irreversibility and time's arrow

Irreversibility and Time’s Arrow Peter T. LANDSBERG

Faculty of Mathematics Studies, University of Southampton, U.K.

<The chapter on <Times’s Arrows, is a confusing blend of speculation and possibly wrong ideas., From a review of M Gell-Mann’s <The Quark and the Jaguar, by P W Anderson in Physics World, August 1994.

This paper is based on a talk given at the meeting of the Portuguese Physi- cal Society 19-23 September 1994 at Covilha, Portugal. In it we review the physicist’s key problems regarding the arrow of time. For this purpose it is convenient to introduce six demons and a short way of characterising them, which will be clarified in the paper:

D1 1812 Laplace: All-knowing (Section 11) D2 1867 Maxwell: Anti-heat conduction (Section 4) D3 1869 Loschmidt: Velocity reversal (Section 4) D4 1936 Eddington: Particle count (Section 16) D5 1937 Dirac: Decrease of the gravitational <<constant>> (Section 17) D6 1970 Landsberg: Expansion or Contraction? (Section 14)

1. Time-symmetric mechanics

Ludwig Boltzmann was born 150 years ago, and struggled for much of his professional life with the problem of irreversibility. What is this problem? A billiard ball bounces off the side of a table at a point P, giving a trajectory APB. The reversed trajectory BPA is also allowed by the laws of mechanics. Since it is the same as the original trajectory, but with time reversed, one can say that the laws are TIME-SYMMETRICAL. A most important question is that of the boundary conditions. How was the ball projected: from A or from

D i a 1 e c ti c a Vol. 50, N o 4 (1996)

248 Peter T. Landsberg

B? That decides on which of the two senses the ball follows its trajectory. In any case, note that also for collisions among the hard balls (or particles) of mechanics the laws which govern them are time-symmetrical. [Elastic balls would be subject to compression and heat would be dissipated, thus involving irreversible thermodynamics.]

2. Anti-thermodynamics

Consider next a box of gas whose ends are at different temperatures. It will by HEAT CONDUCTION attain a uniform temperature after a little while (if otherwise insulated). How does this happen? By molecules colliding with each other, and the faster molecules from the hotter part losing energy to the slower molecules of the cooler part. But each collision is time-symmetrical, so that the reversed process, a kind of ANTI-HEAT CONDUCTION, is also possible. But, and this was Boltzmann’s problem, the reversed process is anever, (?) seen. Thus the laws of mechanics suggest the sequence.

should be matched by

One knows that entropy goes up with the equilibration just discussed. It must therefore go down under anti-heat conduction; we can call this behaviour ANTI-THERMODYNAMIC. The Boltzmann problem can therefore be ex- pressed as follows: Why do we not see anti-thermodynamic behaviour?

temperature difference - collisions - uniform temperature

uniform temperature - collisions - temperature difference.

3. Anti-diffusion

Again, if a gas is confined by a partition to one half of its container, then, upon removal of this partition, one has a non-equilibrium state, which gives rise to DIFFUSION and thereafter to an equilibrium state in which the gas fills the volume uniformly. The time-symmetrical collisions are again responsible, and we ask: Why is the spontaneous contraction of the gas into a part of its container, a kind of ANTI-DIFFUSION, <never, (?) seen? This is another variant of Boltzmann’s problem.

4. Maxwell’s and Loschmidt’s demon

For the anti-heat conduction we have a <Maxwell demon, (which I shall call crD2,) who, starting with an equilibrium gas, opens a trap door in the gas, so as to let through the fast molecules to the left (say), but does not let the slow molecules through, and hence establishes a temperature difference: the left-

Irreversibility and Time’s Arrow 249

hand side of the box is now hotter, and an anti-thermodynamic process has been performed (Daub, 1970; Leff and Rex, 1990). For anti-diffusion we have a cLoschmidt demon, (D3) (see, e.g. Batterman 1990) who precisely reverses all molecular velocities at a certain instant, and so sends all the molecules back into the original corner of the box.

