# Can a system have negative entropy?

We know that the entropy is zero for reversible processes and always positive for irreversible processes. Can there exist a system which may have negative entropy?

• I think you are talking about entropy change in a process, correct? – Chet Miller Aug 4 '17 at 14:42
• Why entropy of the whole system is zero or greater than zero but not negative? – Pooja Aug 5 '17 at 2:41
• What do you define as the"whole system?" – Chet Miller Aug 5 '17 at 3:04
• Means our object of interest – Pooja Aug 5 '17 at 3:29
• like the entropy of universe is always increasing but why ? – Pooja Aug 5 '17 at 3:31

The entropy $S$ of a system is related to the number of possible microstates $\Omega$ that a system can adopt in the following way:

$$S = k_B \log \Omega$$

Note that $\Omega$ must always be an integer, and it must always be at least 1; hence, $S$ is always greater than or equal to zero.

In the zero-entropy case, the object is a perfect crystal at zero temperature, which has only one possible microstate. (Thus, the above definition is made possible by the Third Law of Thermodynamics.) Any other situation has more than one possible microstate, so the entropy must be greater than zero.

• Can we corelate the entropy with randomness – Pooja Aug 4 '17 at 8:47
• Assuming a uniform probability distribution of the system being in any one microstate, then yes, the approximate "randomness" of a system is related to its total number of microstates, and thus to the entropy. – probably_someone Aug 4 '17 at 8:48
• We know what happened at absolute zero but what will happen below 0K – Pooja Aug 4 '17 at 8:58
• It depends on your definition of temperature. If you relate it to average particle kinetic energy, then it is impossible, as kinetic energy is always positive. If you define temperature as 1/(the amount of entropy added to the system when a given amount of energy is added), then negative temperatures are possible in systems that become more orderly (i.e. have fewer microstates) when energy is added. Most practical examples of such systems are generally quite hot, though, so this notion of temperature is somewhat non-intuitive. – probably_someone Aug 4 '17 at 9:10
• If an upper limit exists on the amount of energy a particle can have, then adding energy to a system pas a certain point serves to pack more and more particles into (for bosons) the highest energy state, or (in the case of fermions) the highest available energy state. A bunch of indistinguishable degenerate particles (in the case of bosons; in the case of fermions, a bunch of indistinguishable particles who are essentially locked in one energy state) is much less random than a bunch of particles which have many possible energy states. Thus, higher-energy states have less entropy. – probably_someone Aug 4 '17 at 9:24

I think what you mean is that the entropy doesn't change for reversible processes, but increases for irreversible processes. In this sense your question would be if the entropy of a system can decrease. Yes, absolutely! The entropy can decrease for a system that is not closed. For example, Earth receives the solar energy prom the Sun and dissipates in into space as heat. The entropy of the whole (closed) system (Sun, Earth, and space) always increases. However, the entropy on Earth alone can indeed decrease. Entropy is often referred to as a measure of chaos, so order would be the opposite of entropy. In this sense biological life and evolution represent a highly organized matter and therefore a low entropy. Such a reduction of the entropy as the emergence of life and its evolution on Earth was possible exactly because Earth alone is not a closed system, but a conduit of a tremendows entropy increase of the solar energy dissipating as heat. Without this constant entropy increase life on Earth would be impossible. It is exactly the entropy increase in the entire system that allowed the entropy in the part of the system to decrease thus producing life, evolution, and ultimately intelligence.

• Even in a closed system, the entropy can decrease. Just remove heat from a body, for example. – Chet Miller Aug 4 '17 at 14:45
• @Chester Miller: Could you please provide a link or reference to the idea that the entropy of a closed system can decrease? – safesphere Aug 4 '17 at 17:45
• Well, every thermodynamics text book has the equation $dS=dq_{rev}/T$. What would you conclude if I told you that $dq_{rev}$ is negative for a particular process (such as isothermal compression of an ideal gas or cooling a solid)? – Chet Miller Aug 4 '17 at 19:22
• @Chester Miller: Your examples are not closed systems and they do not answer my question. I am not asking for ideas or conclusions. I am asking if you can provide a reference specifically stating that "the entropy of a closed system can decrease". The reason I am asking is that such a system would violate the law of the entropy increasing in a closed system and I haven't heard of any violations of this law. So if you have any actual reference (other than your own deductions), I'd be interested to learn. – safesphere Aug 4 '17 at 19:46
• I think we have a terminology issue here. When a physicist talks about a closed system, what he means is one in which there is no exchange of mass, heat, or work with the surroundings; this is what we engineers call an isolated system. In engineering (and most thermo books), a closed system is one in which there is no exchange of mass with the surroundings; exchange of heat and work are allowed. See the following link: google.com/… – Chet Miller Aug 4 '17 at 20:06

Yes. Reverse the velocity of all the particles in the universe and entropy will only go down.

https://youtu.be/yRvbEoHHx4M?t=39m14s

• Can you describe it more clearly? – Pooja Aug 5 '17 at 3:36
• [link] (youtu.be/yRvbEoHHx4M?t=36m42s) – Scott Weinblatt Aug 5 '17 at 3:46
• Interestingly, this is incorrect. The law of the entropy increase is a theoretical result of the statistical physics. It is a well known law, but its other side is virtually unknown. In classical physics time is reversible. This leads to an almost shocking conclusion: if you reverse time (essentially the essence of your suggestion), the entropy would still increase. The law of the entropy increase does not state that the entropy increases from the past to the future. It states that the entropy will increase from the present to anywhere you go, the future or the past. – safesphere Aug 5 '17 at 4:38
• @safesphere Then why did the past have lower entropy then now? Are you suggesting that the past doesn't even exist? – Scott Weinblatt Aug 5 '17 at 4:59
• @Scott Weinblatt: The entropy is lower in the past, because we are moving from the past to the future and the statistical physics requires the entropy to increase. This becomes interesting IF you reverse the direction of time, such as (simplified) by reversing the direction of all moving particles, as Scott suggested above. You'd think intuitively that everything would just go backwards and the entropy would decrease. This is not true. Things would get messed up (e.g. by quantum fluctuations or whatever) and the entropy would still increase. So entropy increases for anti-matter as well. – safesphere Aug 5 '17 at 5:49

## protected by Qmechanic♦Aug 5 '17 at 5:11

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).