# B What is Quantum Statistical Mechanics?

1. Mar 14, 2016

### fxdung

Is Quantum Statistical Mechanics being the application of Quantum Mechanics on the separate particles of bulk matter or the application of QM on whole agregate matter?

2. Mar 14, 2016

### dextercioby

In short, it's quantum mechanics done right, even though, from my point of view, it uses less interesting mathematics than the "orthodox" quantum mechanics.

3. Mar 14, 2016

### fxdung

Then what is the difference between non-relativistic QFT and quantum statistical mechanics?

4. Mar 15, 2016

### Demystifier

Quantum statistical mechanics (QSM) introduces additional uncertainties that are not already present in "pure" quantum mechanics (QM). In QM you assume that the whole system is in a known pure state. In QSM you don't assume that, so you describe the whole system by a mixed state. Mixed states appear also in QM, but only as descriptions of subsystems, by partial tracing of the pure state for the whole system. By contrast, in QSM you start from the mixed state from the very beginning.

Generalizing QM to QFT makes no difference. If you start from a pure state in QFT, then it's "pure" QFT. If you start from a mixed state in QFT, then it's statistical QFT.

5. Mar 15, 2016

### atyy

The most important concepts in quantum statistical mechanics that are not present in pure quantum mechanics are the thermodynamic concepts like temperature, and the thermodynamic limit.

6. Mar 15, 2016

### vanhees71

Non-relativistic QFT is one way, in my opinion the most clever way, to describe quantum many-body systems. NRQFT is comprehensive, i.e., it includes all of the "1st quantized formalism" but it allows for more, e.g., the description of many-body systems in terms of "quasi particles", where the "quasiparticles" are not particles in the usual sense but can be quantized collective excitations of a many-body system, e.g., the lattice vibrations (sound waves) in a solid. The mathematics is pretty similar to a QFT description of particles and that's why one calls such excitations, treated in this way mathematically, "quasiparticles". They also have names like "phonons" for the lattice vibrations.

7. Mar 15, 2016

### fxdung

In classical physics,temperature is defined as average kinetic energy of molecules.But what is temperature of radiation,of QFT,of QM?

8. Mar 15, 2016

### A. Neumaier

The inverse of the coefficient of the Hamiltonian in the exponent of the assumed form of the density operator, e.g., in the canonical ensemble $\rho=e^{-H/kT}$.

9. Mar 16, 2016

### Demystifier

Only in the thermodynamic equilibrium. Statistical physics (either classical or quantum) studies also systems far from thermodynamic equilibrium, in which case the concept of temperature does not make sense.

10. Mar 16, 2016

### vanhees71

For an introduction, see the marvelous papers by Danielewicz:

Danielewicz, P.: Quantum Theory of Nonequilibrium Processes I, Ann. Phys. 152, 239, 1984
http://dx.doi.org/10.1016/0003-4916(84)90092-7 [Broken]

Danielewicz, P.: Quantum Theory of Nonequilibrium Processes II. Application to Nuclear Collisions, Ann. Phys. 152, 305–326, 1984
http://dx.doi.org/10.1016/0003-4916(84)90093-9 [Broken]

Last edited by a moderator: May 7, 2017
11. Mar 16, 2016

### A. Neumaier

???

The concept of temperature makes perfect sense in nonequilibrium thermodynamics, as a local temperature field. One can see this from the Navier-Stokes equation, which are a nonequilibrium system derivable from statistical mechanics. See, e.g., the book on statistical physics by Linda Reichl.

12. Mar 16, 2016

### vanhees71

Well, also hydrodynamics is an approximation of limited applicability. It's valid for small Knudsen numbers

https://en.wikipedia.org/wiki/Knudsen_number

For large Knudsen numbers we have Boltzmann(-Uehling-Uhlenbeck) equations. It's amazing, how far you can stretch hydro to be valid also for Knudsen numbers at the order of 1. Then, however you need "higher-order moments", i.e., approximations beyond Navier-Stokes (in relativistic hydro you need at least 2nd order hydro anyway, because of the famous causility trouble with Navier-Stokes; the most simple realization is Israel-Stuart based on a relaxation time approximation).

Have a look at the following thesis:

http://fias.uni-frankfurt.de/helmholtz/publications/thesis/Gabriel_Denicol.pdf

Note that the thesis is in English. Only the summary is in German!

13. Mar 16, 2016

### Demystifier

In this case you have a local thermodynamic equilibrium. But sometimes you don't have even that.

14. Mar 16, 2016

### atyy

15. Mar 16, 2016

### A. Neumaier

If the region is dense enough, local temperature makes sense. In particular, close to the big bang.

16. Mar 16, 2016

### Demystifier

Suppose, for instance, that the density matrix of a system is
$$\rho =\frac{A}{{\rm Tr}A}$$
where
$$A=e^{-\beta_1 H}+e^{-\beta_2 H}$$
$H$ is the Hamiltonian and $\beta_1 \neq \beta_2$. What is the temperature of that system? No temperature at all!

17. Mar 16, 2016

### A. Neumaier

Did you want to suggest that the early universe had such a density matrix?

For large systems, most density matrices are not realized in Nature.

18. Mar 17, 2016

### Demystifier

No.

True! Similarly, for small systems, most pure states are also not realized in Nature. Yet quantum mechanics, as a theory, is a theory of all such pure states, whether realized in Nature or not. Likewise, quantum statistical mechanics, as a theory, is a theory of all such mixed states, whether realized in Nature or not. That's the only point I wanted to make.

19. Mar 17, 2016

### A. Neumaier

No. It is a theory of only such states that conform to assumptions found in reality, otherwise one could never obtain irreversibility from unitarity. Only concentrating on states representing reality rather than arbitrary states makes quantum statistical mechanics work.

20. Mar 17, 2016

### Demystifier

Then I am using a wider definition of statistical mechanics than you do. For example, with my definition I can study the possibility of Boltzmann brains, the hypothetical and never observed phenomenon which, in some sense, violates "the principle of macroscopic irreversibility". For some recent research on Boltzmann brains see e.g.
http://arxiv.org/find/grp_physics/1/ti:+AND+Boltzmann+brain/0/1/0/all/0/1

Last edited: Mar 17, 2016