# Achieving the minus degree, but in a hotter way

Last year in a post, I discussed interpretations of a few quantities in our physical world, viz. energy, time, and mass. At the end I asked the readers to think about how we should understand negative temperature in absolute scale.

Recently in the ultra cold atoms in an optical lattice, such a negative temperature has been realized. The work has been published in the prestigious journal Science. Undoubtedly this is a remarkable achievement, but such a realization is not for the first time! E. M. Purcell (The NMR Guru) and his team found negative temeparature in the nuclear spin systems in LiF.

Instead of going deep into these experiments, let’s discuss how conceptually
we can think of temperature going to become negative.

Suppose we have a $N$ number of non-intercating spins in a magnetic field $B$.

At absolute zero temperature we expect the lowest energy state, i.e. when all the spins are aligned in the direction of the magnetic field (say, down) and the total energy will be $E=-N \mu B$, where $\mu$ is the dipole moment due to each spin. In this state (we call the ground state) all spins are aligned and hence ordered. So the entropy becomes zero. Another way to argue the same by can be done by looking at the number of possible configurations: $\Omega$. Since all spin alignment is unique, $\Omega=1$, hence entropy: $S =k_B\ln\Omega=\ln 1=0$ (The formula is popularly known as Boltzmann’s eq., where $k_B=1.38\times 10^{23}$ J/K is the Boltzmann constant).

Now suppose we provide an extra energy $\mu B$ so that one spin can flip from down to up. Now the total energy will be $E=(-N+1)\mu B$. Now since there is a disorder due to one flip, we start gaining a finite entropy. Following the alternative argument, we can see that now $\Omega=N$ since each among the $N$ spins equally qualify to flip. Thus the entropy becomes finite, i.e. $S= k_B \ln N \ge 0$. Now if more energy is provided then the entropy will start increasing, say, for two spins to flip, number of configurations will be $^N C _2=N(N-1)/2$. Thus entropy will be $S=k_B \ln N (N-1)/2$. Thus entropy will progess as $S= k_B \ln N (N-1)(N-2)/3$, and so on as we flip more and more spins. However, this progress will reach to a maximum when half of spin becomes remain down and the half gets flipped (up), i.e. $S_\text{max}=k_B \ln ^N C _{N/2}$. This maximum entropy value can be also thought as the middle block of a Pascal’s pyramid representing the middle ($N/2$-th) coefficient of a binomial expansion. Since the entropy becomes maximum at this point (i.e. an extremal point), differential change of entropy with energy becomes zero: $\frac{\partial S}{\partial E}=0$. And following the thermodynamic relation: $1/T=\frac{\partial S}{\partial E}$, we get $T=\infty$.

But trust me, infinity is not The End! So far we have added energy to the system and it became hotter and hotter. And we may always get tempted to say, the system becomes hottest at $T=\infty$. But surprisingly, if we add a little more energy to the system, we will find that entropy changes its mind: it starts decreasing now, as our configuration count moves away from the middle of the Pascal’s pyramid. Since entropy change becomes negative (implying decreasing trend), we start getting negative temperature, starting from a negative infinity, just after the positive (see the Fig. below). And now you may leave your skepticism and say that this negative $T$ regime is far hotter (In a patriarch pun, one can make an analogy: Black girls are far hotter than the white girls).

Sorry as I could not discuss about the recent optical lattice experiment as I’m always afraid that cold atoms are not my regular cup of tea. However, the basic idea (decreasing entropy) is the same, and I must confess that the Fig. that I have put above was stolen from the Science paper.

References:

The Science article: Negative Absolute Temperature for Motional Degrees of Freedom by Braun et al.

A Nuclear Spin System at Negative Temperature by E. M. Purcell and R. V. Pound, Phys. Rev. 81, 279–280 (1951).