What is Quantum Statistical Mechanics?

[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?

 

[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.

 

[fxdung]

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

 

[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.

 

[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.

 

[vanhees71]

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

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.

 

[fxdung]

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

 

[A. Neumaier]

But what is temperature of radiation,of QFT,of QM?

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 $\rho=e^{-H/kT}$.

 

[Demystifier]

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.

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.

 

[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

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

 

[A. Neumaier]

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.

?

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.

 

[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!

 

[Demystifier]

The concept of temperature makes perfect sense in nonequilibrium thermodynamics, as a local temperature field.

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

 

[atyy]

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

Yes, of course you are right.

But, just for fun, can we say there is always a temperature if we average over large enough regions? Eg. these guys assign a temperature that seems to go all the way back to the big bang:
http://www.astro.ucla.edu/~wright/BBhistory.html
http://www.globalchange.umich.edu/globalchange1/current/lectures/universe/bigbang.jpg.

 

[A. Neumaier]

can we say there is always a temperature if we average over large enough regions?

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

 

[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!

 

[A. Neumaier]

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!

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.

 

[Demystifier]

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

No.

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

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.

 

[A. Neumaier]

quantum statistical mechanics, as a theory, is a theory of all such mixed states

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.

 

[Demystifier]

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.

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

 

[A. Neumaier]

hypothetical and never observed phenomenon

You may study them. But I prefer to study (like all mainstream work in statistical mechanics) what can be observed. This dramatically reduces the weirdness of QM.

 

[Demystifier]

You may study them. But I prefer to study (like all mainstream work in statistical mechanics) what can be observed. This dramatically reduces the weirdness of QM.

Fair enough!

 

[vanhees71]

Well, I'd never say "never". Who knows, what kinds of "weird" states the experimentalists are able to prepare in the future? It's just amazing what they can do know concerning entangled states of many-photon states, photons and atoms, and even "macroscopic" objects.

On the theory side the only restriction of possible states are superselection rules (mostly based on fundamental symmetries).

 

[Demystifier]

On the theory side the only restriction of possible states are superselection rules (mostly based on fundamental symmetries).

But we never know whether a symmetry is fundamental or merely emergent, valid approximately on large distances only.

Concerning superselection rules, I find the explanations of those by decoherence (instead of symmetry) much more appealing. In that sense, superselection rules (SR) appear to be an artefact of statistics, making SR's not more fundamental than the second law of thermodynamics.

 

[A. Neumaier]

Who knows, what kinds of "weird" states the experimentalists are able to prepare in the future?

I restricted my interest just to what is observable. I don't think experimentalists will ever be able to prepare unobservable states...

 

[vanhees71]

What do you mean by "unobservable states"? It's just a matter of technical development what we are able to prepare or not, right?

 

[Demystifier]

I don't think experimentalists will ever be able to prepare unobservable states...

I believe they prepare them all the time, but they just don't know it because the states are ... well, unobservable.

 

[ddd123]

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://lanl.arxiv.org/find/grp_physics/1/ti:+AND+Boltzmann+brain/0/1/0/all/0/1

Link doesn't seem to be working...

 

[Demystifier]

Link doesn't seem to be working...

Thanks, I think I fixed the problem now:
http://arxiv.org/find/grp_physics/1/ti:+AND+Boltzmann+brain/0/1/0/all/0/1

 

[A. Neumaier]

What do you mean by "unobservable states"? It's just a matter of technical development what we are able to prepare or not, right?

Whatever will be preparable any time in the future must be a state that comes from partial tracing of the state of a bigger system including its environment. Thus it is covered by my definition of the states that can appear in Nature. Whereas one cannot observe a state that doesn't occur in Nature, and one cannot prepare such a state.

 

[vanhees71]

I believe they prepare them all the time, but they just don't know it because the states are ... well, unobservable.

Hm, I still don't know, what you mean by "unobservable states". If a state is (in principal) unobservable, then it's not a state. So an "unobservable state" seems to be a constradictio in adjecto. The only thing, I'm aware of are superselection rules which forbid certain states, but that means that they simply do not exist (e.g., the charge superselection rules forbidding superpositions of states of different charge or the angular-momentum superselection rule forbidding superpositions of half-integer with integer angular momenta).

 

[vanhees71]

Whatever will be preparable any time in the future must be a state that comes from partial tracing of the state of a bigger system including its environment. Thus it is covered by my definition of the states that can appear in Nature. Whereas one cannot observe a state that doesn't occur in Nature, and one cannot prepare such a state.

Well, If I prepare a cup of coffee and leave it at rest a while on my desk, I don't trace anything but prepare a thermal-equilibrium state by just waiting long enough. That's an easy preparation without in any way tracing out anything.

 

[A. Neumaier]

Well, If I prepare a cup of coffee and leave it at rest a while on my desk, I don't trace anything but prepare a thermal-equilibrium state by just waiting long enough. That's an easy preparation without in any way tracing out anything.

To be able to say what you said you traced out the whole universe except for the coffee in the cup. For only that part is in equilibrium. If you wait longer, maybe the bigger system consisting of coffee, cup and desk will be in thermal equilibrium. But as long as you are in the room, the whole room will not be in equilibrium. Thus you need to trace out at least yourself. And the outside of the building your desk is in. And your computer if it is running...

 

[vanhees71]

Wow, I'm a very mighty being, being able to trace out the whole universe by cooking a cup of coffee ;-)), but when I come to thermal equilibrium with my environment, I'm dead. So I better don't trace myself out...