The Direction of Time
Prof. J. C. Baird
The enthalpy, or ``heat energy", has to do with the "energetics" of a process, the system going to states of lower ``energy," but it is just one of the factors in nature determining the direction of chemical change. We also intuitively know that systems tend to disorder-ice melts spontaneously at room temperature, or a gas separated from an evacuated volume by a valve expands into it when the valve is opened "never" to spontaneously return to the original volume. The word never is placed in quotes to indicate that the probability of all the gas molecules returning to the original volume, leaving an evacuated chamber, is very, very slight. The entropy is the measure of such spontaneous, natural tendencies to disorder.
If we look at the equations of dynamics [1] of both chemistry and physics we find that time reversal is possible. That is, we may run systems, like a movie, forwards and backwards. What then gives us our concept of time, both psychologically and in physics, as an ever increasing thing? Obviously, a clock is one thing [2] and the concept of time another. Does entropy, the measure of spontaneous tendencies, give time its direction?[3] Is time connected with entropy? Entropy is an equilibrium concept (a thermodynamic state function) defined at a particular state of the physical system. Going to the next equilibrium, physical state of the system may be a non-equilibrium process where entropy is not well defined in the classical thermodynamics context. But the change in the entropy may well indicate to us "the direction of time." Entropy is an equilibrium concept, but the beginning of the universe is a non-equilibrium system. Can the idea of entropy apply to the universe? Can we create a quantity called the "entropy production" and somehow relate this to time and its evolution? Does time depend on the existence of matter in the universe? Consider the early universe, just 10^-32 sec. after the big bang. Is there time? Does this implied definition of time have to do with evolution of elementary particles and matter? The causal laws of physics are symmetric in time and a direction in time must, we suppose, require many particles and therefore a statistical approach. Is time and its direction connected with complexity?
How about living systems and their chemical evolution?[4] The observed fossil record seems to show a "forward" direction. That is, species do not seem to evolve forward and then backwards.[5] How is this "irreversibility" of evolution applied to the individual evolutionary steps? Is the flow of time somehow involved?
In the August 11, 1995 Chronicle of Higher Education, page A9, is an article titled "Replaying `Life's Tape.'" The researchers, Richard Lenski and Michael Travisano at Michigan State University, are attempting to explore the impact of chance on evolution by growing 12 genetically cloned and identical populations of e-coli bacteria over a span of 10,000 generations starting in a new environment (i.e., a new type of food). The ability of the bacteria to reproduce varied among the twelve lines of descendants indicating random events are involved.
The paleontologist, Stephen J. Gould, asserts that "playing life's tape" over and over again would consistently lead evolution down a pathway radically different from the road actually taken."[6]
The idea may be that there are many physical paths from many starting points that lead to some final physical state. To go back from this final state to a particular starting point is of low probability due to the large number of potential paths leading to physically realizable starting states. [7] This may give time its direction and may be illustrated by Ehrenfest's Dog-Flea Problem.
We can see how the approach to equilibrium works by considering a simple yet realistic
model problem. [8] Consider two dogs, one containing 300
fleas numbered from 1 to 300 and the other without any. The dogs come close to each
other and the fleas start to jump every 
; seconds. The fleas jump randomly and this is determined
by drawing, randomly, their number (from 1 to 300). A number drawn means a flea jumps
to the neighboring dog. If d represents the number of times a drawing is made then
the total elapsed time is t = d x . The number of fleas on "Fido" is NFido and on "Whimsey"
is NWhimsey so that NFido + NWhimsey = 300 and NFido - NWhimsey = k = the difference
in the number of fleas between the two dogs. It is pretty clear that at the start
there are more fleas on Fido than on Whimsey and that there will be a greater transfer
to Whimsey from Fido. As time progresses and more draws are made some fleas on Whimsey
begin to jump back. After a time equilibrium is approached. A simple computer program
can be made to do this and the output is shown in the figure below.
If Fido had two fleas and one jumped to Whimsey then the next jump might be back
to Fido and the initial state would have occurred after some time related to the
jump time. On the other hand, if Fido starts out with 300 fleas then after some time
there are a lot of fleas on Whimsey and it is obviously unlikely that the original
state with 300 fleas on Fido and none on Whimsey will occur. It has been found that
the average recurrence time for a given number of fleas 2R and a given state n is
TR,n = ; (R +
n)!(R °ree; n)! (2R)! 2 2R with ; the jump time. For our case 2R = 2 and 2n = 2 T1,1 = 4; and T1,0 = 2; . For R = 150, n = 150 T150,150
= 2.03 £ 10 90 ;
While for R = 150, n = 0 T150,0 º 21.73; . For a state with a long recurrence time the process
will appear to be irreversible. If the recurrence time is small then irreversibility
has little meaning. We can see that for chemical and biological systems where the
number of states is very much larger than this example we can expect irreversibility.
Thus, the direction of time does seem to depend on complexity.
We can probe the nature of this problem by varying the number of fleas. We might start with five and then run a new game with ten to guage just how the number of states changes with the number of fleas.