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I read that heat cannot flow from cooler body to hotter because for that entropy of the system becomes negative. Why is that so? Why we cannot have negative entropy?

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The 2nd Law of Thermodynamics is based on an overwhelmingly extensive body of empirical evidence on how thermodynamic systems behave. There are many different statements of the 2nd Law, and all of them are equivalent to one another. Once one of these has been specified, all the other follow. One such statement says that heat cannot flow spontaneously from a hot body to a colder body. Another statement, which is much more mathematically based, defines entropy, how its change should be calculated for a change in state of a closed thermodynamic system, and says that, for an isolated system experiencing a change in thermodynamic state, the entropy of the system can only increase or remain constant. So, if you calculate the change in entropy of an isolated system consisting of a hot body and a cold body placed in contact with one another and allowed to equilibrate, you will find that the combined entropy of the hot body and cold body increases.

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  • $\begingroup$ I could still remember your one of the first answers at chem SE that provided this intuitive link: physicsforums.com/insights/… $\endgroup$
    – user36790
    Commented Jan 14, 2016 at 13:50
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    $\begingroup$ Thanks very much. I'd like to think that the article helped some students overcome their thermodynamics confusion so that they could do their homework. $\endgroup$
    – Chet Miller
    Commented Jan 14, 2016 at 14:33
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The reason why heat cannot flow from cold to warm is that the change in entropy will become negative, and that doesn't happen in a closed system. Negative entropy is by definition not possible. Here's why:

Alternative example: gas in a box

I think entropy gets a little more intuitive if we think of it in terms of statistical mechanics. If we imagine a box with a number of ideal gas particles, these particles can be arranged in many different ways. One way they can be arranged is to have all the particles gathered in one corner of the box, and the rest of the box being empty. There are many different permutations of the individual particles which would all give this result - the state described as all the particles being in the upper left back corner is called a macrostate (describes the system macroscopically but doesn't care about which individual particle sits where), while each distinct "seating arrangement", or permutation, of the particles which all yield the same macrostate are called its microstates. We can easily observe the macrostate, while the microstate is much harder to observe.

It should be intuitively clear that there are many microstates which yield the same macrostate; but also that there are enormously many more microstates that yield a macrostate in which the gas particles are more or less evenly distributed in the box. This is actually the statistical-mechanics definition of entropy:

The entropy of a macrostate is proportional to the logarithm to the number of microstates which yield the given macrostate.

(Under the assumption that each microstate is equally probable). Mathematically, this is given as:

$$E = k_b \ln \Omega $$

With $k_b$ being the Boltzmann constant and $\Omega$ being the number of possible microstates of the given macrostate. This also explains why we cannot have negative values of entropy: There is no such thing as "a negative number of microstates".

We then assume that the system starts in a low-entropy state and evolves through random fluctuations, or a random walk, in configuration space - that is, the "space" made up of all possible microstates (not confined to any particular macrostate). You can imagine that each macrostate occupies a region in configuration space, the size of which is determined by the number of microstates it has - by its entropy. As the system evolves, once it has randomly walked out of its low-entropy initial configuration, it is very unlikely that its random path will ever take it back there again, simply because it is so tiny compared to the immensely larger number of microstates of a high-entropy state. In fact, once the system has settled into its maximum entropy state, it is very unlikely that it will fluctuate out of this state again, due to simple statistics.

Heat transport

Now going back to the example of the original question, we can see the analog to the gas particle example, only instead of the geometrical position of the particles, the state is defined by the distribution of their kinetic energies. Again, it is perfectly possible to have a configuration, in which all the kinetic energy is distributed over particles in the left side of the box, while all the particles on the right side are at rest. However, it should also be intuitively clear that there are immensely many more microstates in which the energy is evenly distributed over all the particles. So if our system starts in the low-entropy configuration in which one subsystem is hot and the other is cold, and is then allowed to evolve through random fluctuations through configuration space, again it will quickly leave the tiny "patch" of its initial macrostate and venture into regions of higher entropy - more microstates - from which it is very unlikely to ever return.

NOTE, though, that it actually is entirely possible for the system to randomly fluctuate back into a low-entropy state - to have a negative change in entropy. It is possible to stir your cappuccino into a configuration that separates milk and coffee, or to suddenly gather all air particles in one corner of the room - it is just extremely unlikely, and thus we never observe it in real life.

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