Graduate Is there a quantum uncertainty to the number of atoms in a marble?

[Dale]

The thread is reopened for discussion with a reminder that interpretations are off topic for this forum.

 

[A. Neumaier]

The grand canonical ensemble is not applicable to the case considered here, because we just discuss a marble as a closed system, if I understand the OP right. So if anything thermal applies here at all, it's the canonical ensemble with a fixed number of particles.

How do you define pressure for a marble in a canonical ensemble? The marble has finite extent, so the thermodynamic limit is not applicable. Surely one can exert pressure on a marble and thereby affect its thermodynamic state.

 

[DrDu]

Although I have not read all threads, and I don't know whether this has already been mentioned, there is a superselection rule which states that it is not possible to have a superposition of states containing even and odd numbers of Fermions.

Edit:

Furthermore, in non-relativistic quantum field theory, there is a superselection rule for states of different mass. As the speed of all atoms in a marble are way below the speed of light, so that nuclear reactions are not possible, a superposition of states containing different numbers of atoms is not possible, too.

 

[PeterDonis]

the speed of all atoms in a marble are way below the speed of light, so that nuclear reactions are not possible

Nuclear reactions don't always require particles moving at relativistic speeds. For example, uranium-235 fission happens (and even works best) with thermal neutrons at temperatures where they are non-relativistic. So a marble made of U-235 at room temperature in the presence of a slow neutron source would not have a conserved number of atoms.

 

[A. Neumaier]

Furthermore, in non-relativistic quantum field theory, there is a superselection rule for states of different mass.

This superselection rule only holds for a Galilei invariant system, not for a marble which is only rotation invariant and breaks translation and boost invariance. The total mass of a nonrelativistic universe is superselected, not masses of pieces of it!

 

[vanhees71]

How do you define pressure for a marble in a canonical ensemble? The marble has finite extent, so the thermodynamic limit is not applicable. Surely one can exert pressure on a marble and thereby affect its thermodynamic state.

The thermodynamical potential corresponding to the canonical ensemble is the Free Energy in terms of its natural variables, $F=F(T,V,N)$, where $T$ is the temperature (i.e., $\beta=1/T$ is the Lagrange parameter to fix the average energy of the system) and $V$ is the volume and $N$ the fixed particle number as external parameters.

Now you have
$$\mathrm{d} F = \mathrm{d}(U-T S)=T \mathrm{d} S-p \mathrm{d}V -\mathrm{d} T S + \mu \mathrm{d} N - T \mathrm{d} S=-p\mathrm{d} V -S \mathrm{d} T+\mu \mathrm{d} N,$$
from which
$$p=-(\partial_V F)_{T,N}, \quad S=-(\partial_T F)_{V,N}, \quad \mu=(\partial_N F)_{T,V}.$$
The relation to the canonical partition sum is (all with $k_{\text{B}}=1$)
$$F=-T Z_N.$$

 

[DrDu]

This superselection rule only holds for a Galilei invariant system, not for a marble which is only rotation invariant and breaks translation and boost invariance. The total mass of a nonrelativistic universe is superselected, not masses of pieces of it!

True, you have to assume that the marble is in an asymptotically free state. I.e. long after the process of it's creation or collision with another marble. So the marble is it's own universe (why does this remind me of "man in black"?). However, the explicit breaking of translational or boost symmetry I don't see as a problem. The point is that performing a series of boost and time reversal operations which in total are equivalent to doing nothing, will return the marble to it's initial state, up to a phase, which depends on the mass of the marble. So there can be no superposition of mass states with a definite phase relation between the different mass states. Hence a superposition of mass states will always be a mixture, not a pure state.

 

[vanhees71]

Since when is translational and boost symmetry broken for a marble considered as a closed system (i.e., a lonely marble in an inertial frame)?

 

[A. Neumaier]

True, you have to assume that the marble is in an asymptotically free state. I.e. long after the process of it's creation or collision with another marble.

... and neither visible (interacting with light) nor supported (interacting with the matter on which it rests), nor surrounded by air! This is not an ordinary marble as in the OP.

So the marble is it's own universe (why does this remind me of "man in black"?). However, the explicit breaking of translational or boost symmetry I don't see as a problem. The point is that performing a series of boost and time reversal operations which in total are equivalent to doing nothing, will return the marble to it's initial state, up to a phase, which depends on the mass of the marble. So there can be no superposition of mass states with a definite phase relation between the different mass states. Hence a superposition of mass states will always be a mixture, not a pure state.

The state even of a completely isolated marble (a macroscopic object in local equilibrium) is anyway a mixture, not a pure state.

 

[atyy]

The state even of a completely isolated marble (a macroscopic object in local equilibrium) is anyway a mixture, not a pure state.

What about pure state statistical mechanics along the lines of
https://arxiv.org/abs/1302.3138https://arxiv.org/abs/1309.0851

 

[A. Neumaier]

What about pure state statistical mechanics along the lines of
https://arxiv.org/abs/1302.3138https://arxiv.org/abs/1309.0851

What would be the temperature of a system in a pure state?

 

[A. Neumaier]

The state even of a completely isolated marble (a macroscopic object in local equilibrium) is anyway a mixture, not a pure state.

What about pure state statistical mechanics along the lines of
https://arxiv.org/abs/1302.3138https://arxiv.org/abs/1309.0851

Here the mixedness is hidden in the random pure state to which the exponential is applied. The thermodynamics is reproduced only when averaging over this randomness. Thus the true state is the mixture determined by the distribution of the random pure state.

Despite its name, a ''random pure state'' in the sense given is not a pure state but the family of all pure states, equipped with a probability distribution. Just like a random number is not a single number but the family of all numbers, equipped with a probability distribution.