# Can entropy be equal to zero?

I've searched for it but I only found contradicting answers from "scientists":

Dr. David Balson, Ph.D. states: "entropy in a system can never be equal to zero".

Sam Bowen does not refutes the following affirmation: "It is know[n] that entropy is zero when a pure crystalline substance is at absolute zero". He says there's residual motion in particles, but he doesn't say clearly that 0 entropy isn't possible.

I this link is a long discussion about that, it isn't really conclusive, but most people seem to agree that entropy can be zero.

I'm not very receptive to the idea of zero entropy.

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If we use the definition of entropy $$S=-k_B \sum_i P_i ln P_i$$ where $P_i$ is the probability of the i'th microstate, then at 0K, we know the system is certainly in the ground state, $P_0=1$ so the definition returns zero.

The only "residual motion" in the system at this state is that due to the uncertainty principle in the ground state, but this does not change the fact that we know for certain which state the system is in, and therefore the entropy is zero.

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You say "at 0K" the entropy is 0. The article however says the reason entropy can never be 0 is becuase 0K can never be reached. – Mew Dec 23 '12 at 15:08
Oh yes, that's absolutely right in realizable systems. My answer only applied to the idealized case at 0k, since I assumed that the OP's "residual motion" was referring to zero point fluctuations. – twistor59 Dec 23 '12 at 15:18
Yep I see. I would up-vote your answer but I have ran out of votes for the day. – Mew Dec 23 '12 at 15:21
The temperature need not be zero. As long as $kT$ is significantly smaller than the energy spacing between the ground state and the first excited state, $S = 0$. – Johannes Dec 23 '12 at 15:42
@emarti: at the (nondegenerate) ground state of a system, that partial derivative is ill-defined, since $U$ is at a minimum because it's hit the edge of its domain. – Jerry Schirmer Dec 11 '13 at 6:43

I don't see a disagreement: both authors seem to agree that entropy goes to zero as temperature does. At finite temperature, the entropy is always nonzero. You'll find that this is true by the definition of temperature (or entropy).

The much more interesting question is: when does entropy NOT go to zero when temperature goes to zero? For that to be possible, the number of configurations at zero temperature must be extrinsic (order $N$). These systems are called frustrated and show very unusual thermodynamic behavior. In quantum systems things are much stranger, where the usual example is the spin liquid. I'm not sure whether quantum spin liquid recently observed by Young Lee's group at MIT is an example, but it might be.

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At finite temperature the entropy can be zero. The problem is that classical thermodynamics does not work very well at low temperatures. Why? Because it assumes that the energy is continuous but it isn't near absolute zero. So as Johannes pointed out, one can obtain zero entropy with a small but non-zero temperature. – Paul J. Gans Dec 31 '12 at 3:30

## protected by Qmechanic♦Feb 9 '14 at 17:11

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