I think a useful starting point is the definition of temperature itself:

1/T = dS/dU,

where S = entropy, U = total energy, T = temperature

To to have positive temperature simply means that dS/dU > 0, which means that adding energy to the system increases its entropy. (I think this makes physical sense for most systems - you'd normally expect your bunch of gas molecules to get more "disordered" if you give it more energy to run around, and conversely sucking energy out will cause your molecules to move slower and less.)

However, it's possible to imagine situations where increasing the total energy actually causes a decrease in entropy (dS/dU < 0, so T < 0). For instance, consider the classic textbook example of an ensemble of two-state systems that do not interact with each other.

For concreteness, let's imagine having N distinct, non-interacting electron spins in an external magnetic field, where spin up has energy +1, and spin down has energy 0.

For a given total energy E, one can derive that the entropy is S = k log (W), where W = N! / { E! (N-E)! }. It's possible to show mathematically that when E > N/2 you'll get dS/dU < 0, but you can argue equally well from physical intuition:

When you start off with everything spin down, adding energy will greatly increase the total number of ways you can allocate your energy to the N electrons - at very small E, you'll pretty much have almost N ways of allocating each energy quantum you have. However, as you approach E = N/2, you get less and less extra ways of allocating energy with each extra quantum...

And when you reach E > N/2 you basically have more spin ups than spin downs - you'll be stuck with having more and more spin ups, and hence less ways of "allocating spin downs", as you gain more energy. The total entropy of the system thus decreases as you add more energy, when E > N/2, giving rise to negative temperature.

PS: The wikipedia article on negative temperature brings out an interesting point: T < 0 objects will behave as if they're hotter than anything with T > 0 (-1 > +infinity!!) when they're in thermal contact.

I was referring to the possibility of a negative temperature with reference to the zero kelvin.atleast mathematically if not from a practical sense?

I don't really read that article, but one post about it. Anyway, I think it's useful in this topic.

Equilibration Rates and Negative Absolute Temperatures for Ultracold Atoms in Optical Lattices

Akos Rapp, Stephan Mandt, Achim Rosch

Physical Review Letters

Vol.: 105, 220405

DOI: 10.1103/PhysRevLett.105.220405

In physics certain systems follow negetive temperature that is thermodynamic temperature.negetive temperatures can be expressed in negetive numbers in kelvin scale.temperatures that are expressed in -ve temperature on familiar celsius or fahrenheit are simpliy colder than zero points of these scales

for more information refer Edward samuel F.J.bryan taylor-negetive temperature states of two dimensional plasmas and vortex fluids

The absolute temperature in the thermodynamic limit (of an infinite or macroscopic system size) cannot be negative, but in some special cases an effective negative T (in Kelvin scale) can be defined: http://en.wikipedia.org/wiki/Negative_temperature. Instead, T=0 is a well defined concept in quantum mechanics (only zero-point motion remains), but cannot be experimentally realized due to laws of termodynamics.

Hi Tapio! I agree with your comment. I think one could even state that the (Kelvin scale) temperature of any system (of any size) in equilibrium cannot be negative. The reason is that (by the 0th law of thermodynamics) this system is in equilibrium with a reservoir at the same temperature, and since the nature of the reservoir makes no difference, it could well be a (classical) ideal gas. But since the temperature of the latter is proportional to the kinetic energy per molecule, it can't be negative. So the original system has a nonnegative temperature. I agree that sometimes it's convenient to define a negative *effective* temperature (as in a laser with a "population inversion") but I prefer to avoid this, as it tends to generate confusion!

I see no problem with negative pressure as a (metastable) equilibrium for liquids and solids. And gravitational systems can even have negative heat capacities!

The Kelvin temperature T is a physical property defined operationally by the zeroth law of thermodynamics. Unlike spin, temperature is by definition a non-negative quantity as well as being a classical concept in the sense that it does not depend on Planck’s constant.

Negative temperature is a mathematical trick showing up as a consequence of reversing the Hamiltonian function, H for –H, for a given system made up for spin population, for instance. This spin population after formally replacing H with –H looks as if the temperature of the spins has become negative. In this context, temperature is not a directly measurable quantity.

It is worth noticing that the concept of negative temperature relies on restricted symmetry features underlying the Hamiltonian for a given physical system. More specifically, negative temperature is inapplicable to translational motion.

One can recommend the following books on the concept of temperature:

http://pt.scribd.com/doc/83802856/2011-Is-There-a-Temperature-Conceptual-Challenges-at-High-Energy-Acceleration-and-Complexity-Springer

http://pt.scribd.com/doc/83801008/1984-Smorodinsky-Temperature

Bolivar.

I think a useful starting point is the definition of temperature itself:

1/T = dS/dU,

where S = entropy, U = total energy, T = temperature

To to have positive temperature simply means that dS/dU > 0, which means that adding energy to the system increases its entropy. (I think this makes physical sense for most systems - you'd normally expect your bunch of gas molecules to get more "disordered" if you give it more energy to run around, and conversely sucking energy out will cause your molecules to move slower and less.)

However, it's possible to imagine situations where increasing the total energy actually causes a decrease in entropy (dS/dU < 0, so T < 0). For instance, consider the classic textbook example of an ensemble of two-state systems that do not interact with each other.

For concreteness, let's imagine having N distinct, non-interacting electron spins in an external magnetic field, where spin up has energy +1, and spin down has energy 0.

