# Does entropy decrease if time is reversed?

Entropy increases if we let newton's equation work its magic.

Since newton's equation is time reversible, I would assume that in a closed isolated system, solving the differential equation and running time backwards would increase (and NOT decrease) the entropy of the system. Is that true?

• No. As far as I understand the arrow of time (if indeed time actually exists) it is the direction of increasing entropy. If in a closed isolated system (which does not exist except theoretically) if you managed to reduce entropy you would actually, for that system, have achieved a time reversal. Or if you managed to run time backwards, the entropy would indeed reduce. – PaulD Sep 10 '17 at 21:24

Simple answer: in our universe, definitely no.

You're hitting here on an idea known as Loschmidt's Paradox: given that microscopic laws are time reversible, entropy should have the same tendency to increase whether we run a system forwards or backwards in time, exactly as you understand.

The fact that this understanding is manifestly against experimental observation can be explained if we observe that the universe began (i.e. found itself at the time of the big bang) in an exquisitely low entropy state, so that almost any random walk in the universe's state space tends to increase entropy. Likewise, in the everyday world, things "happen" when a systems are not in its maximum entropy state: they spontaneously wander towards these maximum entropy states, thus changing their states and undergoing observable changes. Sir Roger Penrose calls this notion the "Thermodynamic Legacy" of the big bang and you could read the chapter entitled "The Big Bang and its Thermodynamic Legacy" in his "Road to Reality". In summary, we have a second law of thermodynamics simply by dint of the exquisitely low entropy state of the early universe.

 Loschmidt's own name for it is "reversal objection" (umkehreinwand), not "paradox". Paradoxes, i.e. genuine logical contradictions cannot arise in physics, otherwise they could not be experimentally observed.

Suppose we measure the entropy change of a spontaneous process that takes place within an isolated system. We find that the entropy increases.

Spontaneous processes are regarded as irreversible in classical physics. This means that once a spontaneous process has come to its end, it never goes back to the initial state. If it were to go back then changes in thermodynamic state functions would reverse, including entropy. That answers your question.

Following the work of Boltzmann it was realised that the second Law of thermodynamics is really only statistical in nature. Think of a bottle of gas. The bottle is opened to the laboratory and the gas spreads out. It has never been observed that the gas goes back into the bottle and we might be tempted to say that this will never happen and that it cannot happen. However, imagine that there are only three molecules in the bottle. It is now not impossible that we would observe the three molecules jumping back into the bottle, if we were patient. However, if there are millions of molecules (or more) in the bottle it would be statistically so unlikely that this reversal would take place that we would never ever see. If our three molecules did jump back into the bottle, the entropy change would be reversed but time would not change. Time would carry on moving forward.