Classification of thermodynamic phase transitions relies on the analiticity of the Helmholtz free energy (or the corresponding thermodynamic potential, depending on the ensemble). As it is widely known, first-order phase transitions are characterized by a discontinuity in the first derivative of the thermodynamic potential with respect to the relevant intensive variable, while in continuous phase transitions the thermodynamic potentials are continuous and differentiable, but high-order derivatives may be undefined.
Imagine now that one considers a system udergoing a phase transition for given values of temperature, pressure, etc. Is it possible to infer that such a system will exhibit a thermodynamic phase transition by looking at the microcanonical density of states (DOS), instead of the thermodynamic potentials? Does the DOS carry some signature of the phase transitions? If yes, what traits indicate the order of the transition ?
By studying some classical papers (like those on random energy models and trap models by Derrida), one can infer, for instance, that a DOS with edge states can be linked to a thermodynamic freezing (glass) transition. Are there similar signs in the simpler cases of first and second order phase transitions? I intuitively believe there must be, since, for instance, the canonical partition function is simply the Laplace transform of the microcanonical DOS. But I don't know what those signs may be.
I think the DOS would show a signature.
If we look at the system as an ion-electron mixture and write its partition function, there will be both ionic (phonon in case of solid) and electronic DOS. From the electronic DOS, the metal-insulator transition can be figured out. From the phonon DOS in a solid the phase transitions like structural transitions may be figured out (see phonon softening).
Dear Sai Venkata Ramana Ambadipudi and Gert Van der Zwan
,Thank you both for your answers. Sai Venkata Ramana Ambadipudi , although I'm more interested in classical systems, you're so right that perhaps there's much to learn from quantum systems. Indeed, the study of quantum phase transitions is well developed, and in many cases what happens to non-thermal quantum systems undergoing a quantum phase transition has some effect on thermodynamic properties at non-zero temperatures.
Gert Van der Zwan
, I indeed can try to look at it myself. The reason why I decided to open a discussion is to avoid "rediscovering warm watter" , I wanted to make sure first that there weren't established and explicit results along these lines.Allow me now to divagate a little bit. In the case of a second order phase transition, there's ergodicity breaking in the thermodynamic limit. Basically, when being in a state with a given energy, there're states with the same energy that are not accessible because they corresponds to a different value of the order parameter, and the order parameter being generically extensive, cannot spontaneously change its value due to a fluctuation because it would imply jumping over an infinitely high (free)energy barrier in the thermodynamic limit. Now, given that the zero-temperature limit of the canonical free energy is nothing but the ground-state energy of the system, such a picture is compatible with a degenerate ground state, where there're dynamically disconnected sets in phase space compatible with the same energy. How does this translates to the full DOS, and, as you also ask, is it possible to read some information about critical exponents directly from the DOS? I precisely wonder if someone has worked out an answer to these questions.
When the thermodynamic phase transition is of first order, I don't know better. I could probably say, using again that the ground-state enegy is the zero-temperature limit of the free energy, that there's a finite jump in the derivative of the ground-state energy with respect to some parameter(s) of the Hamiltonian, but I don't know precisely how to translate that to the full DOS. What do you think?
Best wishes,
Reinaldo.
The connections are very obvious. In the microcanonical ensemble entropy S is defined as S=k_B ln (\Omega(E,V,N)/[N!h^{3N}]), with \Omega the density of states, integrated up to E. The second derivative of S with respect to E, at constant V and N equals ( -1/T^2) times the inverse specific heat at constant volume, which will be non-analytic at higher order phase transitions. This holds both classically and quantum mechanically (apart from the factor N!h^{3N}).
Dear Henk van Beijeren ,
Thank you for your answer. I also thank Gert Van der Zwan
for his previous answer too. Sorry about the delay, I've been very sick last days.Indeed, this is obvious, but somehow my question is much simpler and the answer is less obvious (I think). Note that, following the prescription you give, the second derivative of the logarithm of the density of states is non-analytic for systems undergoing a second order phase transition, but this does not directly hint at the shape of the graph of the DOS. It may be that the DOS either vanishes or some non-linear combination of its derivatives diverges at a finite, non-zero energy.
My question is basically qualitative. Think on the example I give in the original question. When a DOS exhibits an edge for high absolute values of the energy, I know for sure the system exhibits an ideal glass transition.
The reason of this qualitative (and very precise) question is just that I need it for something else, but using all your suggestions I can try to see if I can build the general statement I'm looking by myself.
Thank you again.
Best wishes,
Reinaldo.
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