5. Anti-diffusion: statistics

The word <never, is a very strong word, and it led us to make incorrect statements. Thus for a gas of TWO molecules it is certainly true that it will oc- casionally and spontaneously contract into a portion of its container. Even for FIVE molecules this is true. It is because we implicitly assumed that the gas contains many, many molecules that there is a Boltzmann problem at all. We now see that the circumstance that there is a problem is connected with the statistics of particles. Even if the number of molecules is astronomically large, a gas in equilibrium WILL return to a microscopically defined initial state, but one may have to wait a long time (which one can calculate). If that time ex- ceeds the period from the last big bang to the present, i.e. the age of the universe, then one can safely dismiss the possiblity of actually seeing such a return.

6. Statistical weight

A card shuffling example is often given in this context (e.g. see M Shallis, 1983). We take a pack of 52 cards, which is a tiny number compared with the millions and millions of molecules even in a cubic centimeter of a typical gas. The probability of regaining the original arrangement of the cards after a proper shuffle is nevertheless tiny: less than one in a million, million. . . (the word <million, appears eleven times i.e. 10 66). If every person on earth were to shuffle a pack once every second, it would still take statistically much longer than the age of the universe to come to the original arrangement (10 50 as against loll years). This gives one an idea of the effect of the large numbers involved.

7. Problems: I Anti-processes; 11 Constant fine-grained entropy.

Let us regard anti-diffusion, anti-heat conduction, and therefore the problems posed by the demons of Maxwell and Loschmidt, as the FIRST problem of irreversibility. The SECOND problem can then be formulated in terms of a phase space for n particles. This is an imagined many-dimensional space in


which all 3n position coordinates (three per particle) and all 3n momentum components are the axes. The state of the whole system is represented by a point in this space. This point moves (as the state of the system changes) according to classical or quantum mechanics which both yield time-symmetrical equations of motion. If one considers many copies of the same system, but with different initial conditions, there are then many points in phase space, giving rise to <<phase space densities>> of these points. It turns out, as can be seen from most books on statistical mechanics, that the non-equilibrium entropy, which is defined in terms of such phase space densities, is constant in time, whereas thermodynamics requires it to be non-decreasing with time. It involves an interesting history (too long to be discussed here) of the search for a function in classical mechanics (and its phase space) which can represent the entropy.

8. I: Computer experiment on velocity reversal; statistical weight.

The FIRST problem (of anti-diffusion, etc.) can be understood by accept- ing the fact that velocity reversal a la Loschmidt does decrease the entropy S (or, equivalently increases Boltzmann’s H-function, H = -S). Boltzmann knew, when he challenged Loschmidt <<YOU reverse all the molecular velocities)), that it could not be done. But now WE can do it - because of the master toy of the twentieth century: the fast computer. And indeed the entropy CAN decrease in computer experiments. Thus initial conditions do exist from which entropy decreases (e.g. Aharony, 1971). The initial conditions for entropy decrease, introduced by velocity reversal are, however, so extraordinar- ily delicate that the slightest error brings one back to entropy increase. One may call this an instability of the boundary condition. [In Aharoni’s paper this is made clear by means of a simple diagram, not needed here.] The card experiment mentioned above gives one some idea of the vast number of possi- bilities generated by even quite restricted situations. This number becomes even larger when one considers possible initial conditions for anti-diffusion. These strange, but in principle possible, situations are not seen because of the instability of the boundary conditions involved. The Loschmidt theoretical experiment convinced Maxwell that no purely dynamical proof (i.e. one based on classical mechanics) of the second law was possible. Further, since he possibly regarded his demon as a representation of God, he did not consider that the demon might need energy to <<see>> the molecules. That realisa- tion came later (Szilard, Brillouin). After all, photons have to bounce off molecules to make them visible and this leads to energy dissipation and en-

Irreversibility and Time’s Arrow 25 1

tropy increase. This was ONE way of showing that the second law was not in fact violated. Other ways came later (see, for example, Leff and Rex, 1990).

