# What exactly is the physical picture of Time Reversal Symmetry

1. What's the exact meaning for time reversal symmetry in classical mechanics and quantum mechanics, respectively?
2. Why is it right for just single or just a few particles but wrong for many particles or macroscopic system?
3. And what's the exact number or some upper limit for "many particles"?

I mean the physical picture rather than just the mathematical details.

To illustrate this, consider an imaginary plane dividing a box into the left (L) and right (R) regions. If we have two particles traveling around in the box and subject to the laws of classical mechanics, we can easily find the particles (1,2) as (L,L), (R,R), (L,R) and (R,L). These are the microstates of the system. We can define the macrostates as (L), (R) and (M) which means, two particles in the left, two particles in the right and one particle each side. These macroscopic states are the ones cared by Thermodynamics and constrained by the Second Law. All microstates are equally probable but since there are two microstates corresponding to the macrostate (M) it is more probable that we observe the particles one each side of the box. Anyhow, it is pretty easy to find all the particles at one side. It means that spreading out from one side or congregating to it are "easily possible" (in a probabilistic sense) and therefore we say the system is reversible. However as we increase the number $N$ of particles, the number of microstates grows exponentially, $2^N$, whereas the number of microstates corresponding to the macrostates (L) or (R) does not change, it is one. Thus the probability of finding all particles to the left is $1/2^N$ - the same as obtaining $N$ heads after tossing a coin $N$ times. For a gas, with typically $10^{23}$ particles, this probability is nearly zero. The time we would have to wait in order to see all particles at one side would be possibly longer than the age of the Universe. In this sense, this system is not reversible - you will never see all the air in the room spontaneously going to the left side.