Entropy  increase without energy increase?
Many everyday examples of entropy increase involve a simple energy increase. This energy increase is usually evident from a rise in temperature. 
We can analyze many simple situations in terms of energy and entropy. Why does ice melt in a warm room? A first approximation is easy. The faster moving ("hotter") molecules in the room can disperse their energy by making the slower moving ("colder") molecules in the ice speed up. This would be a following of the second law and therefore it should be a spontaneous process involving an increase in entropy in the ice as it melts to form water. A more sophisticated view includes the fact that liquid water can have many more ways of dispersing energy than ice -- water molecules in the liquid form can rotate and move far more freely than in ice. Therefore, if water more effectively disperses energy than ice, when they are together at an equilibrium temperature, liquid water will be favored because it better disperses the energy available in the system.
Conversely, why do snowflakes form when moisture (water) is in air that is colder than water's freezing temperature? The water will disperse its energy to the colder air and then the water's temperature will drop to freezing and the water will begin to form crystals of ice that we recognize as snowflakes.
Some more difficult evaluations of energy and entropy are involved even in mundane situations encountered daily. However, with a few hints we can arrive at general answers for all such events.
1. Why do gases mix spontaneously? The same basic question is expressed in "Why could you quickly smell perfume that is released in one corner of a large room in the far corner even if the room air could be 'absolutely perfectly' still?" (There is NO change in energy in the process and yet it is spontaneous. Where is any energy dispersal here that the second law says is characteristic of all spontaneous happenings?)

2. Why do liquids mix spontaneously? Same question, "Why does cream mix with coffee at the same temperature?" (NO change in energy. Where is any kind of energy dispersal?!)

3. Why would perfume vapor or oxygen or nitrogen or helium spontaneously and instantly flow into an evacuated chamber? (NO change in energy. Where's the second law here?)

There is a broad range of speed and kinds of motion in any group of molecules that is above absolute zero. Molecules move (translate), tumble around (rotate) and vibrate (atoms in the molecules act as though they were connected with springs, back and forth, or wig-wag vibration). All of these motions increase as energy content increases (indicated by the temperature). Each type of motion is associated with specific energy levels ranging from lower to higher energy content.  These levels are discrete, i.e., molecules cannot be in any in-between energy state. Energy is "quantized" and treating their energy relationships is part of quantum mechanics.
The more energy levels that are occupied by energetic molecules, the more widely energy can be dispersed and the greater is the entropy. But in the many cases we have talked about, additional energy levels could only be occupied if the system were heated so the slower molecules would be speeded and there would be many more fast moving molecules to occupy the possible higher levels. However, this is not the only way that additional energy levels can be made available.
When molecules are allowed to expand into a larger volume (in three-dimensional space) , quantum mechanics shows that an interesting change in possible energy levels takes place: the energy levels become closer together. (Technically, we must say that the density of occupiable levels in any selected energy range is greater.) This means effectively that molecules, if allowed to occupy a larger volume even without any increase in their energy, can spread out to occupy many more energy levels. This means greater dispersal of energy and an increase in entropy simply by there being a greater three-dimensional volume in which the molecules can move. (Further, because any change in which entropy increases is a spontaneous change. It happens without any outside aid, energy input, etc.)
How does that apply to (1), perfume in a room? It spontaneously mixes with the gases in the large room because its energy is redistributed among more energy levels than in the small vapor space of the bottle. This is the same as having greater energy dispersal = an increase in entropy = spontaneity.

And (2), cream in coffee? (Or any other kinds of liquids mixing?) Same as above. Because of an increase in volume, the energy of the cream, or of any liquid mixing with another, is redistributed among more energy levels in the greater volume than alone by itself = greater energy dispersal = increase in entropy = spontaneous mixing.

(3) A gas spontaneously rushing in to a space that was a vacuum? Same explanation as above. Increase in volume = more energy levels available for a substance with the same energy as in a smaller volume = redistribution of energy among more energy levels = increased energy dispersal = increase in entropy = spontaneous process.

In this example of a gas being "allowed" to go into an evacuated bottle, box, or chamber, our feelings are that this should not only be spontaneous (happen by itself) but instantaneous (happen very fast). But feelings aren't reliable. Science demands reasons (and, as we are aware, the second law makes predictions only about the spontaneity of events, not about their rates or speed of their taking place). Fortunately, there is now a firm theoretical basis for our practical gut feeling that "of course a gas would automatically and instantly fill a vacuum!". Quantum mechanics provides unquestionable calculations that are the reasonable basis, not just for a gas expanding into a vacuum, but for all the results of the second law presented in this article. In science, that's even better than feeling.