University during a s tay sponsored by N O R D I T A and by the l~ational Science Foundation.

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17. WA6ONER, R. V., FOWLEB, W. A. and H o ~ , F., Ap. J. 148, 3 (1967). 18. LAYZ~B, D., Ann. Rev. Aatr. and A~. 2, 341 (1964). 19. LAYZ~, D., Mere. 8oc. Roy. So./d?. XV, 33 (1967). 20. HAm, E. L., Phys. Rev. 80, 580 (1950). 21. BERGSOn, H., Evolution Cr~t~e, 52nd edn., Presses Universltaires de France, Paris, 1940. 22. SCrgRODI~O~.R, E., What Is Life? Cambridge Umversity Press, New York, 1963.

~OTE

The reader may be interested to compare Professor Layzer's above treatment with that of Professor Herman Zanstra in Vt~as in Astronomy, ed. A. Beer, 10, 23--43 (1968).

cosmology and the arrow of time

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