People also began to realise that equilibration proceeds in a physical system because the number of available states for equilibrium is vastly greater than the number of states available for a non-equilibrium state. So the second law of thermodynamics was recognised to be <<merely>> a statistical law.

9. ZZ: Coarse-graining

We can look at the SECOND problem of irreversibility (that the statistical mechanical entropy does not increase), by noting that if all the available classical or quantum mechanical information has been used, one obtains

dS/dt = 0. We speak of the FINE-GRAINED entropy in that case. Thus, given that we work in a Hilbert space,

Fine-grained entropy implies dS / dt = 0. Hence, by a rule of logic dS/dt > 0 implies entropy has to be COARSE- GRAINED. This means that we confine attention to groups of classical or quantum mechanical states, and say whether or not our system is one or other of these. Now entropy can increase with time, though this is purchased at the expense of a loss of knowledge of the system. But in thermodynamics it is quite usual that one is interested only in GROUPS of states.

10. Spin-echo and coarse-graining

But coarse-graining as a basic procedure in statistical mechanics has also been criticised, for example by reference to the so-called spin-echo experiments of the 1950ies. In these a total nuclear magnetic moment M (at low temperatures) disappears slowly because a not entirely homogeneous magnetic field is applied in which the nuclear moments precess at different rates (because of the lack of homogeneity). The entropy has increased and equilibrium has apparently been approached. Now by applying a magnetic field which flips the spin once, individual spins continue to precess at the same rate, but in the opposite direction. The original magnetic moment is then even- tually almost fully restored with apparently an entropy decrease. This is some- what like the Loschmidt velocity reversal. But there is an important difference: A non-equilibrium state was masquerading as an equilibrium state and a coarse-grained view would have been quite misleading. These and more modern (e.g. polarization echo) experiments have been performed now for more than 40 years (see e.g. S. Zhang et al. 1992).


11. Time’s arrow as an illusion; Laplace’s demon

The increase of entropy furnishes one of the key physical indicators of the progress of time. If it can be extracted from mechanics only by coarse-graining, the question must be faced whether the progress of time itself is only an illusion. To consider this matter, let us call in Laplace’s demon (1812) often called the Laplacian calculator. He is our third demon, but, as he hails from 1812, we shall call him D 1. He is a dematerialised intelligence, a kind of God who knows of all collisions, can distinguish all microstates in a fine-grained phase space, and for his powerful mind all calculations of future and past states (in so far as allowed by quantum mechanical uncertainty) are performed instantaneously. For him all elementary processes are therefore time- symmetrical (the well-known time symmetry violation by kaon-decay apart). As he knows only of these elementary processes, how would we communicate with Laplace’s demon? Maybe time would not exist for him. The concept of a table is far too rough for him: where we see a surface, he still sees a swarm of molecules which mix with those of the surrounding air. We would have to tell him to discard information which he has at his disposal, so as teach him our language which attributes words to rough concepts; and again to arrive at coarse-graining so as the extract the direction of time. Thus one wonders if time’s arrow - the uniform forward movement of events according to the coarse human perception - is perhaps just a crutch for the weak human brain, an illusion manufactured by us in order to enable us to make sense of our sur- roundings without overloading our brains. One is reminded of an often- quoted remark of Einstein’s to his close friend Michele Besso, who kept en- quiring about the nature of irreversibility. Einstein considered it to be an illusion produced by improbable initial conditions. Further, on Michele’s death Einstein wrote to his window that <<For us convinced physicists the distinction between past, present and future is an illusion . . .>> (Prigogine and Stengers 1984).

