# Does the second law of thermodynamics tell me how the entropy changes?

In thermodynamics I can e.g. compute the properties of ideal gases with certain energies $U_1,U_2$ in boxes with certain volumes $V_1$ and $V_2$. Say I have two such boxes and they have some specific parameter values, like say I know their temperatures $T_1,T_2$. Now I put them together, the gases can interact and I thereby construct a new box with volume $V_1+V_2$ and the energy is $U_1+U_2$. Using the laws of thermodynamics, I can compute everything else again now. Say I find a new temperature $T_{\text{new}}$.

One says that the temperatures of the gases in the systems changed by putting the boxes together. However, on the computational side what I did was just considering a system of new specifications. To get the new result, I didn't have to enter the real world staring conditions except for the variables which also were necessary to compute their respecive properties - effectively, the theory didn't have to tell me how the system changed, just what the restrictions are - and I basically entered into a new system with these values. For example, it's not relevant which gas had with specifications before. I just did a little trivial algebra and computed what has to be. When then talk about a change of entropy and how the gas behaves, but that seems to be only decorative.

If I say that the second law of thermodynamics tells me that heat flows from a system with high temperatures to a system of low temperature I'll be able to derive rules for the entropy say, how the combined system has to look like if I state their variables and insist on extensivity of certain variables and so on - but it's not about change in the sense that I compute how the system developes from one point to the other in details. Rather I just compute how the end configuration has to be.

My question is

Is the starting configuration of a thermodynamics system ever relavant?

And secondly,

In thermodynamics, if I compute "the change a system" in the sense of the above example, do I always induce the necessarity for doing such a thing by stating "now we bring system one and system two in contact"?

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(i) Yes, the starting configuration is relevant. It determines the later state of the thermodynamic system.

(ii) Changes in a system happen automatically whenever temperature, pressure, or chemical potential are not uniform in the system. (The gradients of the corresponding fields generate forces changing the sytem, until equilibrium = a maximaum entropy state) is achieved.
''Bringing two systems into contact'' is only a particular way to prepare initial conditions of very simple form.

(iii) To compute how entropy changes (this is the subject of nonequilibrium thermodynamics), you need a more specific model than just generakities about thermodynamics.
For example, the nonequilibrium thermodynamics of water is given by the Navier-Stokes equations. In addition to conservation of mass, energy, momentum, and angular momentum, one can derive from the equations a formula for the entropy production, whichis nonnegative at each point, and is a term in the differential equation for entropy that implies that entropy increases globally in an isolated system.

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Okay, are states with gradients not already non-equilibrium systems? Do the three or four laws even suffice to describe such systems? I was thinking of equlibrium thermodynamics here - do the answers change? – NikolajK Mar 25 '12 at 20:32
@NickKidman: indeed gradients are formally non-equilibrium. However, we have a general physics principle that things which are close to equilibrium are going to behave in a way which is close. Many gradients are conceptually taken care of with the concept of local equilibrium, and imagining breaking up the system into lots of little systems which exchange heat and particles. In reality nothing is at equilibrium; but we expect the equilibrium statements to be accurate in the right regimes. What those regimes are is a matter of real (and sometime unclear) physics. – genneth Mar 26 '12 at 2:23
@NickKidman: In equilibrium, entropy is constant (already maximal entropy), and only reversible transformations are permitted. (In reality, these are of course also slightly dissipative, but arguments idealize the situation.) - Thus your question makes sense only in a nonequilibrium setting. Local equilibrium is the simplest of the nonequilibrium settings, and the second law takes the form of nonnegative entropy production. – Arnold Neumaier Mar 26 '12 at 8:07

Depending on what you're interested in, your starting configuration may or may not be relevant.

For a statistical system, one has to measure the system for a finite time to make some meaningful measurement. One usually sets the time to begin and end taking these measurements to some time at infinity, so the starting configuration doesn't matter. This is often used so an arbitrary starting configuration can be chosen, usually one that makes your problem easier to solve.

In equilibrium thermodynamics, you must bring two systems together to change a system. This is, again, because all measurements are done at infinity so all changes that will happen within the system have happened, ie it is in equilibrium. This is, for example, why there is usually no dependence on time in thermodynamics questions, only on time direction (and reliance on time is usually something trivial like "a current was run through a resistor for x seconds" to provide some amount of heat energy).

Once you go beyond equilibrium thermodynamics, you're looking at systems where the laws of thermodynamics you have previously encountered don't necessarily hold in the same way, only on average.

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Is the starting configuration of a thermodynamics system ever relavant?

The initial state is surely relevant for the question that is actually addressed by the second law, namely a comparison of the initial state and the final state. The second law "only" says that the final state never has a lower entropy than the initial state. The second law is too general and universal so it doesn't say exactly what the value of the entropy is (a detailed macroscopic or microscopic model of the physical systems is needed for that) but regardless of the context, the entropy of the final state is promised to be higher than the entropy of the initial state which is a nontrivial statement, one that underlies the thermodynamic arrow of time (future-past asymmetry in the behavior of warm or macroscopic objects).

Your calculations of the final entropy or final temperature may be detailed but these detailed calculations are not a component of the second law. The second law is about something that you apparently didn't do at all – the comparison of the initial and final entropy. The actual quantification of the increasing entropy needs some detailed model. For example, Boltzmann's H-theorem is a quantitative refinement of the second law that contains the calculation of the actual rate of entropy increase. One may want to use more realistic, quantum versions of the H-theorem in the real world.

In thermodynamics, if I compute "the change a system" in the sense of the above example, do I always induce the necessarity for doing such a thing by stating "now we bring system one and system two in contact"?

It's someone hard to decode what is necessary for what. At any rate, the mutual contact between two or more subsystems is needed for heat transfer. If two subsystems that are separately in equilibrium are not in contact, their entropies (and therefore the total entropy) may stay constant (marginally obeying the second law). The entropy will only starts to (strictly) increase if they get in touch. Whether the system starts to change at that moment depends on whether or not the systems already agree in all the intensive quantities that may be balanced, especially temperature (but also pressure, chemical potentials etc.). If there is some inequality in those that may be balanced by the mutual interaction of these subsystems, this inequality will be balanced and the entropy will strictly increase along the way.

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