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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?

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Maybe I'm missing something, but I don't see how $S=k\ln\Omega$ requires spatial dimension. –  KennyTM Nov 16 '10 at 12:48
    
Swapping timelike and spacelike coordinates, while trivial-looking, is a very dangerous thing. Hyperbolic PDE's become Elliptical, for one thing. For another, the key property of Entropy, its non-decreasing nature, becomes invalid as one increases in a spacelike direction. People studying black hole thermodynamics have made progress using "null affine time" to define entropy, though. –  Jerry Schirmer Dec 5 '10 at 19:20
    
@Jerry: mathematician in me agrees but the physicist says "who cares?". I haven't ever come upon any area of physics where the use of Wick's rotation causes any problems. In particular, relation between characteristic function and moment generating function and between Wiener process and path integral. –  Marek Dec 5 '10 at 20:30
    
Well, I would say the construction offered above is a different thing than Wick rotation, which is just a choice of complex time. The OP isn't trying to talk about defining entropy on an Euclidean section of a generalized complexified spacetime--(s)he's trying to define thermodynamics on a timelike slice and then "evolve" along a spacelike vector. The former is reasonably well-defined. The latter is likely to cause a lot of trouble, starting with defining what 'future-pointing' and 'past-pointing' mean in a spacelike sense. –  Jerry Schirmer Dec 6 '10 at 4:48
    
@Jerry, okay, I misunderstood your comment. You were talking about swapping time-like and space-like coordinates. That is precisely what Wick rotation is doing. –  Marek Dec 6 '10 at 11:28
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7 Answers

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

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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:

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To clarify, $\Omega$ is the number of microstates, or the multiplicity. –  Noldorin Nov 16 '10 at 16:48
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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

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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.

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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.

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What? So you think the amount of information in the system depends on how fast it moves relative to you? I say this is nonsense. Also, what would the other three components of the entropy four-vector be? –  Marek Dec 6 '10 at 11:04
    
The quantity would be density of entropy (or entropy per nucleon) and the three components of the four vector would be the flux of entropy density in the fluid. It's really hard to make sense of the First Law if you don't generalize entropy that way. –  Jerry Schirmer Dec 6 '10 at 15:11
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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.

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Okay, this answer begs a question (and I am going to ask it, because it bugs me for a long time). Namely that it assumes conservation of energy (in order to propose first law). Therefore it also assumes that the space-time is stationary (as in all the basic BH solutions). But then the definition of entropy is clearly three-dimensional: you choose one time-slice and work there forever. –  Marek Dec 6 '10 at 11:07
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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.

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