thermodynamics - Irreversible expansion and time reversal symmetry

Suppose there are N non-interacting classical particles in a box, so their state can be described by the $\{\mathbf{x}_i(t), \mathbf{p}_i(t) \}$. If the particles are initially at the left of the box, they can eventually occupy the whole box according to the Newtons law. In this case, we call the gas expand and this process is irreversible.

Nevertheless, Newtons laws also have time-reversal symmetry, so we should be able to construct an initial condition such that particles occupy the whole box (i.e. not all particles in the left chamber) will all move to the left chamber. Here are the simple questions:

1. 
   

   How to select the initial condition $\{\mathbf{x}_i(0), \mathbf{p}_i(0) \}$ if you can solve the set of equation of motion.

   
   


2. 
   

   Why the gas expansion is irreversible even though you can select the initial condition above.

   
   

Answer

1/ OK, let's start from an initial condition where all the particles are made to fit a tiny little corner of the room and their initial velocities are chosen randomly, according to a Maxwell-Boltzmann distribution for instance. As we let the system evolve, the gas will expand, that is true because it corresponds to the behaviour the Maxwell-Boltzmann predicts, that of an ideal gas near equilibrium and therefore the gas will try to fill the room.

Now, suppose at some moment we stop the evolution of the system, for instance after the gas has filled the entire room. Now, if we reverse all the velocities of the particles but do not change their positions, it is clear that by construction, we have found a solution that will evolve backwards to the corner where the gas was originally. So, it is indeed possible to find an initial configuration and in fact several initial configurations that satisfy your requirement. (But it takes a Maxwell Demon to realize them in practice.)

2/ Now, why doesn't that invalidate the macroscopic irreversibility of the system? Well, we should look at how many microstates corresponding to the macrostate "box entirely filled" do go back to the corner. Say the box has volume $V$. Now, assuming we have $N$ particles, the total number of microscopic configurations compatible with the box filled will be proportional to $V^N$. (I am omitting the velocities in this analysis, they will only make the analysis less simple to follow.)

How many of these configurations go back to the corner? Well, to compute that, remember that we can easily construct the solutions which go back to the corner by taking the configurations that start from the corner and expand to fill the entire room and then reverse the velocities. So, let's say we call the volume of the corner $W$. So the amount of trajectories that go back to the corner will be proportional to $W^N$. So, the fraction of trajectories that go back to the corner among the trajectories that fill the entire volume will be $W^N/V^N$. Since $W/V<1$ and $N$ is a large number, of the order of the Avogadro number, you can see why you never actually observe these trajectories in real life.

But, if you make your system small enough, less than 10 particles, and your corner is a half of the room, then the probability is about $1/2^{10} \approx 0.001$, I'd say it is worth the wait. ;p

OK, I wanted to add something to this explanation. The apparant paradox that is proposed in the OP is originally attributed to Loschmidt. The paradox Marek is talking about which is related to the ergodicity of the system is due to Zermelo, they are different paradoxes and require different answers.