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Loschmidt Paradox

Consider the time evolution of a system $N$ particles. In the $6N$-dimensional phase space, a point $(q_i, p_i)\equiv(q_1,p_1,q_2,p_2,\cdots,q_{3N},p_{3N})$ denote one microstate. Loschmidt argued that for a system starting from $(q_i(t_0),p_i(t_0))$ and evolving to $(q_i(t_f),p_f(t_f))$, if we reverse the momenta of all particles at $t_f$—i.e., if we can prepare a system described by $(q_i(t_0),-p_i(t_0))$—then the system should evolve back to the initial state at a later time because the laws of classical mechanics are time reversible. If true, this implies the amazing phenomenon of a gas spontaneously returning to a smaller volume from a larger volume.

Boltzmann solved this paradox by noting that:

An overwhelmingly large number of microstates are compatible with a macrostate, specified by a few state quantities like pressure, volume, and temperature. That means the initial state typically evolves to one of numerous possible microstates which cannot be distinguished macroscopically.
Moreover, all the microstates with same total energy can be found with equal probability.
That means the exactly one final state that allows the system to go back to the initial state is the actual final state can occur with a miniscule probability.
As a concrete example, the number of microstate of a system with $N$ particles is proportional to $V^N$. So if you halve the volume, the available microstates decrease by $1/2^N$. So, in the thermodynamic limit, the probability of a gas spontaneously collapsing to half of its volume is $1/2^N \rightarrow 0$.

Loschmidt Paradox

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