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
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.