Ising lattice seen by an inertial observer

[Heidi]

We have an Ising lattice on the x-axis . on every site there is an atom which can be up or down. i suppose that there are N atoms (repetedly with the same values). Each sequence of spins has an energy H with a probability exp(-H/k T)
i suppose that there is a device attached to each atom indicating the value every second. he as
Far from the axis an observer has a single atom (not coupled to the others). an amplitude of probability is assigned to it.
it is exp (-i Ht/h)
every second he sees his amplitude and the sequence of up and down on the axis.

Let us suppose nox that he is no more facing the origini of the x but in an inertion movement parallel to the x.
My question is not about relativity but i accept the fact that he will see on x a sequence of events in space time.

It would be the same if he was not moving but that the devices showed the values a different times
How can we transform the exp(-H/k T) to describe this new case for the observer.

i repeat that here there is no limit for his speed . it may be greater then c and tend to infinity.

 

[Jarek 31]

Temperature e.g. of a gas is proportional to mean energy of its atoms/molecules - mainly kinetic energy ( https://en.wikipedia.org/wiki/Kinetic_theory_of_gases ).

Moving e.g. gas tank with some velocity, you change velocity of all particles inside - increasing their mean kinetic energy.
However, does a moving gas tank (e.g. in a rocket) have higher temperature than a resting one?

My point is that, while special relativity doesn't emphasize any velocity, thermodynamics does - e.g. a mean velocity of atoms of this gas.
Shouldn't we calculate temperature as kinetic energy after subtracting this mean velocity?

However, from Boltzmann ensemble perspective e.g. Ising model, moving everything with the same velocity increases kinetic energy by the same value ({E_i} -> {E_i + d}), what doesn't change Boltzmann distribution.

 

[Heidi]

in the first case we have a timeless process.
in the second case the observer sees a sequence of measurements at different times (on a line of spasce time). if v tends to infinity it looks like a time motion on the x axis.

 

[Jarek 31]

Without time, the system should already approach Boltzmann distribution among possibilities.

However, with time the speed of such approach is a very difficult question, especially that hidden dynamics is often time(/CPT)-symmetric ... to break such time symmetry we need some uniformity approximation generally called stosszahlansatz e.g. in https://en.wikipedia.org/wiki/H-theorem or much simpler Kac ring: https://www.maths.usyd.edu.au/u/gottwald/preprints/kac-ring.pdf

 

[hutchphd]

Shouldn't we calculate temperature as kinetic energy after subtracting this mean velocity?

Temperature is the ensemble rms velocity and so we already do.

 

[Jarek 31]

Isn't RMS (root-mean squared) velocity just sqrt(E[v^2])?
To neglect e.g. velocity of rocket this gas tank is in, shouldn't we do as in variance instead: sqrt(E[v^2]-E[v]^2) ?

 

[hutchphd]

Yes and in practice that is the definition...semantics notwithstanding.

 

[Heidi]

these answers are not exactly what i am looking for:
I have an observer on the line x = vt . it is an oriented line.
if v = 0 it is x = 0 so the observer in on the t axis and the orientation is from past to future. he has an atom on him and exp(-iH) relates amplitudes before and after just like exp(-H) relates the probability on the x-axis for a site to the site on its left.
we have t = x/v so if x tends to infinity t tends to t = 0 and the observer moves on the x axis. his atom is still governed by the same rule (an amplitude rule) but there is also the Boltzmann probability rule on the x axis.
the problem is that in this case the x-axis is also his time trajectory?
how to solve that?
in this case before and on the left is the same thing.

 

[Jarek 31]

I am not certain if I properly understood, but you have asked about Ising chain moving with constant velocity.
Even without subtracting mean velocity as we have discussed, kinetic energy should increase by a constant this way: {E_i} -> {E_i +c}.
As already mentioned, this kind of change of absolute energy does not modify Boltzmann probability distribution from - depending only on relative energies.

Including relativistic effects, e.g. Lorentz contraction reducing lattice constant and so changing ~1/r^3 spin-spin energy ( https://en.wikipedia.org/wiki/Magnetic_dipole–dipole_interaction ), electromagnetism is Lorentz-invariant hence such corrections should be compensated e.g. by electric field induced by moving magnetic field.

 

[Demystifier]

@Heidi I think you mixed several things up.

First, "spin" in the Ising model is not a quantum spin. It is a classical discrete variable. Hence quantum physics should be irrelevant.

Second, the Ising Hamiltonian does not have the kinetic energy part (proportional to the momentum squared). This implies that spins do not change with time.

In view of this, either you have to reformulate your question, or the answer is trivial.

 

[Heidi]

thank you
I will take your advice and open another thread starting from the quantum point of view.