Is there a four-dimensional definition of entropy?

Question

It seems odd that entropy is usually only defined for a system in a single 'slice' of time or spacelike region. Can one define the entropy of a system defined by a 4d region of spacetime, in such a way that yields a codimension one definition which agrees with the usual one when the codimension one slice is spacelike?

Answer 1

You are thinking about Boltzmann's definition of entropy, I guess?

In Boltzmann's definition, entropy is just the logarithm of the amount of possible states associated with certain macroscopic variables. In its generality, therefore, it doesn't seem to me to exclude the possibility of counting states with different time coordinates. Or in your more general context, on different time-slices. The question is, what does this correspond to? Does it make sense to do that? You would have to specify the time-development of the macroscopic variables and count the number of microscopic trajectories compatible with those macroscopic trajectories.

As a matter of fact, there exist so-called dynamical entropies. In a heuristic sense, what they do is counting the density of phase-space trajectories of a system, whereas Boltzmann entropy just counts the amount of accessible states under certain macroscopic constraints.

http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Sinai_entropy#Measure-theoretic_entropy

Answer 2

As far as know, entropy works in systems with Hamiltonian dynamics, that is, when there is explicit dependence on time.

In classical mechanics (where position and momentum depend on time), there is Boltzmann entropy $S = k_b \ln \Omega$ ($\Omega$ - 'number' of states).

In (non-relativistic) quantum mechanics (where wavefunction depend on time), there is von Neumann Entropy $S = -k_b \langle \rho \ln \rho \rangle $ ($\rho$ - density matrix).

Though in general there is information-theoretic quantity Shannon entropy $S = \sum_i p_i \ln p_i$ ($p_i$ are probabilities that the system is in the $i$-th state). Maybe in Quantum Field Theory there is some kind of 4-d entropy, but I am not sure. Anyway, the fundamental property 'entropy is non-decreasing function of time' has any meaning only if $S$ is a function of time.

For a broader discussion see great review/didactic paper:

• A Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221–260 (1978)

Answer 3

Do you mean the entropy flux density 4-vector, and its 4 divergence, along with their projections onto the plane orthogonal to the energy transport? See e.g. http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1989A%26A...211..476O&db_key=AST&page_ind=0&data_type=GIF&type=SCREEN_VIEW&classic=YES

Answer 4

Entropy doesn't need a four dimensional definition. It is a scalar amount, since the number of states is just a number. These states could be parts of a n-dimensional phase-space, but it doesn't change entropy.

Answer 5

Well, if one knows the entropy in the microcanonical ensemble, then one can find the internal energy. But we already know that the energy of a system depends on the reference frame in which it is observed. Thus, it is more appropriate to think of entropy in the same terms one thinks of energy--the 0-component of a four-vector, rather than as a 4-scalar.

Answer 6

See the wiki page for Black Hole Entropy. If I recall correctly from my GR course, the area of a certain hyper-surface is always increasing in one direction of time.

Answer 7

Very late to this, but entropy is a functional of the phase space DoFs, which, at least for a field theory, is essentially dependent on a choice of a space-like hypersurface (or hyperplane). The thermodynamic dual of entropy is of course temperature, and the all-important $kT$ has dimensions of energy because this determines the probability density $\exp{(-E/kT)}$, and thereby the amplitude of classical thermal fluctuations.

A 4-dimensional equivalent of $kT$ would be an action. Planck's constant can also be understood to determine the amplitude of fluctuations, which are of course quantum fluctuations (which are distinct from thermal fluctuations essentially because they are invariant under the action of the Lorentz group). So, a 4-entropy would perhaps be the thermodynamic dual of the amplitude of quantum fluctuations.

This is essentially allusive, however, since I think this does not allow us, AFAIK, to construct a 4-entropy. There's something on all this in my Phys. Lett. A 338, 8-12(2005), or, on arXiv, http://arxiv.org/abs/quant-ph/?0411156.