12. Proposed origins of time’s arrow.

Much has been written about the problems just discussed and much advanced mathematics has been developed to deal with them (developments of ergodic theory, Liouville equations, mixing flows, subdynamics, maximum entropy methods, etc., see e.g. Garrett, 1991; Dougherty 1994). However, these problems cannot yet be regarded as fully understood. Still the above discussion stresses at least some essential points.


Views on the origin of time’s arrow which have been put forward in the brief span of the thirteen years 1962- 1975 include (See Appendix of Lands- berg, 1978): a) The universe is young (Hund, 1972) b) Effect of outside world on the system (Hoyle, 1962) c) Gravitation is responsible (Davies, 1974) d) It is the effect of boundary conditions (McCrea, 1975) e) Past and future are primitve concepts (Layer, 1967) f ) The weight of the number of realisable states (Landsberg, 1975) g) Not due to statistics (Bondi, 1962) Many more ideas have of course been put forward in the last twenty years. In particular, cosmology has attracted much professional and popular attention. We turn to this topic next.

13. Different armws of time

The nineteenth century already knew several arrows of time: the psychological one (we remember the past, not the future), the biological arrow (evol- ution), the thermodynamics one (entropy increase) and the electromagnetic one came a little later (the retarded potentials matter more than the advanced potentials). This century brought us the arrow of kaon decay which belongs to subnuclear physics and the cosmological arrow of time which gives us an expanding rather than a contracting universe. Why do these arrows all point in the same direction? Although it has been suggested that the cosmological arrow is primary and impresses its direction on all the others, this is not gener- ally accepted: one is clearly reluctant to believe that milk and coffee get mixed up in one’s cup because of the expansion of the universe.

Less controversial is the view that, since memory traces laid down in the brain are biochemical, this arrow determines that (1) the biological, (2) the psychological and (3) the thermodynamic arrows all point in the same direction. Here, and in many other considerations the arrow of kaon-decay is left as an isolated curiosity, which is probably of limited philosophical importance.

Lastly, one may adopt the no-boundary proposal according to which the universe is without boundary or edges and also without singularities at which the laws of physics might break down (Hawking 1993). But its implications for the arrows of time are still under discussion.

14. Oscillating universes

In the simplest cosmological models of a (smeared out) universe there arises a SCALE FACTOR R (t) which describes the extent of the expansion


of the universe, and in the simple Newtonian version is concave towards the time axis (t). From this one can infer that R has been zero at some time in the past (Landsberg 1986). This we can call the BIG BANG: R(0) = 0. The oc- currence of this zero is a kind of singularity theorem, which has a distinguished analogue in relativistic cosmology, and tells us about the existence of a big bang, provided of course we are philosophically inclined to go that far back in time. Let us consider first a degenerate case which causes rather special philosophical problems. The scale factor can IN PRINCIPLE be symmetrical about some maximum and one could imagine the entropy of this smeared out model universe to return at the BIG CRUNCH to its original Big Bang value: we would have cyclic renewal and the universe would then behave as if some Pythagorean or Mayan philosophy of cyclical time was to beverified. Because entropy is increasing in living processes, life in this contracting universe would then have to be counter-directed to the cosmological contraction (in which the entropy decreases), and we would STILL have to say: <<Ah, the universe is expanding!>> (Fig. 1).

Entropy I

or

R (t)

f--- Time t

A contracting phase follows expansion. The arrows on the curve correspond to the time as it appears to a disembodied intelligence. The arrows below the curve indicate a conjectured direction of human time.

So IS the universe contracting in spite of what we would see? This can be judged only by a dematerialised, demon (D 6) subject to neither entropy changes nor aging. At a 1970 Thermodynamics Conference it was associated with my name (Landsberg, 1970). This demon tells us whether the expansion


of the universe is an illusion or not. It is not shut in a box like Maxwell’s, he is in a sense the universe’s great outsider.

