# 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? –  Nikolaj K. 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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