# Negative absolute temperatures

It’s a famous result in thermodynamics that you can’t reach absolute zero no matter how hard you try. Also the definition of absolute zero makes it obvious that its meaningless to talk about colder temperatures. So, really, what are negative absolute temperatures? Just continue reading because here I’m going to make an attempt in answering that question.

##### Temperature

At first its good to have a reminding discussion about the concept of temperature itself. In thermodynamics, temperature is that quantity which is the same between two systems in thermal equilibrium i.e. two systems which aren’t isolated from each other but don’t exchange heat. But this definition won’t be useful here because of its thermodynamical nature. Thermodynamics treats macroscopic systems regardless of their microscopic structure (that’s why you don’t need to change thermodynamics because of QM) but here we’ll have to work with the microscopic point of view. So we should consider other definitions of temperature. One definition that we’ll find useful, is through the equation ## \frac 1 T \equiv \frac{d S}{dE} ## where S is the system’s entropy function and E is its internal energy. Also it’ll be sometimes useful to consider the association of temperature with the average energy of the constituents of the system distributed among their degrees of freedom.

##### The Boltzmann factor

Now let’s say we want to talk about the statistical mechanics of a system of particles. From QM, we know that each particle can only be in discrete energy levels and can have no energy in between. The relative number of particles occupying the energy levels ## \varepsilon_1 ## and ##\varepsilon_2##, is given by the Boltzmann factor ## \frac{n_2}{n_1}= \exp\left(-\frac{\varepsilon_2-\varepsilon_1}{kT}\right)##. In systems where there is an infinite number of energy levels (we always take it for granted that there is a ground state which bounds the energy from below, so the “energy ladder” actually continues to infinity only upwards), as particles gain more energy, they climb up the ladder and theoretically, this continues as long as you keep injecting energy to the system. There can be no equilibrium state in which the number of particles in a given energy state, is smaller than that of an energy level above it. Combining this fact with the form of Boltzmann factor given above, we understand that in such systems the temperature can only be positive.

##### A simple model

But what about a system with a finite number of energy levels? For simplicity, I’m going to consider a system with two energy levels ##\varepsilon_1## and ##\varepsilon_2## and a constant number of particles n. So we have ## n=n_1+n_2 ## and ##\varepsilon=n_1\varepsilon_1+n_2\varepsilon_2 ##. Note that if we assume the energy to stay constant too, there will be no dynamics because the total energy and the number of particles will uniquely determine the state of the system. So the system is assumed only to have a constant number of particles but it can exchange energy with its surroundings. Now using the definition of entropy (##S=k \ln \Omega ##) and noting that ## \Omega=\frac{n!}{n_1!n_2!} ##, with a little bit of algebra along with Stirling’s approximation, we can find that

## S=k \left[ n \ln n-\frac{\varepsilon-n\varepsilon_1}{\varepsilon_2-\varepsilon_1}\ln{(\frac{\varepsilon-n\varepsilon_1}{\varepsilon_2-\varepsilon_1})}+\frac{\varepsilon-n\varepsilon_2}{\varepsilon_2-\varepsilon_1} \ln{(-\frac{\varepsilon-n\varepsilon_2}{\varepsilon_2-\varepsilon_1})} \right] ##.