The above artificial picture changes if one passes to a two-component universe, containing, say, matter and radiation, each in equilibrium by itself, but not with each other. The different laws of expansion imply that radiation and matter develop different temperatures upon expansion and heat transfer takes places between these components. As a result the entropy goes up ALL the time and so does the energy. [This is possible because energy conservation in the usual sense does not hold in general relativity]. One can visualise this as a kind of <<entropy gap>> which opens up; it is the difference between the maximum entropy at this time which the universe would have if it were in equilibrium and the actual entropy of the universe at that time.

If the model can be arranged to <<bounce>>, successive cycles become longer and R (t) achieves higher maxima. After many cycles one may specu- late that the model universe can only JUST contract. This is equivalent to as- serting that the energy density is close to the critical value which divides theoretical models of oscillation from models of ever-lasting expansion. This picture suggests therefore an answer to the question why our universe seems to be very close to this situation (the so-called <<flatness>> problem): The universe is old and has already gone through many oscillations (Landsberg, Piggott and Thomas, 1992; Smolin, 1992). It can of course be argued that a universe model cannot bounce because of all the irreversible processes which take place in the first half cycle, and because of black hole formation (e.g. Blud- man, 1984). But here we are again in an area of current investigation.

15. Order in the universe

The cosmological and thermodynamic arrows seem to be superficially contradictory since one would expect a high degree of <<disorder>> near the be- ginning of the hot big bang, while <<order>> emerges later with the formation of galaxies, etc. The cosmological process, like the biological one, SEEMS to be anti-thermodynamic. This problem can, however, be resolved by two obser- vations:

(i) Ordinary systems which are driven far away from equilibrium display surprising reservoirs of structure which are not expected if one restricts attention to equilibrium. This applies to fluids (Benard convection) and chemical reactions (chemical oscillations, Nicolis and Prigogine, 1977) just as much as it does to semiconductors (switching transitions, Scholl, 1987). The meaning of order needs to be carefully analysed in such cases.


(ii) The presence of gravitational effects keeps the universe and many of its component parts away from equilibrium, and there is in any case a problem of how to define the entropy in the presence of these effects.

One could define <corder>> in terms of the “entropy gap, introduced in the middle of para. 14, i.e. as the ratio of the entropy of the universe at the present time divided by the maximum conceivable entropy the universe might have. This disorder>> D of any system is then unity if the system has its maximum conceivable entropy. What is this maximum entropy? It is here that the black hole enters.

16. Black holes

The simplest black hole is the Schwarzschild black hole, which has no charge and no angular momentum. Its entropy is extremely large, so that it is very useful for the denominator of the disorder parameter D. The entropy value in terms of Boltzmann’s constant for a black hole of mass M is

where M is the mass of the sun. For comparison, note that the cosmological background radiation at 2.7 K for the whole of the observable universe has an entropy in terms of Boltzmann’s constant given by

5.5 x 1088 Now the 10 baryons in the observable universe give a mass of 10 23 solar masses. If they were to coalesce into one black hole its entropy would be an unimaginably large

On this basis the universe is very orderly. There are, however, some problems. Does a single tremendous black hole make sense? Will it not evaporate? I have shown elsewhere that entropy and order can, in certain circumstances increase together, but this observation is of little help here.

The number of baryons in the universe was first estimated by Eddington in 1936 and the demon who can complete this count I have called Eddington’s demon D4.

1077 ( M / M * ) ~

k. The disorder is then tiny: D - log8/ (Penrose, 1981; Landsberg 1989)

17. A few extra points. [G(t), 2-dim time]

We have now covered several of the interesting problems regarding time. But we still have to explain how it came to pass that scientists imagined that Newton’s gravitational constant might decrease with time, so that one could then write it as G (t). There is no accepted mechanism which could bring about this decrease, so one can attribute this to another demon who I have


called Dirac’s demon D5. For it was Dirac who observed that the age of the universe, which we have already noticed to be of order 10 l7 s, divided by an atomic unit of time e /mc (m is the electron mass) - 10 -23 s is a pure number of order 10 40. It must be expected to increase with time as the universe ages. Next Dirac observed that the ratio of the electrostatic to the gravitational force between the proton and the electron is of the same order of magnitude. If we argue that on equality between two such large numbers can be no accident, and so is not merely characteristic of the present epoch, then this force ratio must ALSO increase with time. The only reasonable way whereby this can happen is if G(t) decreases. There are now limits known on the experimen- tally permitted variation of G(t).

