Recent we felt a disturbance of the force: It has been demonstrated that the absolute temperature of a real system can be pushed to negative values.
The interesting underlying question is: What is temperature really? Temperature seems to be an intuitive everyday concept, yet the explanations of ‘negative temperatures’ prove that it is not.
Actually, atoms have not really been ‘chilled to negative temperatures’. I pick two explanations of this experiment that I found particularly helpful – and entertaining:
- Less Than Absolute Zero by Matt Springer and
- What the Dalai Lama can teach us about temperatures below absolute zero by Aatish Bhatia.
As Matt points out The issue is simply that formally temperature is a relationship between energy and entropy, and you can do some weird things to entropy and energy and get the formal definition of temperature to come out negative.
Aatish manages to convey the fact that temperature is inversely proportional to the slope of the entropy vs. energy curve using compelling analogs from economics. The trick is to find meaningful economic terms that are related in a way similar to the obscure physical properties you want to explain. MinutePhysics did something similar in explaining fundamental forces (cannot resist this digression):
I had once worked in laser physics, so Matt’s explanation involving two-level system speaks to me. His explanation avoids to touch on entropy and thus avoids to use the mysterious term entropy to explain mysterious temperature.
You can calculate the probabilities of population of these two states from temperature – or vice versa. If you manage to tweak the population by some science-fiction-like method (creating non equilibrium states) you can end up with a distribution that formally results in negative temperatures if you run the math backwards.
[In order to allow for tagging this post with Physics in a Nutshell I need to state that the nutshell part ends here.]
But how come that ‘temperature’ ever became such an abstract concept?
From a very pragmatic perspective focussed on macroscopic, everyday phenomena temperature is what we measure by thermometers, that is: calculated from the change of the volume of gases or liquids.
You do not need any explanation of what temperature or even entropy really is if you want to design efficient machines, such as turbines.
As a physics PhD working towards an MSc in energy engineering, I have found lectures in Engineering Thermodynamics eye-opening:
As a physicist I had been trained to focus on fundamental explanations: What is entropy really? How do we explain physical properties microscopically? That is: calculating statistical averages of the properties of zillions of gas molecules or imagining an abstract ‘hyperspace’ whose number of dimensions is proportional to the number of particles. The system as such moves through this abstract space as times passes by.
In engineering thermodynamics the question to What is entropy? was answered by: Consider it some property than can be calculated (and used to evaluate machines and processes).
New terms in science have been introduced for fundamental conceptual reasons and/or because they came in handy in calculations. In my point of view, enthalpy belongs to the second class because it makes descriptions of gases and fluids flowing through apparatuses more straight-forward.
Entropy is different despite it can be reduced to its practical aspects. Entropy has been introduced in order to tame heat and irreversibility.
Richard Feynman stated (in Vol. I of his Physics Lectures, published 1963) that research in engineering contributed two times to the foundations of physics: The first time when Sadi Carnot formulated the Second Law of Thermodynamics (which can be stated in terms of an ever increasing entropy) and the second time when Shannon founded information theory – using the term entropy in a new way. So musing about entropy and temperature – this is where hands-on engineering meets the secrets of the universe.
I tend to state that temperature had never been that understandable and familiar:
Investigations of the behavior of ideal gases (fortunately air, even moist air, is an ideal gas) have revealed that there needs to be an absolute zero temperature – when the volume of an ideal gas would approach zero.
When Clausius coined the term Entropy in 1865
1850 (*), he was searching for a function that allows to depict any process in a diagram such as the figure above, in a sense.
(*) Edit 1 – Jan. 31: Thanks to a true historian of science.
Heat is a vague term – it only exists ‘in transit’: Heat is exchanged, but you cannot assign a certain amount of heat to a state. Clausius searched for a function that could be used to denote one specific state in such a map of states, and he came up with a beautiful and simple relationship. The differential change in heat is equal to the change in entropy times the absolute temperature! So temperature entered the mathematical formulations of the laws of thermodynamics when doing something really non-intuitive with differentials.
Entropy really seems to be the more fundamental property. You could actually start from the Second Law and define temperature in terms of the efficiency of perfect machines that are just limited by the fact that entropy can only increase (or that heat always needs to flow from the hotter to the colder object):
The more we learn about the microscopic underpinnings of the laws that have been introduced phenomenologically before, the less intuitive explanations became. It does not help trying to circumvent entropy by considering what each of the particles in the system does. We think of temperature as something as some average over velocities (squared). But a single particle travelling its path through empty space would not have temperature. Neither would any directed motion of a beam of particles contribute to temperature. So temperature is better defined as the mean deviation of a distribution of speeds.
Even if we consider simple gas molecules, we could define different types of temperature: There is a kinetic temperature calculated from velocities. In the long run – when equilibrium has been reached – the other degrees of freedom (sich as rotations) would exhibit the same temperature. But when a gas is heated up, heat is transferred via collisions: So first the kinetic temperature rises, and then the energy is transferred to rotations. You could calculate a temperature from rotations, and this temperature would be different from the kinetic temperature.
So temperature is a property that is derived from what an incredible number of single particles do. It is a statistical property and it makes only sense when a system had enough time to reach an equilibrium. As soon as we push the microscopic constituents of the system that makes them deviate from their equilibrium behaviour, we get strange results for temperature – such as negative values.
This post was also inspired by some interesting discussions on LinkedIn a while ago – on the second law and the nature of temperature.
(*) Edit 2 – Feb. 2: Though Clausius is known as the creator of the term entropy, the concept as such has been developed earlier by Rankine.