For a given total energy E, one can derive that the entropy is S = k log (W), where W = N! / { E! (N-E)! }. It's possible to show mathematically that when E > N/2 you'll get dS/dU < 0, but you can argue equally well from physical intuition:

When you start off with everything spin down, adding energy will greatly increase the total number of ways you can allocate your energy to the N electrons - at very small E, you'll pretty much have almost N ways of allocating each energy quantum you have. However, as you approach E = N/2, you get less and less extra ways of allocating energy with each extra quantum...

And when you reach E > N/2 you basically have more spin ups than spin downs - you'll be stuck with having more and more spin ups, and hence less ways of "allocating spin downs", as you gain more energy. The total entropy of the system thus decreases as you add more energy, when E > N/2, giving rise to negative temperature.

PS: The wikipedia article on negative temperature brings out an interesting point: T < 0 objects will behave as if they're hotter than anything with T > 0 (-1 > +infinity!!) when they're in thermal contact.

I always understood temperature to be the average kinetic energy of a group of particles. Since the kinetic energy of a particle is 1/2(mass)*(speed)^2, I would understand that to mean it could not have a negative temperature unless it had negative mass or the speed was a complex number that would result in a negative value for v^2. I am not familiar with any situation where that would be the case.

@Tim. The kinetic energy definition indeed means that T>=0, and freely moving particles cannot have a negative T. You need bound states or finite number of energy states to define an *effective* negative T. See Wiki and the postings above for more details.

Dear Colleagues,

I strongly recommend the reading of the scientific literature on the possibility of absolute negative temperatures, such as the preprint on Arxiv (http://arxiv.org/abs/1112.4299):

Quantum Simulators at Negative Absolute Temperatures

Akos Rapp

(Submitted on 19 Dec 2011)

We propose that negative absolute temperatures in ultracold atomic clouds in optical lattices can be used to simulate quantum systems in new regions of the phase diagrams. First we discuss how the attractive SU(3) Hubbard model in three dimensions can be realized using repulsively interacting 173-Yb atoms, then we consider how an antiferromagnetic S=1 spin chain could be simulated using spinor 87-Rb or 23-Na atoms. The general idea to achieve negative absolute temperatures is to reverse the sign of the external harmonic potential. Energy conservation in a deep optical lattice imposes a constraint on the dynamics of the cloud, which will relax towards a T

In respectable scientific literature on negative absolute temperatures, one can read:

"The proof given (...) that the temperature is positive was based on the condition for the system to be stable with respect to the occurrence of internal macroscopic motions within it. But the system of moments here considered is by its nature incapable of macroscopic motion, and so the previous arguments do not apply to it; nor does the proof based on the normalisation condition for the Gibbs distribution, since in the present case the system has only a finite number of energy levels, themselves finite, and so the normalisation sum converges for any value of T.

Thus we have the interesting result that the system of interacting moments may have either a positive or a negative temperature. Let us examine the properties of the system at various temperatures.

(...)

(...) the region of negative temperatures lies not "below absolute zero" but "above infinity". In this sense we can say that negative temperatures are "higher" than positive ones. This is in accordance with the fact that, when a system at a negative temperature interacts with one at a positive temperature (i.e. the lattice vibrations), energy must pass from the former to the latter system; this is easily seen by the same method as that used to discuss the exchange of energy between bodies at different temperatures.

States with negative temperature can be attained in practice in a paramagnetic system of nuclear moments in a crystal where the relaxation time t_2 for the interaction between nuclear spins is very small compared with the relaxation time t_1 for the spin-lattice interaction (E. M. Purcell and R. V. Pound, 1951)".

Landau and Lifshitz (Statistical Physics, pp. 222-224)

Dear Colleagues,

Attached is a draft of my manuscript on

BROWNIAN MOTION AT NEGATIVE ABSOLUTE TEMPERATURES.

Comments, suggestions, and criticisms are wellcome.

Bolivar.

(Brasília, 06 April 2012)

http://pt.scribd.com/doc/88262561/2012-Bolivar-Brownian-Motion-at-Negative-Absolute-Temperatures-First-Version

since T = math toy to get a nr. out of a system => T < 0 for systems where applying the toy blindly leads to such value (i.e. in lasers) ... nothing profound, just another "meter" used outside its range.

The question is, whether Kannan meant negative temperature in the sense negative values in the Kelvin scale (which are possible as extensively discussed here), or in the sense colder than absolute zero (which is not possible).

This leads me to another related question on temperature: classically the temperature can be defined as a measure of the degree of motion of the particles.

This means the higher the temperature, the faster the motion of the particles.

Now the essential question is, is the faster the motion of the particles increasing the temperature or is the increase in temperature accelerating the motion of the particles? (the latter would mean temperature behaves in some sense as a force)

Temperature is measured (indirectly) in 2 theoretical ways:

1) take all molecules, measure their energies of motion, rotation, vibration, then take an average over their number, them multiply by 2/k_B ---- the answer is a number called temperature

2) make a distribution of the entities' energies (say electrons in a crystal), and make a fit with a function exp(-E/k_B X). The parameter X from the fit is called temperature.

You can see in case (2) that the possibility for a negative fit parameter arises. One that is not possible in case (1).

The two are not contradictory, rather express different aspects. In solid state there are more aspects to quantify (eg. - entities sitting on given energy levels, and not occupying the continuum such as in the classical gas).