Unlike the usual entropy functions where the increase in energy will always result in an increase of entropy, this function has a maximum at ## \varepsilon=\frac n 2 (\varepsilon_1+\varepsilon_2) ## which shows that regardless of the amount of internal energy of the system, the dynamics will lead it to a point where the two energy levels have an equal number of particles in them. Now considering the Boltzmann factor, we find that this means the equilibrium temperature of this system is ## T=\pm \infty ## (the reason for ##\pm## will become clear in a moment). Now lets see what happens when we consider the temperature of this system. Taking the derivative of the above function w.r.t. energy, we find the temperature of the system to be ## T= \frac{\varepsilon_2-\varepsilon_1}{k} \left[\ln\left(- \frac{\varepsilon-n\varepsilon_2}{\varepsilon-n\varepsilon_1} \right) \right]^{-1} ##. This function tells us that the temperature is positive for ## \varepsilon<\frac n 2 (\varepsilon_1+\varepsilon_2 ) ## and negative for ## \varepsilon>\frac n 2 (\varepsilon_1+\varepsilon_2 ) ##. But because the equilibrium energy is ## \varepsilon=\frac n 2 (\varepsilon_1+\varepsilon_2) ##, the dynamics of the system will lead it from positive temperatures to ## T=+\infty ## and from negative temperatures to ## T=-\infty ##. Note that the system has no internal dynamics and its only the interaction with the environment that is leading it to equilibrium. But of course its only a property of this simple model. You can do the same calculations for a three level system which will be more interesting.

##### Negative absolute temperatures

From the simple model discussed above, its easy to see that negative temperature states are actually states with higher energy than positive temperature states. Also Boltzmann factor shows us that in such states, there are more particles in higher energy levels than the lower ones. What actually happens is that as the particles gain energy and climb up the ladder, at some point, they reach the highest energy level and there is no higher energy to climb to. So any energy added further, just takes more particles from lower levels to higher levels. At some point the number of particles in different levels may become equal which means the temperature is infinity. Note that there is no difference between plus or minus infinity and we should consider these two to be identical. If energy is again injected, the temperature rises from ## -\infty ## and goes towards 0 from the negative side but again 0 itself can’t be reached. So if we want to have a absolute temperature axis ordered from coldest to hottest, we should have something like below:

## 0^+ , 1 , 2 , … , 273.15 , … , + \infty , -\infty , … , – 273.15 , … , -2 , -1 , 0^- ##

From the above information, it seems more natural to use ## \beta \equiv \ – \frac{1}{kT} ## as temperature. Because it takes both infinities to a single point(##\beta=0##) and also sorts out different temperatures in order of increasing energy. Another advantage is that this way, because ## 0^+ ## goes to ## -\infty ## and ## 0^- ## goes to ## +\infty ##, you can get the two zeroes as far as possible from each other which is the right way of thinking about them. Also this way, its easier to think that you can’t reach these temperatures.

##### Experimental realization

Now that we have some good theoretical knowledge about negative absolute temperatures, its time to talk about the experimental side. How can we have systems with a finite number of energy levels in lab? Experimentalists use a trick to acquire such systems. The point is, although the total energy of any system forms a ladder which is only bounded from below, different degrees of freedom have different energy ladders. There are some degrees of freedom which actually have upper bounds too, like spin degrees of freedom which will have different energies in the presence of a magnetic field. Now what experimentalists do, is that they prepare the situation for the system somehow that the energy exchange between the doubly-bounded degree of freedom and other degrees of freedom becomes so much slower than the dynamics of the doubly-bounded degree of freedom itself. Also the dynamics of other degrees of freedom become slower too. So in this way, they can think of the dynamics of the system being completely due to the doubly-bounded degree of freedom.

After this discussion, you may now expect me to talk about the modifications that we need to make to thermodynamics(specially the 2nd law) because of the existence of negative absolute temperatures. Of course you can find some sources which actually do that. But here I don’t want to do this because as far as I know, physicists still can’t place such systems in a heat engine along with more conventional systems and so I prefer not to speculate about this issue.

Interesting topic! Nice first entry!

Interesting article, Shyan.

