Hamilton's equations of motion

are invariant under a reversal of the direction of time . Under such a transformation, the positions and momenta transform according to

Thus,

or

The form of the equations does not change! One of the implications of
time reversal symmetry is as follows: Suppose a system is evolved
forward in time starting from some initial condition up to
a maximum time *t*; at *t*, the evolution is stopped, the sign of the
velocity of each particle in the system is reversed, i.e., a time reversal
transformation is performed,
and the system
is allowed to evolve once again for another time interval of length *t*;
the system will return to its original starting point in phase space, i.e.,
the system will return to its initial condition. Now from the point of
view of mechanics and the microcanonical ensemble, the initial conditions
(for the first segment of the evolution) and the conditions created
by reversing the signs of the velocities for initiating the second
segment are equally valid and equally probably, both being points
selected from the constant energy hypersurface. Therefore, from the
point of view of mechanics, without *a priori* knowledge of which
segment is the forward evolving trajectory and which is the
time reversed trajectory, it should not be possible to say
which is which. That is, if a movie of each trajectory were to be
made and shown to an ignorant observer, that observer should not
be able to tell which is the forward-evolving trajectory and which
the time-reversed trajectory. Therefore, from the point of view
of mechanics, which obeys time-reversal symmetry, *there is not
preferred direction for the flow of time*.

Yet our everyday experience tells us that there are plenty of situations in which
a system seems to evolve in a certain direction and not in the reverse
direction, suggesting that there actually is a preferred direction in time.
Some common examples are a glass falling to the ground and smashing into
tiny pieces or the sudden expansion of a gas into a
large box. These processes would *always* seem to occur in
the same way and never in the reverse (the glass shards never reassemble themselves
and jump back onto the table forming an unbroken glass, and the gas particles
never suddenly all gather in one corner of the large box). This seeming
inconsistency with the reversible laws of mechanics is known as
*Loschmidt's paradox*. Indeed, the second law of thermodynamics,
itself, would seem to be at odds with the reversibility of the laws
of mechanics. That is, the observation that a system naturally
evolves in such a way as to increase its entropy cannot obviously
be rationalized starting from the microscopic reversible laws of motion.

Note that a system being driven by an external agent or field will not be in equilibrium with its surroundings and can exhibit irreversible behavior as a result of the work being done on it. The falling glass is an example of such a system. It is acted upon by gravity, which drives it uniformly toward a state of ever lower potential energy until the glass smashes to the ground. Even though it is possible to write down a Hamiltonian for this system, the treatment of the external field is only approximate in that the true microscopic origin of the external field is not taken into account. This also brings up the important question of how one exactly defines non-equilibrium states in general, and how do they evolve, a question which, to date, has no fully agreed upon answer and is an active area of research. However, the expanding gas example does not involve an external driving force and still seems to exhibit irreversible behavior. How can this observation be explained for such an isolated system?

One possible explanation was put forth by Boltzmann, who introduced the notion of
*molecular chaos*. Under this assumption, the momenta of two particles
do not become correlated as the result of a collision. This is tantamount to
the assumption that microscopic information leading to a correlation between
the particles is lost. This is not inconsistent with microscopic reversibility
from a probabilistic point of view, as the momenta of two particles
before a collision are certainly uncorrelated with each other.
The assumption of molecular chaos allows one to prove the so called
Boltzmann *H-theorem*, a theorem that predicts an increase in entropy
until equilibrium is reached. For more details see Chapters 3 and 4
of Huang.

Boltzmann's assumption of molecular chaos remains unproven and may or may not
be true. Another explanation due to Poincaré is based on his
*recurrence theorem*. The Poincaré recurrence theorem states that
*a system having a finite amount of energy and confined to a finite spatial
volume will, after a sufficiently long time, return to an arbitrarily small
neighborhood of its initial state.*

**Proof of the recurrence theorem** (taken almost directly from Huang, pg. 91):

Let a state of a system be represented by a point in phase space. As the
system evolves in time, a point in phase space traces out a trajectory that
is uniquely determined by any given point on the trajectory (because of the
deterministic nature of classical mechanics). Let be an arbitrary
volume element in the phase space in the volume . After a time
*t* all points in will be in another volume element
in a volume , which is uniquely determined by the choice of .
Assuming the system is Hamiltonian, then by Liouville's theorem:

Let denote the subspace of phase space that is the union of all for . Let its volume be . Similarly, let denote the subspace that is the union of all for . Let its volume be . The numbers and are finite because, since the energy of the system is finite and the spatial volume occupied is finite, a representative point is confined to a finite region in phase space. The definitions immediately imply that contains .

We may think of and in a different way. Imagine the region to be filled uniformly with representative points. As time progresses, will evolve into some other regions that are uniquely determined. It is clear, from the definitions, that after a time , will become . Also, by Liouville's theorem:

Recall that contains all the future destinations of the points in , which in turn is evolved from after a time . It has been shown that has the same volume as since by Liouville's theorem. Therefore, and must contain the same set of points (except possibly for a set of zero measure).

In particular, contains all of (except possibly for a set of zero measure). But, by definition, all points in are future destinations of the points in . Therefore all points in (except possibly for a set of zero measure) must return to after a sufficiently long time. Since can be made arbitrarily small, Poincaré's theorem follows.

Now consider an initial condition in which all the gas particles are
initially in a corner of a large box.
By Poincaré's theorem, the gas particles must eventually return to their initial state
in the corner of the box. How long is this recurrence time? In order to answer this,
consider dividing the box up into *M* small cells of volume *v*. The
total number of microstates available to the gas varies with *N* like
. The number of microstates corresponding to all the
particles occupying a single cell of volume *v* is . Thus, the probability of
observing the system in this microstate is approximately .
Even if *v*=*V*/2, for , the probability is vanishingly small.
The Poincaré recurrence time, on the other hand, is proportional to the
inverse of this probability or . Again, since ,
if *v*=*V*/2, the required time is

which is many orders of magnitude longer than the current age of the universe. Thus, although the system will return arbitrarily close to its initial state, the time required for this is unphysically long and will never be observed. Over times relevant for observation, given similar such initial conditions, the system will essentially always evolve in the same way, which is to expand and fill the box and essentially never to the opposite.

Wed Jan 26 09:29:17 EST 2000