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I read in a book that Newton's laws, the Schrödinger equation, the special and general theory of relativity, etc., make no distinction between the past and the future.

It then explains that the second law of thermodynamics does make that distinction, but later goes on to say that law is basically molecules bumping into each other, which happens according to the fundamental laws.

Then, does it follow that, at any point in time, if I could locate every particle in the universe and reverse its velocity, the universe would run "backward" and end up at the Big Bang?

AndyD
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What you are talking about is time asymmetry. Pierre Laplace produced a thought experiment called Laplace's Demon which stated essentially what you are talking about, that a vast intellect should be able to calculate the complete positions and velocities of all the particles of the universe.

When statistical mechanics was introduced, it basically eliminated this possibility by showing that it was statistically unlikely for the universe to revert to the big bang, this would be an extremely low entropy state, and due to the second law of thermodynamics, entropy must increase.

There are some issues with simply running everything backwards, take the magnetic field for example, if you took a charge and just reversed its direction, then its magnetic field would curl in the opposite direction (clockwise as opposed to counterclockwise) as opposed to curling in the same direction but backwards (clockwise to clockwise but translating backwards). This is given by the right hand rule.

There is also an issue with the amount of energy you would need to add to the universe in order to completely reverse every trajectory which is on an energetically favorable path at the moment.