Dr. Ulrich Schneider wrote some years ago this interesting article (published by Max Planck Society):"A temperature below absolute zero – Atoms at negative absolute temperature are the hottest systems in the worldJanuary 4, 2013What is normal to most people in winter has so far been impossible in physics: a minus temperature. On the Celsius scale minus temperatures are only surprising in summer. On the absolute temperature scale, which is used by physicists and is also called the Kelvin scale, it is not possible to go below zero – at least not in the sense of getting colder than zero kelvin. According to the physical meaning of temperature, the temperature of a gas is determined by the chaotic movement of its particles – the colder the gas, the slower the particles. At zero kelvin (minus 273 degrees Celsius) the particles stop moving and all disorder disappears. Thus, nothing can be colder than absolute zero on the Kelvin scale. Physicists at the Ludwig-Maximilians University Munich and the Max Planck Institute of Quantum Optics in Garching have now created an atomic gas in the laboratory that nonetheless has negative Kelvin values. These negative absolute temperatures have several apparently absurd consequences: although the atoms in the gas attract each other and give rise to a negative pressure, the gas does not collapse – a behaviour that is also postulated for dark energy in cosmology. Supposedly impossible heat engines such as a combustion engine with a thermodynamic efficiency of over 100% can also be realised with the help of negative absolute temperatures.Hot minus temperatures: At a negative absolute temperature the energy distribution of particles inverts in comparison to a positive temperature. Many particles then have a high energy and few a low one. This corresponds to a temperature which is hotter than one that is infinitely high, where the particles are distributed equally over all energies. A negative Kelvin temperature can only be achieved experimentally if the energy has an upper limit, just as non-moving particles form a lower limit for the kinetic energy at positive temperatures –physicists at the LMU and the Max Planck Institute of Quantum Optics have now achieved this. Zoom ImageHot minus temperatures: At a negative absolute temperature the energy distribution of particles inverts in comparison to … [more]© LMU and MPG MunichIn order to bring water to the boil, energy needs to be added. As the water heats up, the water molecules increase their kinetic energy over time and move faster and faster on average. Yet, the individual molecules possess different kinetic energies – from very slow to very fast. Low-energy states are more likely than high-energy states, i.e. only a few particles move really fast. In physics, this distribution is called the Boltzmann distribution. Physicists working with Ulrich Schneider and Immanuel Bloch have now realised a gas in which this distribution is precisely inverted: many particles possess high energies and only a few have low energies. This inversion of the energy distribution means that the particles have assumed a negative absolute temperature.“The inverted Boltzmann distribution is the hallmark of negative absolute temperature; and this is what we have achieved,” says Ulrich Schneider. “Yet the gas is not colder than zero kelvin, but hotter,” as the physicist explains: “It is even hotter than at any positive temperature – the temperature scale simply does not end at infinity, but jumps to negative values instead.”A negative temperature can only be achieved with an upper limit for the energyThe meaning of a negative absolute temperature can best be illustrated with rolling spheres in a hilly landscape, where the valleys stand for a low potential energy and the hills for a high one. The faster the spheres move, the higher their kinetic energy as well: if one starts at positive temperatures and increases the total energy of the spheres by heating them up, the spheres will increasingly spread into regions of high energy. If it were possible to heat the spheres to infinite temperature, there would be an equal probability of finding them at any point in the landscape, irrespective of the potential energy. If one could now add even more energy and thereby heat the spheres even further, they would preferably gather at high-energy states and would be even hotter than at infinite temperature. The Boltzmann distribution would be inverted, and the temperature therefore negative. At first sight it may sound strange that a negative absolute temperature is hotter than a positive one. This is simply a consequence of the historic definition of absolute temperature, however; if it were defined differently, this apparent contradiction would not exist.Temperature as a game of marbles: The Boltzmann distribution states how many particles have which energy, and can be illustrated with the aid of spheres that are distributed in a hilly landscape. At positive temperatures (left image), as are common in everyday life, most spheres lie in the valley at minimum potential energy and barely move; they therefore also possess minimum kinetic energy. States with low total energy are therefore more likely than those with high total energy – the usual Boltzmann distribution. At infinite temperature (centre image) the spheres are spread evenly over low and high energies in an