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I started reading about entropy and the second law of thermodynamics.

Different sites give different definitions and meanings of this law. A few of them:

  1. Disorder always increases

  2. heat always flows from a hot to a cold object

  3. Unusable energy of a system always increases.

In the first one,what is really meant by disorder? What is the true meaning of disorder? Does disorder depend upon the observation of a conscious mind, or does it have a meaning even without an observer?

Also, are all these three definitions actually interconnected?

The third definition made the most sense to me;but since it is a law,can it actually be proven or is it just accepted because of experimental evidence?Can human find ways to utilize this unusable energy in an efficient way and still not increase the entropy of the system? And what does "unusable" energy mean?Aren't usable and unusable energy only significant for humans?Why does the universe even care if energy can be used or not?

I have just started understanding this topic.An answer without maths and one which gives conceptual clarity would be appreciated.

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The second law of thermodynamics states, that the total entropy of an isolated system never decreases with time. There are a couple of details in this sentence, which are worth pointing out:

  • You can decrease the entropy locally, but not globally: E.g. consider a refrigerator. Here we decrease the temperature locally (=inside the refrigerator), which leads to a local decrease in entropy. However, the entropy of the complete system is increased, by transferring heat from the colder to the warmer place. Hence the words total (entropy), and isolated (system) are important.
  • It's possible to have a process which keeps the total entropy constant. Therefore, the entropy does not have to strictly increase with time. Therefore, in the second law "never decreases with time" is often replaced by "always increases with time". However, none physicists often mistake increasing with strictly increasing.
  • As entropy is linked to disorder by the law of Statistical Physics (see below), the first statement of yours

    Disorder always increases

    is equivalent to the upper definition.

  • Your second definition

    heat always flows from a hot to a cold object

    is a consequence of the definition given above (the idea is presented below). By doing the calculation we find, that the disorder increases, if heat is transferred from the warm to the cold and that the heat transfer stops in thermal equilibrium.

  • Your third definition

    Unusable energy of a system always increases.

    is as well a restatement of the upper definition. However, one has to ask, how we measure the "usability" of energy. This would lead to the entropy.

In Stat. Physics entropy is defined as $$ S = k_B \cdot \ln{\Omega} $$ where $\Omega$ is the number of accessible micro-states (link to a video). In order to understand this concept think of a system with two compartments (left / right), which can exchange energy. Each compartment has 10 distinguishable particles in it and each particles has two states:

  • The first state is the ground state. No energy is needed to place a particle in this state.
  • The second state is an excited state. We need one unit of energy to place a particle in this state.

Now suppose, you are given 10 units of energy and the particles are not allowed to change the compartment. However, the energy is allowed to switch the compartment. What are possible configurations:

  1. One config is that all particles in the left compartment are in the excited state. Then no energy units are left for the right compartment. Hence, all particles in the right compartment are in the ground state. So we have only one possible realization in which the 10 units of energy are in the left compartment. Hence, the number of accessible micro-states, where 10 units of energy are on the left and none is on the right, is 1.
  2. An other configuration is where we have 9 energy units in the left compartment and 1 in the right. How many different micro-states do we have? Well, in the left compartment each particle could be the one which is in the ground state. Hence, here we have 9 different state. In the right compartment each of the particles could be in the excited state. Hence, again we have 9 possible states. In total we have $9\cdot 9 = 81$ different micro-states. Hence, the entropy of this state is much larger than the entropy of the state, where all 10 energy units are in the left compartment.
  3. If you keep going like this, you will find, that the state where the number of accessible micro-states is the larges is "5 energy units are in the left compartment and 5 are in the right". This is the equilibrium state, where both compartments have the same "temperature".

The basic of this logic is the law of "a priori equal probability".

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  • $\begingroup$ A small nitpick: heat can flow from a cold body to a hot one. It cannot flow spontaneously though. Moreover, this is a more fundamental statement of the second law in the sense that the concept of entropy (derived by Clausius) arises from it. $\endgroup$ – Diracology Sep 23 '17 at 15:04
  • $\begingroup$ I don't understand your comment about the spontaneous flow. Could you please elaborate on that. I'm well aware, that the classical laws like "heat flows from the warm to the cold body" become merely a probability statements in the Statistical Physics picture: Although the first configuration discussed above is very improbable, it still is a valid micro-state. Furthermore, if one excludes all but the most probable configuration, one really misses the point: Variation is a key feature of the Stat. Phys. theory. The equal a priori probability replaces the "why". $\endgroup$ – Semoi Sep 23 '17 at 15:37
  • $\begingroup$ By spontaneous I simply mean with no other effect or no work input. A refrigerator for instance transfer heat from a colder to a hotter body. But it needs input work though. $\endgroup$ – Diracology Sep 23 '17 at 15:51
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Here is vigorous motion of water in an electric field, obviously able to produce work - e.g. by rotating a waterwheel:

"The Formation of the Floating Water Bridge including electric breakdowns" https://www.youtube.com/watch?v=17UD1goTFhQ

"The water movement is bidirectional, i.e., it simultaneously flows in both directions." https://www.wetsus.nl/home/wetsus-news/more-than-just-a-party-trick-the-floating-water-bridge-holds-insight-into-nature-and-human-innovation/1

The work will be done at the expense of what energy? The first hypothesis that comes to mind is:

At the expense of electric energy. The system is, essentially, an electric motor.

However, close inspection would suggest that the hypothesis is untenable. Scientists use triply distilled water to reduce the conductivity and the electric current passing through the system to minimum. If, for some reason, the current is increased, the motion stops - such system cannot be an electric motor.

If the system is not an electric motor, then it is ... a perpetual-motion machine of the second kind! Here arguments describing perpetual-motion machines as impossible, idiotic, etc. are irrelevant - the following conditional is valid:

IF THE SYSTEM IS NOT AN ELECTRIC MOTOR, then it is a perpetual-motion machine of the second kind.

In other words, if the work is not done at the expense of electric energy, then it is done at the expense of ambient heat, in violation of the second law of thermodynamics. No third source of energy is conceivable.

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A conceptual, but not mathematically rigorous, summary of the Second Law is that uneven distributions of energy tend to even out over time. This is the result of the combination of chance and the conservation of energy and momentum. It is hard to prove for a general case, but it is easy to posit convincing examples.

To take an example involving KE, if you fire a fast-moving particle into a box of slower ones, you have a very uneven initial distribution of energy. Over time, the fast moving particle will trigger a succession of random collisions in which its excess energy will be lost and the average energy of all the other particles will increase slightly.

To take another involving PE, imagine you have a box into which you dump some sand. The surface of the sand will start-off uneven, with peaks and troughs, so there is an uneven distribution of gravitational PE. If you encourage the sand grains to flow, by agitating the box from side to side, the grains from the peaks will fall into the troughs and the surface will flatten. Once there's no inequalities of PE there is no longer any force to trigger further movement, so the sand settles into a stable state.

I hope you can see that the first example I gave shows how heat will transfer from a hotter body to a colder one, since heat is just a measure of particules' KE.

From both examples you should be able to see that when energy is evened out it is no longer in a form that is capable of having any further effect.

The explanation of entropy as a measure of the degree of disorder in a system has always seemed to me to be one of limited conceptual value because the concept of 'order' has to be interpreted in a very specific way. For example, if you consider my second example, an everyday interpretation of the word 'order' would lead you to conclude that the settled sand appears more neat and orderly, so that the degree of order has increased.

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