# What is Hotter than Infinity?

## What is Temperature?

In simple terms, the temperature is a measure of the degree of hotness or coldness of some object or a body. It is an integral part of our daily life, and a lot of crucial processes depend on the temperature. Our refrigerators maintain a temperature close to 4 degrees in order to preserve food. Air conditioners maintain a steady temperature to make our rooms and indoors a bit more hospitable in extreme weather conditions.

However, the two terms ‘hot’ and ‘cold’ don’t really give us much to work with. A big part of science is numbers. Hence, we arrive at a more formal definition of temperature: the Thermodynamic temperature. It is what we call absolute temperature and is defined as:

“Temperature is the measure of random submicroscopic oscillations and vibrations of the fundamental constituents of matter”

It is an absolute scale of temperature and the lowest point is defined with the help of the Third Law of Thermodynamics. This law, in simple terms, states that the ‘entropy’ or the disorder of a perfectly crystalline substance is zero at absolute zero, the 0 Kelvin. This definition along with the law sets a limit: there cannot be a temperature lower than zero kelvin; simply because the entropy cannot be negative.

## An Even More Refined Approach(?)

As it turns out, the thermodynamic scale of temperature isn’t exactly best suited for our purposes. The scale has been defined using the triple point of water, basically 0 degrees celsius or 273.16 Kelvin and hence it is a bit impractical to use it for temperatures that are farther away from this range.  Apart from this, the Thermodynamic definition of temperature implies that the macroscopic concept of temperature is purely related to the microscopic movements of atoms and molecules.

A different approach was devised by Boltzmann, who gave the relationship between probabilities of gas molecules being in a certain state to the Energy and Temperature of the system. It is given as:

Pi ∝exp(-Ei/kT)

Here, Pi is the probability of the molecules being in an energy state Ei and k is the Boltzmann Constant. This relationship is a part of the Kinetic Theory of Gases and is usually considered accurate for a big domain of science.

If we look at this equation, we can see that as the temperature increases, the probability of molecules being in the energy state becomes less and less; since the molecules begin to occupy higher energy states. Since this also implies that at higher temperatures, molecules occupy higher energy states; it also implies that the molecules are likely to emit certain wavelengths and transition to a lower state.

The disparity between the two definitions of temperature, and the fact that Boltzmann’s definition holds up for a better part of science, we are able to achieve some really interesting configurations of the system. One of them is the following.

## Negative Temperatures

By intuition, and also through science, we can see that the absolute zero is unattainable, and so are the temperatures lower than that. But, as it turns out, the disparity in the definitions actually gives us a way to approach the sub-zero temperatures on the Kelvin scale as well.

A higher percentage of molecules tend to stay in a higher energy state at higher temperatures, and vice-versa. This is the fact that we use it when approaching the negative temperatures. Looking back at the Boltzmann relation, if we change the temperature to the negative region, the relationship effectively modifies to:

Pi ∝exp(Ei/kT)

If the temperature goes up, towards negative infinity, the probability of molecules existing in higher energy levels is higher than molecules existing in a lower state.

## Experimental Setup

The setup consists of a system of lasers that is focussed on the cloud of atoms (or an optical lattice, to be precise) and a potential well is created through the interference of the two beams. The potential initially looks something like the following:

The majority of the molecules are currently trapped at the bottom of this well, somewhere around the origin. This is a characteristic low temperature state at this moment.

Then, in a quick series of adjustments and maneuvers, the potential is quickly changed to something that looks like this:

The majority of atoms are hence all of a sudden trapped at a higher energy state and this is the state of a Negative Absolute Temperature. A larger number of molecules are at a high temperature.

It is noteworthy to understand how this is not just another high energy/high-temperature state; or not a simple flaw in the theory. This is a perfectly valid construct. We can see that if we write the Boltzmann distribution for this system. The energy of the system is not much different before and after changing the potential. We have merely changed the distribution of atoms in a way such that we achieve a configuration in accordance with higher temperature states.

It is also noteworthy that if we bring another system near this particular setup, heat will flow from this system to the second system that we brought in. It gives rise to the counter-intuitive statement that the Negative Temperatures are hotter than infinity.

## What does this all mean?

This idea that we actually warped back to the negative region by traversing through infinite temperature zones is what makes this experiment a wonderful thought experiment as well. Something less than zero can actually be greater than infinity. If we stick to the thermodynamic definition for a moment, we can quickly see that here we have a system that clearly defines the predictions made by the older definitions of temperature.

This, however, does not make the Thermodynamic ideas wrong. All it implies is that all the theories that we have currently devised are bound to a certain domain: if we start to transcend those limits, we start obtaining counterintuitive and incorrect results.

Science