identical landscape. Here, all energy states are equally probable. At negative temperatures (right image), however, most spheres move on top of the hill, at the upper limit of the potential energy. Their kinetic energy is also maximum. Energy states with high total energy thus occur more frequently than those with low total energy – the Boltzmann distribution is inverted. Zoom ImageTemperature as a game of marbles: The Boltzmann distribution states how many particles have which energy, and can be … [more]© LMU and MPG MunichThis inversion of the population of energy states is not possible in water or any other natural system as the system would need to absorb an infinite amount of energy – an impossible feat! However, if the particles possess an upper limit for their energy, such as the top of the hill in the potential energy landscape, the situation will be completely different. The researchers in Immanuel Bloch’s and Ulrich Schneider’s research group have now realised such a system of an atomic gas with an upper energy limit in their laboratory, following theoretical proposals by Allard Mosk and Achim Rosch.In their experiment, the scientists first cool around a hundred thousand atoms in a vacuum chamber to a positive temperature of a few billionths of a Kelvin and capture them in optical traps made of laser beams. The surrounding ultrahigh vacuum guarantees that the atoms are perfectly thermally insulated from the environment. The laser beams create a so-called optical lattice, in which the atoms are arranged regularly at lattice sites. In this lattice, the atoms can still move from site to site via the tunnel effect, yet their kinetic energy has an upper limit and therefore possesses the required upper energy limit. Temperature, however, relates not only to kinetic energy, but to the total energy of the particles, which in this case includes interaction and potential energy. The system of the Munich and Garching researchers also sets a limit to both of these. The physicists then take the atoms to this upper boundary of the total energy – thus realising a negative temperature, at minus a few billionths of a kelvin.At negative temperatures an engine can do more workIf spheres possess a positive temperature and lie in a valley at minimum potential energy, this state is obviously stable – this is nature as we know it. If the spheres are located on top of a hill at maximum potential energy, they will usually roll down and thereby convert their potential energy into kinetic energy. “If the spheres are at a negative temperature, however, their kinetic energy will already be so large that it cannot increase further,” explains Simon Braun, a doctoral student in the research group. “The spheres thus cannot roll down, and they stay on top of the hill. The energy limit therefore renders the system stable!” The negative temperature state in their experiment is indeed just as stable as a positive temperature state. “We have thus created the first negative absolute temperature state for moving particles,” adds Braun.Matter at negative absolute temperature has a whole range of astounding consequences: with its help, one could create heat engines such as combustion engines with an efficiency of more than 100%. This does not mean, however, that the law of energy conservation is violated. Instead, the engine could not only absorb energy from the hotter medium, and thus do work, but, in contrast to the usual case, from the colder medium as well.At purely positive temperatures, the colder medium inevitably heats up in contrast, therefore absorbing a portion of the energy of the hot medium and thereby limits the efficiency. If the hot medium has a negative temperature, it is possible to absorb energy from both media simultaneously. The work performed by the engine is therefore greater than the energy taken from the hotter medium alone – the efficiency is over 100 percent.The achievement of the Munich physicists could additionally be interesting for cosmology, since the thermodynamic behaviour of negative temperature exhibits parallels to so-called dark energy. Cosmologists postulate dark energy as the elusive force that accelerates the expansion of the universe, although the cosmos should in fact contract because of the gravitational attraction between all masses. There is a similar phenomenon in the atomic cloud in the Munich laboratory: the experiment relies upon the fact that the atoms in the gas do not repel each other as in a usual gas, but instead interact attractively. This means that the atoms exert a negative instead of a positive pressure. As a consequence, the atom cloud wants to contract and should really collapse – just as would be expected for the universe under the effect of gravity. But because of its negative temperature this does not happen. The gas is saved from collapse just like the universe.

Why you haven’t made specific examples of systems having negative absolute temperaure, as a Laser pump system where population invertion is in act? While internal energy increases, in that system, entropy decreases.

Regards,

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Lasers

are notexamples of what I described. The population inversion in lasers happens as a non-equilibrium state while negative temperature states are equilibrium states.Ok, thank you!

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