The notion of a multidimensional time has also been raised, but has not been taken very far.

I am grateful to Dr. A Burri, University of Bern, Switzerland, for helpful comments on the manuscript.

REFERENCES

AHARONY, A., Time reversal symmetry violation and the H-theorem, Physics Letter 37A, 45-46 (1971).

BATTERMAN, R. W., Irreversibility and statistical mechanics: A new approach, Philosophy of Science 57,395-4/9 (1990).

BLUDMAN, S. A., Thermodynamics and the end of the closed universe, Nature 308,319-322 (1984).

DAUB, E. E., Maxwell’s demon, Studies in the Hist and Philosophy of Science 1, 213-227 (1970).

DOUGHERTY, J. P., Foundations of non-equilibrium statistical mechanics, Phil. Trans. R. SOC. Lond A 346,259-305 (1994).

GARRETT, A. J. M., Macroirreversibility and microirreversibility reconciled: the second law, in Maximum Entropy in Action, Eds. B. Buck and V. A. Macaulay (Oxford: Clarendon Press, 1991), p. 139-186.

HAWKING, S. W., The no-boundary proposal and the arrow of time, Vistas in Astronomy 37, 559-568 (1993).

LANDSBERG, P. T. (Ed.), International Conference on Thermodynamics, Cardiff, 1970, p. 543, remark by H. S. Robertson. See also Pure and Applied Chemistry 22, No. 3/4 (1970).

LANDSBERG, P. T., Thermodynamics, Cosmology and Physical Constants in The Study of Time ZZZ, Eds. J. T. Fraser, N. Lawrence and D. Park (New York: Springer, 1978), p. 115-138.

LANDSBERG, P. T., Mechanics via cosmology, in Mathematical Modelling Methodology, Models andMicros, Eds. J. S. Berry, D. N. Burghes, I. D. Huntley, D. I. G. James, A. 0. Moscardini (Chichester: Ellis Horwood, 1986) p. 142-148.

LANDSBERG, P. T., The Physical Concept of Time in the Twentieth Century, in Physics in the Making, Eds. A. Sarlemijn and M. J. Sparnaay, (Elesevier, 1989).

LANDSBERG, P. T., PIGOTT, K. D. and THOMAS, K. S., Many-cycle effects in irreversible oscillating universe models, Astro. Lett. and Communications 28,235-246 (1992).

LEPP, H. S., REX, A. F., Maxwell’s Demon (Bristol: Adam Hiker, 1990) NICOLIS G.. and PRIGOGINE I., Self-Urganhtion in Non-Equ&brium jystems (New York:

Wiley, 1977).


PENROSE, R., Time-Asymmetry and Quantum Gravity in Quantum Gravity 11. (Eds. C. J.

PRIGOGINE, I. and STENGERS, I., Order out of Chaos (New York: Bantam Books, 1984), p. 294. ROTHSTEIN, J., Physical Demonology, Mehodos 11. (no. 42), 99-119 (1959). SCHOLL, E., Non-equilibrium Transitions in Semiconductors (Berlin: Springer, 1987). SHALLIS, M., On Time (Penguin Books, 1983). SMOLIN, L., Does the universe envolve? Class. Quantum Grav. 9, 173-191 (1992). ZHANG, S., MEIER, B. H und ERNST, R. R., Polarization Echoes in NMR, Phys. Rev. Lett. 69,

Isham, R. Penrose and D. Sciama, Oxford University Press, 1981).

2149-2151 (1992).

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